Chapter 10
New Growth Theories and Monopolistic Competition

The previous two chapters deal with issues related to growth with endogenous human capital and knowledge. Although knowledge is not treated as an exogenous variable, these models have a common limitation if one wants to know in the microeconomic level about what are the motives for private companies to make innovation and for individuals to get educated. For instance, in the previous chapter, knowledge stock, Z(t), receives no compensation, and every individual is assumed to be free to exploit the entire stock of Z(t). Although these models are congruous with that technological change drives economic growth and the knowledge is a nontrivial good, they don't explain why profit-maximizing private firms would make efforts to generate technological change.

Adam Smith emphasizes interdependence between economic productivity and division of labor. The division of labor is one of the sources for economies to operate under increasing returns. Malthus examined the economic dynamics with endogenous population. He shows that the system exhibited decreasing returns due to population dynamics. Marshall recognizes that if an economic system did not satisfy some constant or decreasing returns postulate, competition itself is not dynamically stable. He argues that industries in which particular processes exhibit increasing returns to scale must rapidly become monopolized. The contemporary economic reality in developed economies is rarely purely competitive or purely monopolistic. Growth theory based on perfect competition may be proper for revealing complexity of economic growth on highly aggregated - sectorial, interregional, national, international - levels; it tends to lose validity if one wants to explain driving forces of economic growth on levels of individual firms. Because big companies have increasingly - globally as well as locally - dominated the scene of economic life, it is reasonable for the contemporary mainstream of growth theory, dubbed as the new growth theories, to swap the paradigm of monopolistic competition for that of perfect competition of the neoclassical growth theory.

As ideas and knowledge can be applied to different aspects of production and consumption. New knowledge can not only produces new goods, but also better production techniques and improve quality of older goods like automobiles and household appliances. Through schooling, education, learning by doing, learning by consuming, and R&D activities, people are able to use ideas and techniques to create new ideas, effectively diffuse and distribute knowledge, and produce more and better goods. Since the late 1980s endogenous growth theory has caused great attention of economists. The 'new' endogenous growth attempts to explain technical change as the outcome of market activity in response to economic incentives. In the new growth theory, technological change does not take place in a predetermined fashion without any social and economic costs. An important development in the new growth theory has been to broaden the definition of capital. It not only includes physical capital, but also research and development, human capital, government-financed infrastructure, and more generally, the institutions needed to protect liberty and property rights. The new growth theory has modeled endogenous knowledge accumulation through many channels, including formal education, on-the-job training, basic scientific research, learning by doing, process innovations, industrial innovations, and product innovations. The crucial assumption that leads to sustainable endogenous growth is the existence of increasing returns to scale in economic production under monopolistic competition.

The concept of monopolistic competition and modeling frameworks associated with type of imperfect competition have been applied to various problems in macroeconomics, international and interregional economics, economic growth and development. Monopolistic competition is characterized as follows:

• (i) The products are differentiated. It consists of many buyers and sellers. Unlike perfectly competitive firms, firms are characterized by significant product differentiation. Consumers view firms' products as imperfect substitutes for each other.
• (ii) The number of firms is so large that each firm ignores its strategic interactions with other firms.
• (iii) Entry is unrestricted and takes place until the profits of incumbent firms are driven down to zero. Any firm can hire the inputs, such as labor and capital, needed to compete in the market, and they can release these inputs from employment when they do not need them.

The character of imperfect competition is often emphasized for describing decentralized allocations in the presence of increasing returns. Its competitive feature allows us to avoid complexity of strategic interactions among firms (like in oligopoly models). The modeling framework with monopolistic competition makes it possible to endogenize entry-exit processes and the range of products supplied in the market through these processes. In determining their prices in the short term, monopolistic competitors behave much like the differentiated products oligopolists. Taking the prices of other firms as given, each firm face a downward-sloping demand curve - the downward sloping is held because of product differentiation. Each firm maximizes its profit at the point at which its marginal revenue equals marginal cost. In a short-run equilibrium, the price chosen by a firm may exceed the typical firm's average cost at the prevailing output level. This situation will attract new entrants into the industry. As firms enter the monopolistically competitive market, a typical firm's demand curve shifts. At a long-run equilibrium, a typical firm sets the profit-maximizing price equal to the average cost, making zero profit.

This chapter is concerned with the new growth theories with monopolistic competition and endogenous knowledge. The chapter is organized as follows. Section 10.1 reviews Romer's economic development model with monopolistic competition. In Section 10.2, we examine the growth model with product variety proposed by Grossman and Helpman. In Section 10.3, we study the growth model with variety of consumer goods proposed by Barro and Sala-i-Martin. Section 10.4 represents Aghion and Howitt's economic development model with creative destruction. Section 10.5 studies a growth model with improvement in quality of products. In Section 10.6, we represent Young's growth model with learning by doing and research. Section 10.7 points out that the new growth theories can sustain positive steady growth rates perhaps because presumed linear or linearized knowledge growth. Without linearization in the key equations of knowledge growth, it is difficult to sustain unlimited economic growth.

10.1 Development with Monopolistic Competition

Schumpeter emphasizes market power in explaining dynamics of innovation. Although in 1973 Shell constructed a model with a single monopolist who invests in technological change, the assumption of permanent monopoly is not realistic for approaching competition in modern economies.¹ Romer developed a model of economic growth with monopolistic competition in 1990.² The Romer model revised below is a synthesis of different approaches - except the neoclassical growth theory, the model of monopolistic competition in consumption goods formulated by Dixit and Stiglitz in 1977, the dynamic framework by Judd in 1985, and the model with differentiated inputs in production by Ethier in 1982.³

The Romer model considers four basic inputs - physical capital, human capital, labor, and an index of the level of technology. Capital, denoted by K(t), is measured in units of consumption goods. Let N stand for the population and the labor force. The number N is assumed to be constant in this model. It can be seen that an exogenous growth rate of labor force will make the analysis more complicated. The model separates the rival component of knowledge (also called human capital), H, from the nonrival, technological component, A(t). The nonrival knowledge A(t), measured in the number of designs - can grow without bound. It is assumed, for the technical reason, that the total stock, H, of human capital in the population and the fraction supplied to the market are fixed.

The economy is composed of three sectors. The research sector produces new knowledge i.e., new designs) with human capital and the existing stock of knowledge as inputs. An intermediate-goods sector uses the designs from the research sector together with capital to produce producer durables that will be used in final-goods production. A final-goods sector uses labor, human capital, and the set of producer durables. The output of this sector can be either consumed or saved.

The sector that produces producer durables cannot be described by a representative firm. Let there be a distinct firm i for each durable good i. A firm must purchase or produce a design for good i before starting production by converting η units of final output into one durable unit of good i. Once the firm has produced a design for durable i, it obtains an infinitely lived patent on the design. If it manufactures x(i) units of the durable, it rents those durables to final-output firms for a rental rate p(i). Since it is the only seller of durable i, it will face a downward-sloping demand for the good. Neglecting any possible depreciation, we see that the value of one unit of durable i is the present discounted value of the infinite stream of rental income.

Final output Y(t) is produced by combination of physical labor N, human capital devoted to final output H_Y, and physical capital. Here, the physical capital consists of an infinite number of distinct types of producer durables xindexed by a continuous variable i ∈ [0, ∞). It should be remarked that only a finite number of these potential inputs (that have already been invented and designed) are available for use at any time t. We adopt the Cobb-Douglas production function for final output:

$Y\mathrm{(}\mathit{t}\mathrm{)}=H_Y^{\alpha }{\mathit{N}}^{\mathit{\beta }}\underset{0}{\overset{\infty }{\int }}\mathit{x}\mathrm{(}\mathit{i}{\mathrm{)}}^{1-\mathit{\alpha }-\mathit{\beta }}\mathit{d}\mathit{i}\mathrm{,}\mathrm{1}\mathrm{>}\mathit{\alpha }\mathrm{,}\mathit{\beta }\mathrm{>}\mathrm{0.}$(10.1.1)

The production function is homogenous of degree one. Output in the final-goods sector can thus be described in terms of the actions of a single, aggregate, price-taking firm. The specified form of ${{\displaystyle \int }}_0^{\mathrm{\infty }}\mathit{x}\mathrm{(}\mathit{i}{\mathrm{)}}^{\mathrm{1}\mathrm{-}\mathit{\alpha }\mathrm{-}\mathit{\beta }}$ dt implies that all durables have additively separable effects on output. We omit possibility of complementarity and of mixtures of types of substitutability among producer durables.

As usual, it we neglect depreciation, capital K(t) evolves according to the following equation:

$\dot{\mathit{K}}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{Y}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{C}\mathrm{(}t\mathrm{)},$

where C(t) is aggregate consumption. According to the definitions of η and x(i,t), we have:

$\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{\eta }\underset{0}{\overset{\mathrm{\infty }}{{\displaystyle \int }}}\mathit{x}\mathrm{(}\mathit{i}\mathrm{,}\mathit{t}\mathrm{)}\mathit{d}\mathit{i}\mathrm{.}$

To specify the process for the accumulation of new designs, A(t), we assume that research output depends on the amount of human capital devoted to research and on the stock of knowledge available to a person doing research in the following way, Ȧ(t) = κH_AA(t), where H_A, is the total human capital employed in research. As κ and H_A are constant, this specification means that A(t) will grow with a fixed rate of κH_A and will become infinite as time passes. We also have:

${\mathit{H}}_A+{\mathit{H}}_Y=\mathit{H}\mathrm{.}$

We choose price of current output to be unity at any point of time. Let

• r(t) = the interest rate on loans denominated in goods;
• p_A(t) = the price of new designs;
• w_H(t) = the rental rate per unit of human capital.

Because any one engaged in research can freely take advantage of the entire existing stock of designs in doing research to produce designs, from Ȧ(t) = κH_AA(t) we should have:

$w_H\left(t\right)=p_A\left(t\right)kA\left(t\right)\mathrm{.}$

The final-output firm is faced with a price list {p(i): i ∈ R₊} for all the producer durables, including infinite prices for the durables that have not been invented. The firm maximizes profits in selecting x(i). Given values for H_Y and N, profits for the representative final-output firm are:

$\underset{x}{\underset{}{\max }}{\displaystyle \underset{0}{\overset{\infty }{\int }}}\mathrm{[}{H}_Y^{\alpha }{\mathit{N}}^{\mathit{\beta }}\mathit{x}\mathrm{(}\mathit{i}{\mathrm{)}}^{1-\mathit{\alpha }-\mathit{\beta }}\mathrm{-}\mathit{p}\mathrm{(}\mathit{i}\mathrm{)}\mathit{x}\mathrm{(}\mathit{i}\mathrm{)}\mathrm{]}\mathit{d}\mathit{i}\mathrm{.}$

The first-order condition for the above maximization problem is:

$p\mathrm{(}\mathit{i}\mathrm{)}=\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\alpha }\mathrm{-}\mathit{\beta }\mathrm{)}{H}_Y^{\alpha }{\mathit{N}}^{\mathit{\beta }}\mathit{x}\mathrm{(}\mathit{i}{\mathrm{)}}^{-\mathit{\alpha }-\mathit{\beta }}\mathrm{.}$(10.1.2)

The demand curve defined by this equation is what the producer of each specialized durable takes as given in choosing the profit-maximizing price to set. A firm that has already incurred the fixed-cost investment in a design will choose a level of output x to maximize its profit. That is:

$\mathit{\pi }=\underset{x}{\underset{}{\max }}\mathit{p}\mathrm{(}\mathit{x}\mathrm{)}\mathit{x}\mathrm{-}\mathit{r}\mathit{\eta }\mathit{x}=\underset{}{\underset{x}{\max }}\mathrm{[}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\alpha }\mathrm{-}\mathit{\beta }\mathrm{)}{\mathit{H}}_{\mathit{Y}}^{\alpha }{\mathit{N}}^{\mathit{\beta }}{\mathit{x}}^{\mathrm{1}-\mathit{\alpha }-\mathit{\beta }}\mathrm{-}\mathit{r}\mathit{\eta }\mathit{x}\mathrm{]}\mathrm{.}$

The flow of rental income is p(x)x: the cost equals the interest cost on the ijx units of output needed to produce x durables. In this model, the only sunk cost is the initial expenditure on the design. The monopoly pricing problem defined above is that of a firm with constant marginal cost that faces a constant elasticity demand curve. The resulting monopoly price is a markup over marginal cost, i.e.:

$p^{*}=\frac{\mathit{r}\mathit{\eta }}{\mathrm{1}\mathrm{-}\mathit{\alpha }\mathrm{-}\mathit{\beta }}\mathrm{.}$(10.1.3)

Each producer of specialized durables charges the monopoly price. Under this pricing strategy, the maximal profit is π = (α + β)p*x*, where x* is the quantity on the demand curve implied by the price p*.

The decision to produce a new specialized input depends on a comparison of the discounted stream of net revenue and the cost P_A of the initial investment in a design. Because the market for designs is competitive, the price for designs will equal the present value of the net revenue that a monopoly can extract, i.e.:

$\underset{}{\overset{\mathrm{\infty }}{{\displaystyle \underset{t}{\int }}}}{\mathit{e}}^{-\underset{t}{\overset{\tau }{\int }}r\left(s\right)ds}\mathit{\pi }\mathrm{(}\mathit{\tau }\mathrm{)}\mathit{d}\mathit{\tau }={\mathit{P}}_A\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$

Assuming P_A to be constant and taking derivatives of the above equations with respect to t, we obtain:

$\pi \mathrm{(}\mathit{t}\mathrm{)}=\mathit{r}\mathrm{(}\mathit{t}\mathrm{)}{\mathit{P}}_A\mathrm{.}$(10.1.4)

This equation tells that at any point of time, the instantaneous excess of revenue over marginal cost must be just sufficient to cover the interest cost on the initial investment in a design.

To close the model, we describe behavior of consumers. Consumers in the Romer model are endowed with fixed quantities of labor and human capital. At the beginning, consumers own the existing durable-goods-producing firms and net revenues of these firms are paid to consumers as dividends. Final-goods firms earn zero profits and own no assets. We specify Ramsey consumers with discounted, constant elasticity preferences:

$\mathit{U}=\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{{\displaystyle \int }}}\frac{C^{\mathrm{1}-\mathit{\sigma }}}{\mathrm{1}\mathrm{-}\mathit{\sigma }}{e}^{-\rho t}dt\mathrm{,}\mathit{\sigma }\mathrm{\in }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{\infty }\mathrm{)}\mathrm{.}$

We have already known that intertemporal optimization condition for a consumer faced with an interest rate r(t) is:

$\frac{\dot{\mathit{C}}\mathrm{(}\mathit{t}\mathrm{)}}{\mathit{C}\mathrm{(}t\mathrm{)}}=\frac{\mathit{r}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{\rho }}{\mathit{\sigma }}\mathrm{.}$(10.1.5)

This is only the equation that consumers' preferences enter the solution of the model.

The Romer model, like most of the models in the new growth theory, defines a mathematically complicated system. Since such a system is difficult to analyze, one has to be only concerned with steady states of the model. We now examine properties of balanced growth equilibrium.

An equilibrium point for the model is paths for prices and quantities such that:

• (1) consumers make decisions with interest rate as given;
• (2) holders of human capital decide whether to work in the research sector or the manufacturing sector with A, P_A, and the wage rate in the manufacturing sector w_A as given;
• (3) final-goods producers choose labor, human capital, and a list of differentiated durables with prices as given;
• (4) each firm that owns a design and manufactures a producer durable maximizes profit by setting prices with interest rate and the demand curve as given;
• (5) firms contemplating entry into the business of producing a durable take prices for designs as given; and
• (6) the supply of each good is equal to the demand.

We now try to find an equilibrium in which the variables A, K, and Y grow at constant exponential rates.

First, we note π = (α + β)p*x*. By this condition and condition (10.1.4):

${\mathit{P}}_A=\frac{\mathit{\pi }}{\mathit{r}}=\frac{\mathrm{(}\mathit{\alpha }+\mathit{\beta }\mathrm{)}{\mathit{p}}^{*}{\mathit{x}}^{*}}{\mathit{r}}\mathrm{.}$

Inserting equation (10.1.2) into the above equation yields:

${{\mathit{P}}_{}}_A=\frac{\mathit{\alpha }+\mathit{\beta }}{\mathit{r}}(\mathrm{1}\mathrm{-}\mathit{\alpha }\mathrm{-}\mathit{\beta })H_Y^{\alpha }{N}^{\beta }{x}^{*1-\alpha -\beta }\mathrm{.}$(10.1.6)

In this model, A is also used as a measure of the number of designs or durables in use at time t. Consequently, at equilibrium, $\mathit{K}=\mathit{\eta }{\int }_0^{\infty }\mathit{x}\mathrm{(}\mathit{i}\mathrm{)}\mathit{d}\mathit{i}=\mathit{\eta }\mathit{A}{\mathit{x}}^{*}$.

It is assumed that in labor market the wages paid to human capital in each sector must equal. By w_H = P_AκA and the wage for human capital is equal to its marginal product in the final-output sector, i.e.:

${\mathit{w}}_Y=\mathit{\alpha }{H}_Y^{\alpha -1}{\mathit{N}}^{\mathit{\beta }}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\mathit{x}\mathrm{(}\mathit{i}{\mathrm{)}}^{\mathrm{1}-\mathit{a}-\mathit{\beta }}\mathit{d}\mathit{i}=\mathit{\alpha }{\mathit{H}}_Y^{\mathit{\alpha }-1}{\mathit{N}}^{\mathit{\beta }}\mathrm{A}{\mathit{x}}^{*\mathrm{1}-\alpha -\mathit{\beta }},$

the condition of w_H = w_Y is given by:

${{\mathit{P}}_A}_{}\mathit{\kappa }=\mathit{\alpha }{H}_Y^{\alpha -1}{\mathit{N}}^{\mathit{\beta }}{\mathit{x}}^{*1-\alpha -\beta }\mathrm{.}$

Substituting equation (10.1.6) into the above equation solves H_Y as:

${\mathit{H}}_{\mathit{Y}}=\frac{\mathit{\alpha }}{\mathrm{(}\mathit{\alpha }\mathrm{+}\mathit{\beta }\mathrm{)}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\alpha }\mathrm{-}\mathit{\beta }\mathrm{)}}\frac{r}{\mathit{\kappa }}\mathrm{.}$(10.1.7)

For a fixed value of H_A = H - H_Y, the implied exponential growth rate for A is κH_A. It is straightforward to check that x* is constant if r is. The final output is:

${\mathit{Y}}^{*}\mathrm{=}{\mathit{H}}_Y^{\mathit{\alpha }}{\mathit{N}}^{\mathit{\beta }}\stackrel{}{\underset{0}{\overset{\infty }{{\displaystyle \int }}}}x{\left(i\right)}^{1-\alpha -\beta }\mathit{d}\mathit{i}=H_Y^{\alpha }{\mathit{N}}^{\mathit{\beta }}A{x}^{*1-\alpha -\beta }\mathrm{.}$

Consequently, output Y and capital stock K must grow at the same rate as A if N, H_Y, and x* are fixed. Denote this rate by g_Y. Since K/Y is a constant, the ratio is:

$\frac{C}{\mathit{Y}}=\frac{\mathit{Y}\mathrm{-}\dot{\mathit{K}}}{\mathit{Y}}=\mathrm{1}\mathrm{-}{\mathit{g}}_Y\frac{\mathit{K}}{\mathit{Y}},$

must be constant. Consequently, the growth rate of C is the same as that of Y. Thus:

${\mathit{g}}_Y=\frac{\dot{C}}{C}=\frac{\dot{Y}}{\mathit{Y}}\mathrm{=}\frac{\dot{K}}{\mathit{K}}\mathrm{=}\frac{\dot{A}}{\mathit{A}}=\mathit{\kappa }{{\mathit{H}}_A}_{}\mathrm{.}$

From equation (10.1.7), H_A = H - H_Y, and the above equation, we obtain:

${\mathit{g}}_{\mathit{Y}}=\mathit{\kappa }{\mathit{H}}_A=\mathit{\kappa }\mathit{H}\mathrm{-}\mathrm{\Lambda }\mathit{r}\mathrm{,}$(10.1.8)

where Λ ≡ α/(α + β)(1 - α - β).

It is necessary to require H_A being nonnegative. To determine the growth rate g_Y, we use equation (10.1.5), i.e., g_Y = (r - ρ)/σ. Together with equation (10.1.8), we find:

${\mathit{g}}_Y=\frac{\mathit{\kappa }\mathit{H}\mathrm{-}\mathit{\rho }\mathrm{\Lambda }}{\mathit{\sigma }\mathrm{\Lambda }\mathrm{+}\mathrm{1}}\mathrm{.}$(10.1.9)

For the integral in the consumer's preferences to be finite, the rate of growth of current utility (1 - σ)g_Y, must be less than the discount rate ρ. Thus, for σ ∈ [0, 1), it is necessary to have: (1 - σ)κH < ρ(Λ + 1).

Equation (10.1.9) is the main result of the Romer model. Because it shows continued positive growth of the main aggregated variables as well as per-capita consumption, the model is often cited as the 'symbol' of the new growth theory. The model shows that neither N nor η affects the long-run growth rate. As an increase in N increases the demand factor faced by each monopolistic firm and a reduction in η reduces the cost of the monopolist and increases output x*. In either case, the stream of net revenue generated by a new design rises. Nonetheless, the amount of human capital devoted to research is not affected by the two parameters. It should be remarked that since we consider the system as a whole, a change that increases the return to one activity can raise the return to some other activity that competes with the first activity for resources. An increase in N or reduction in η raises the return to human capital in the two types of activity in this model. For the functional forms used here, these two effects exactly cancel. This explains invariance of the growth rate with respect to these two parameters. As Romer mentioned, this idiosyncratic property will disappear if some functional forms are generalized.

From equation (10.1.9), we immediately see that the growth rate of consumption and income are endogenously only when the total human capital H is exogenously fixed (not like in the Solow model, labor force grows at a fixed rate). Since H is fixed and the growth rate of A is equal to a proportion of the component of H (through Ȧ - κH_AA) devoted to research, we see that the Romer model is essentially as exogenous as the Solow model. In fact, given the extremely rich literature of the neoclassical growth theory, the key contribution of the new growth theory may be the introduction of monopolistic competition into the neoclassical growth theory, rather than explanation of endogenous economic growth.

10.2 Product Variety and Growth

As an alternative to the Romer model - in which growth is achieved through the production of an increasing variety of goods, we can use some of the same apparatus to illustrate a similar model proposed by Grossman and Helpman.⁴ The model studies endogenous growth based on intentional industrial innovation. In this approach, research is treated as an ordinary economic activity that requires the input of resources and responds to profit opportunities. Returns to R&D are possible because of the monopoly rents from imperfectly competitive products markets. We will neglect investment in physical and human capital partly because analysis will become too complicated and partly because as we are interested in the key ideas in this approach.⁵

Industrial research may be involved either in process innovation to reduce the cost of producing existing goods or product innovation to invent entirely new goods. Product innovation is further classified according to whether the newly invented goods bear a vertical or horizontal relation to existing products. The former provides greater quality but performs similar functions to those by existing goods; the latter increases product variety. This section is concerned with expanding product variety and the next section with endogenous quality upgrading. There are unlimited potentials for developing new products and entrepreneurs invest resources to find unique goods. No diminishing returns in the creation of knowledge are assumed. Nevertheless, growth ultimately stops because the economic return to invention may decline as the number of available products increases.

The representative household maximizes utility over an infinite horizon. Intertemporal preferences are described by:

$\mathit{U}\mathrm{(}t\mathrm{)}=\underset{}{\overset{\mathrm{\infty }}{{\displaystyle \underset{t}{\int }}}}{\mathit{e}}^{-\mathit{\rho }\mathrm{(}\mathit{\tau }-1\mathrm{)}}\log \mathrm{(}\mathit{D}\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{)}\mathit{d}\mathit{\tau }\mathrm{,}$(10.2.1)

where D(τ) stands for an index of consumption at time τ and ρ is the subjective discount rate. The natural logarithm of the consumption index measures instantaneous utility at a moment of time. Since we are interested in variety of goods, the index D should reflect households' tastes for diversity in consumption. We take the product space to be continuous. We represent consumers' preferences over an infinite set of products, indexed by j ∈ [0, ∞]. At any moment only a subset of these varieties is available in markets. Households can purchase at time t all brands, denoted by the interval [0, n(t)], existing prior to t, where n(t) is the measure of products invented before time t. The variable n is called the number of available varieties. Following Dixit and Stiglitz,⁶ the index D is now is specified in such a way that it exhibits a constant and equal elasticity of substitution between every pair of goods:

$\mathit{D}=[\underset{0}{\overset{n}{\int }}\mathit{x}\mathrm{(}\mathit{j}{\mathrm{)}}^{\mathit{\alpha }}dj{\mathrm{]}}^{1/\mathit{\alpha }},\mathrm{0}\mathrm{<}\mathit{\alpha }\mathrm{<}\mathrm{1},$(10.2.2)

where x(j) denotes consumption of brand j. This specification has found many applications in the contemporary literature of economic growth with high product differentiation. The Dixit-Stiglitz preferences accommodate increasing diversity in consumption and yield aggregate demand functions that have a particular form convenient for analysis. The single parameter a characterizes different tastes for variety. It should be noted that the specification implies that innovative products are in no way superior to older varieties. As shown by Dixit and Stiglitz, with these preferences, the elasticity of substitution between any two products is ε = 1/(1 - α)> 1, and a household spending an amount E (which equals ${{{\displaystyle \int }}_{\mathrm{0}}}^{\prime \prime }$ p(j)n(j) dj, where p(j) is the price of brand j ∈ [0, n]) maximizes instantaneous utility by purchasing:

$\mathit{x}\mathrm{(}\mathit{j}\mathrm{)}=\frac{\mathit{E}\mathit{p}{\mathrm{(}\mathit{j}{\mathrm{)}}^{}}^{-ϵ}}{{{\displaystyle {\int }^n}}_{\mathrm{0}}^{}\mathit{p}\mathrm{(}\mathit{j}\prime {\mathrm{)}}^{\mathrm{1}-ϵ}\mathit{d}{\mathit{j}\prime }^{}}\mathrm{.}$(10.2.3)

The demand functions features a constant price elasticity of f and unitary expenditure elasticity for each product. We may also express equation (10.2.3) in an alternative form given by Ethier:⁷

$x\mathrm{(}\mathit{j}\mathrm{)}=\mathit{D}\mathit{p}{\mathrm{(}\mathit{j})}^{-\epsilon }{[\underset{o}{\overset{n}{\int }}p{(j\prime )}^{1-\epsilon }dj\prime ]}^{-1/\alpha }\mathrm{,}j\in [0,n],$(10.2.4)

where we use:

$\mathit{D}=\frac{\mathit{E}}{{\mathit{p}}_{\mathit{D}}}\mathrm{,}{\mathit{p}}_{\mathit{D}}=\mathrm{[}\underset{\mathrm{0}}{\overset{}{{\displaystyle {\int }^n}}}\mathit{p}\mathrm{(}\mathit{j}{\mathrm{)}}^{\mathrm{1}-\epsilon }dj{\mathrm{]}}^{\mathrm{1}/\mathrm{(}\mathrm{1}-\epsilon \mathrm{)}}$

Substituting D = E/p_d into equation (10.2.1), we have:

$U\mathrm{(}\mathit{t}\mathrm{)}=\stackrel{\infty }{{\displaystyle \underset{t}{\int }}}{\mathit{e}}^{-\mathit{\rho }\mathrm{(}\tau -t\mathrm{)}}\mathrm{[}\log \mathrm{(}\mathit{E}\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{)}\mathrm{-}\log \mathrm{(}{\mathit{p}}_{\mathit{D}}\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{)}\mathrm{]}\mathit{d}\mathit{\tau }\mathrm{.}$(10.2.5)

The weak separability of indirect utility in the level of spending and the price index in equation (10.2.5) simplifies the maximization problem. The household now solves its optimization problem into two stages. The first stage chooses the composition of given levels of spending to maximize instantaneous utility. The second stage optimizes separately the time path of spending. It can be shown that the solution for the latter problem is given by:

$\frac{\dot{\mathit{E}}}{\mathit{E}}=\mathit{r}\mathrm{-}\rho \mathrm{.}$(10.2.6)

Prices can be normalized in such a way that nominal spending remains through time. Consequently, equation (10.2.6) becomes:

$$\begin{gathered} \mathit{E}\mathrm{(}\mathit{t}\mathrm{)}=\mathrm{1}\text{for all }\mathit{t}\mathrm{,} \\ \mathit{r}\mathrm{(}\mathit{t}\mathrm{)}=\rho \text{for all }\mathit{t}\mathrm{.} \end{gathered}$$(10.2.7)

The final decision is to allocate their wealth across available assets according to the conditions given by equation (10.2.3).

We now examine behavior of the producers. The economy is endowed with a single primary factor of production - labor. By an appropriate choice of units, the input-output coefficient is set to one. Producers create blueprints for new goods and manufacture the products previously innovated. The up-front R&D expense can be regarded as a fixed cost in the production cycle of a given commodity. Each known variety of the differentiated product is manufactured by a single, atomistic firm, subject to a common constant-return-to-scale technology. Under these specifications, the unique supplier of variety j maximizes operating profit:

$\pi (j,t)=p(j,t)x(j,t)-w\left(t\right)x(j,t),$(10.2.8)

where w(t) is the wage rate at time t. Faced with the demand function (10.2.3), the firm maximizes the profit by charging a price p(j) = w/α. Henceforth we suppress the time arguments when no confusion arises from doing so. In the momentary equilibrium all varieties are priced equally at p, where p = w/α. With symmetric demand and E = 1, this pricing strategy yields per brand operating profits of:

$\mathit{\pi }=\frac{\mathrm{1}\mathrm{-}\mathit{\alpha }}{\mathit{n}}\mathrm{.}$(10.2.9)

The profits are one component of the return to the owners of firms and are continuously paid to shareholders as dividends. Let v(7) stand for the value of a claim to the infinite stream of profits that accrues to a typical firm operating at time t. In the brief time interval between t and t + dt, the total return to the owners of this firm equals πdt + v̇dt. It is assumed that arbitrage in capital markets ensures equality between this yield and that on a riskless loan. The latter return for an investment of size v is rvdt. Thus the equilibrium in the capital market requires:

$\pi +\dot{v}=rv\mathrm{.}$(10.2.10)

It is assumed that the stock market value at time t of a firm equals the present discounted value of its profit stream subsequent to t:

$\mathit{v}\mathrm{(}t\mathrm{)}=\underset{}{\overset{\mathrm{\infty }}{{\displaystyle \underset{t}{\int }}}}{\mathit{e}}^{-\mathrm{[}\mathit{R}\mathrm{(}\tau \mathrm{)}-\mathit{R}\mathrm{(}t)\mathrm{]}}\mathit{\pi }\mathrm{(}\mathit{\tau }\mathrm{)}\mathit{d}\mathit{\tau },$(10.2.11)

where R(t) represents the cumulative discount factor applicable to profits earned at time t. Differentiation of equation (3.2.10) yields equation (3.2.9).

We now describe the technology for product development. An entrepreneur can add incrementally to the set of available products by devoting a given finite amount of labor to R&D for a brief interval of time. Let l denote units of labor devoted to R&D for a time interval of length dt. New products dn produced in the interval is given by dn = (l/a)dt. The total cost of such a research venture is wldt and the effort creates value for the entrepreneur of v(1/a)dt. Value maximization by entrepreneurs implies that / will be chosen as large as possible if v/a > w (which cannot arise in general equilibrium because it implies an unbounded demand for labor by research enterprises) and be set to zero if v/a < w. Consequently, we have:

$\mathit{w}\mathit{a}\mathrm{\ge }\mathit{v}\text{with equality whenever}\dot{\mathit{n}}\mathrm{>}\mathrm{0}\mathrm{.}$0.2.12)

The combination of free entry and constant returns to scale in research means no excess returns for entrepreneurs.

The representative variety of differentiated product bears a price p and aggregate demand E = 1. So each firm sells 1 / np units and demands 1 / np units of labor. The total demand for labor from n manufactures is equal to 1 / p. Consequently, the labor market equilibrium requires:

$\mathit{a}\dot{\mathit{n}}+\frac{\mathrm{1}}{\mathit{p}}=\mathit{N}\mathrm{,}$(10.2.13)

where aṅ is the total employment in R&D. From the above equation we see that p ≥ 1/N. This completes the specification of the model. We now examine qualitative properties of the system.

First we are interested in a time interval over which new brands are being developed. During this period, v - aw by equation (10.2.12). Combining this free-entry condition with pricing condition p = w/α and p ≥ 1/N, we see that R&D is profitable only when the reward for successful research is sufficiently high; that is dn/dt > 0 implies v > v₀ where v₀ = αa / N. Summarizing these discussions and using equation (10.2.13), v - aw and p = w/α, we see that the number of varieties evolves according to:

$\dot{\mathit{n}}=\{\begin{array}{c}\frac{L}{a}-\frac{\alpha }{v}v>v_0,\\ 0v\le v_0\mathrm{.}\end{array}$(10.2.14)

Next, we obtain an equation for the change in firm value. From equations (10.2.7), (10.2.9), and (10.2.10), we obtain:

$\dot{\mathit{v}}=\rho \mathit{v}\mathrm{-}\frac{\mathrm{1}\mathrm{-}\mathit{\alpha }}{\mathit{n}}\mathrm{.}$(10.2.15)

The dynamics are represented by the two differential equations (10.2.14) and (10.2.15). The dynamics are illustrated in Figure 10.1.

The hyperbola vv shows combinations of n and v for which the value of the typical firm remains constant. The curve is downward sloping. We interpret this as follows. An increase in v raises the opportunity cost of holding shares in the representative firm; then capital markets will expect zero capital gains only if the dividend rate is high; a higher dividend rate requires higher profits and so a smaller set of competing brands. The number of differentiated products is constant at points or below the horizontal level v = v₀. This occurs because production innovation requires a sufficiently high firm value to justify the large costs of R&D. The system is stationary at point A and at all points along vv below this point.

10.3 Variety of Consumer Goods and Growth

The previous section is concerned with the growth model of variety of products. This section introduces growth model with a variety of consumer products. This section is referred to Barro and Sala-i-Martin.⁸ The idea is to introduce a variety of consumer goods into the utility function that parallels the treatment of a variety of intermediate products in the production function as in the previous section. We can address these issues with the framework proposed in this book. If we treat variety of consumer goods as a function of knowledge, we can model variety of consumer goods as our utility function is flexible to treat variety as an endogenous variable.

Let us assume that there consumers care about variety consumer goods, which is measured by an index for consumer i:

${\mathit{c}}_i={{(}^{}\underset{j=1}{\overset{M}{\mathrm{\Sigma }}}{c}_{ij}^{\epsilon })}^{1/\epsilon },\mathrm{0}\mathrm{<}\mathit{\epsilon }\mathrm{\le }\mathrm{1}\mathrm{,}$(10.3.1)

where c_ij is household i's consumption of goods of type j, M is the number of types available at the current time. The household i's utility is given by:

$U_i\mathrm{=}\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{{\displaystyle \int }}}\left(\frac{{\mathit{c}}_i^{\mathrm{1}-\mathit{\theta }}\mathrm{-}\mathrm{1}}{\mathrm{1}\mathrm{-}\mathit{\theta }}\right){\mathit{e}}^{-\mathit{\rho }t}\mathit{d}t\mathrm{.}$

To see why the formation of utility function captures the idea that consumers like variety, suppose that c_ij are measured in a common physical unit and the quantities consumed of each type are the same, c_ij = c_i M. We have:

$\frac{c_i^{1-\theta }\mathrm{-}\mathrm{1}}{\mathrm{1}\mathrm{-}\mathit{\theta }}=\frac{{\mathit{M}}^{\mathrm{(}1-\epsilon \left)\right(1-\theta )/\epsilon }{c}_i^{1-\theta }\mathrm{-}\mathrm{1}}{\mathrm{1}\mathrm{-}\mathit{\theta }}\mathrm{.}$

Hence, the flow of utility rises as M increases for a fixed c_i This shows a positive relation between consumers' utility and variety.

The invention of a new product - an increase in M is assumed to cost η units of goods. The inventor retains a perpetual monopoly in the production of the associated nondurable consumer good, C_j. The marginal cost of production of each consumer good is 1, and the producer determines the consumer price, P_j, maximizing the flow of monopoly profit. To determine P_j, we need to know the demand function. Let a(t) stand for assets per person. Then:

$\dot{\mathit{a}}\mathrm{(}t\mathrm{)}=\mathit{w}\mathrm{(}t\mathrm{)}+\mathit{r}\mathrm{(}\mathit{t}\mathrm{)}a\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\sum _{j=1}^M{\mathit{P}}_j\mathrm{(}t\mathrm{)}{c}_{ij}\mathrm{(}t\mathrm{)},$

where w and r are respectively the wage rate and rate of interest. The Hamiltonian associated with consumers' utility maximization is defined by:

$\mathit{J}\mathrm{=}\left\{\frac{{\left(\sum _{j=1}^M{c^{}}_{ij}^{\epsilon }\right)}^{(1-\theta )/\epsilon }-1}{\mathrm{1}\mathrm{-}\mathit{\theta }}\mathrm{\right\}}{e}^{-\rho t}\mathrm{+}\mathit{v}(\mathit{w}+\mathit{r}\mathit{a}\mathrm{-}\underset{j=1}{\overset{M}{\mathrm{\Sigma }}}{\mathit{P}}_j{\mathit{c}}_{ij})\mathrm{.}$

The first-order condition with respect to c_ij yields:

$\frac{C_{ij}}{{\mathit{C}}_{ik}}={\left(\frac{{\mathit{P}}_j}{{\mathit{P}}_{\mathit{k}}}\right)}^{-1/(1-\epsilon )},\mathit{j}\mathrm{,}\mathit{k}=\mathrm{1}\mathrm{,}\dots ,M\mathrm{.}$

This condition enables us to derive the consumer's demand function for the jth good:

$C_{ij}=\mathrm{\left\{}\frac{{\mathrm{\Sigma }}_{k=\mathrm{1}}^M{\mathit{P}}_{\mathit{k}}{\mathit{c}}_{ik}}{{\mathrm{\Sigma }}_{k=\mathrm{1}}^M{p}_k^{-\alpha /(1-\alpha )}}\right\}{P}_j^{-1/(1-\epsilon )}\mathrm{.}$(10.3.2)

We assume that M is sufficiently large so that the producer of good j can neglect the effect of P_j on the households' total spending per variety of good, that is, on the ratio of sums given in equation (10.3.2). Consumer demand then has the constant elasticity - 1/(1 - ε) with respect to P_j Hence, the monopoly producer of good j determines the consumer price with a markup on the unit marginal cost of production, P_j = 1/ε. As all prices are equal, we denote the prices by P. Since the prices of all consumer goods are equal and the goods enter symmetrically into the utility function, the quantities consumed are the same: c_ij - c_i / M. If we can determine c_i and M, then C_ij are determined.

To determine the evolution of c_i, we use the remaining optimization conditions associated with the Hamiltonian. Setting the derivative of J with respect to c_g equal to zero and then substituting c_ij = c_i / M for all good, we find:

$\mathit{v}\mathit{P}={\mathit{M}}^{\mathrm{(}\mathrm{1}-\epsilon \left)\right(1-\theta )/\epsilon }\mathrm{(}{\mathit{c}}_i{\mathrm{)}}^{-\mathit{\theta }}{\mathit{e}}^{-\mathit{\rho }t}\mathrm{.}$(10.3.3)

The second condition is associated with the state variable a:

$\dot{v}=-rv\mathrm{.}$(10.3.4)

The two conditions, (10.3.3) and (10.3.4) determine:

$\frac{{\dot{\mathit{c}}}_i}{{\mathit{c}}_i}=\frac{\mathrm{1}}{\mathit{\theta }}\mathrm{\{}\mathit{r}\mathrm{-}\rho \mathrm{+}\mathrm{[}\frac{\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\epsilon }\left)\right(\mathrm{1}\mathrm{-}\mathit{\theta }\mathrm{)}}{\mathit{\epsilon }}\mathrm{\left]}\frac{\dot{\mathit{M}}}{\mathit{M}}\mathrm{\right\}}\mathrm{.}$

Using this equation, we obtain:

$\frac{\mathit{d}\mathrm{(}{c}_i\mathrm{/}\mathit{M}\mathrm{)}}{\mathit{d}\mathit{t}}\mathrm{=}\frac{\dot{\mathit{M}}}{\mathit{M}}\mathrm{-}\frac{{\dot{\mathit{c}}}_i}{{\mathit{c}}_i}=\frac{\mathrm{1}}{\mathit{\theta }}\mathrm{\{}\mathit{r}\mathrm{-}\mathit{\rho }\mathrm{+}\mathrm{[}\frac{\mathit{\theta }+\mathit{\epsilon }\mathrm{-}\mathrm{1}}{\mathit{\epsilon }}\mathrm{\left]}\frac{\dot{\mathit{M}}}{\mathit{M}}\right\}\mathrm{.}$(10.3.5)

We now analyze invention. The net present value, V(t), for an inventor of a new consumer good at time t is:

$V\mathrm{(}t\mathrm{)}=\underset{0}{\overset{\mathrm{\infty }}{{\displaystyle \int }}}({\mathit{P}}_j\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{-}1)C_j\mathrm{(}\mathit{\tau }\mathrm{)}{\mathit{e}}^{-\overline{r}(\tau ,t)(\tau -t)}d\tau ,$(10.3.6)

where:

$\overline{\mathit{r}}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}\mathrm{\equiv }\frac{\mathrm{1}}{\mathit{\tau }\mathrm{-}\mathit{t}}{\displaystyle \underset{t}{\overset{\tau }{\int }}}\mathit{r}\mathrm{(}\mathit{\omega }\mathrm{)}\mathit{d}\mathit{\omega }\mathrm{,}$

is the average interest rate between times t and τ. If the interest rate is constant, then the present-value factor simplifies to e^-r(τ-t) The equation shows that the fixed cost η for discovering a new good can be recouped only if P_j exceeds the marginal cost of production for at least part of the time after date t.

We assume free entry into the business of being an inventor. This implies that anyone can pay the R&D cost η to secure the present value. If V(t) > η, then an infinite amount of resources would be channeled into R&D at time t, hence V(t)> η cannot hold at equilibrium. If V(t) < η, then no resources would be devoted at time t to R&D. If r is constant, the free-entry condition is:

$\mathit{\eta }\mathrm{\ge }\mathit{V}\mathrm{(}t\mathrm{)}=\frac{\mathrm{1}\mathrm{-}\mathit{\epsilon }}{\mathit{\epsilon }}\stackrel{\infty }{\underset{t}{{\displaystyle \int }}}{\mathit{C}}_j{\mathit{e}}^{-r(\tau -t)}\mathit{d}\mathit{\tau },$

where we use P_j = 1/ε. The condition holds with equality if Ṁ = 0. In this case, C_j is constant and given by:

$C_j=\frac{C}{\mathit{M}}=\frac{\mathit{r}\mathit{\eta }\mathit{\epsilon }}{\mathrm{1}\mathrm{-}\epsilon }$

If both C and population are constant, then c_ij / M must also be constant; hence the growth rate of c_i / M must be equal to zero. Equation (10.3.5) yields:

$\frac{\dot{C}}{C}=\frac{\dot{\mathit{M}}}{\mathit{M}}\mathrm{=}\frac{\mathit{\epsilon }}{\mathit{\theta }+\mathit{\epsilon }\mathrm{-}\mathrm{1}}\mathrm{(}\mathit{r}\mathrm{-}\mathit{\rho }\mathrm{)},\mathit{\theta }+\mathit{\epsilon }\mathrm{\ne }\mathrm{1}\mathrm{.}$(10.3.7)

It is assumed θ + ε > 1. This assumption guarantees that the growth rate of C is positively proportional to r - ρ.

We will not further analyze behavior of the model because, as shown by Barro and Sala-i-Martin,⁹ the model does not provide new insight, given the growth model with variety of products defined in Section 10.2. Further analysis is referred to Barro and Sala-i-Martin.

10.4 The Schumpeterian Creative Destruction

As mentioned in Chapter 1, the year 1883 is special for the history of the world economy and the history of economic analysis. The year saw the death of Karl Marx (1918-1883), the birth of John Maynard Keynes (1883-1946), and the birth of Joseph Alois Schumpeter (1883-1950).

In the Theory of Economic Development published in 1911, Schumpeter argues that development should be understood as only such changes in economic life as are not forced upon it from without but arise by its own initiative, from within. Schumpeter holds that successful carrying out of new combinations of productive services is the essence of this process. It is spontaneous and discontinuous changes in the channels of the flow, disturbance of equilibrium, which alters and displaces the equilibrium state previously existing. The carrying out of new combinations is innovation, which consists of the following five cases:

• (i) introduction of a new good or a new quality of an old good;
• (ii) introduction of a new method of production;
• (iii) the opening of a new market for a product;
• (iv) the conquest of a new source of raw materials or half-manufactured articles; and
• (v) the carrying out of a new organization of an industry like the creation of a monopoly position or the breaking up of a monopoly position.

Profits are temporary in the sense that they emerge at one point in the economy and accrue to the innovator, then start dwindling as they are shared by an increasing number of innovators on one hand and eaten up by rising costs and falling prices on the other, till they finally disappear. By this time the development achieved in the form of the newly introduced method has been generalized, completely replacing the old inferior methods. Profits are thus both the child and the victim of economic development. Once again the competitive process establishes cost-price equality all round. Factorial rewards equal marginal productivities. Wages and rents are now higher and prices of consumer goods lower. But the circular flow once again comes into its own. A dynamic flow is again disturbed by some new innovation. Creative destruction is continued by competition over profits. The competitive process will strike not at the margins of the profits of existing firms but at their veiy existence. The fear for survival encourages operating efficiency. The dynamic competition among the entrepreneurs creates innovations and destroys old equilibria by establishing new combinations. The process of such creative destruction brings about progress in productivity.

Schumpeter's ideas about development and creative destruction have recently been modeled. This section represents such a model by Aghion and Howitt.¹⁰ The Aghion-Howitt model of economic growth is based on Schumpeter's process of creative destruction. Growth results exclusively from technological progress, which in turn results from competition among research firms that generate innovations. The model assumes that individual innovations are sufficiently important to affect the entire economy. Each innovation consists of a new intermediate good that can be used to produce final output more efficiently than before. Research firms are motivated by the process by the prospect of monopoly rents that can be captured when a successful innovation is patented. But those rents in turn will be destroyed by the next innovation, which will render obsolete the existing intermediate good. Equilibrium is determined by a forward difference equation, according to which the amount of research in any period depends upon the expected amount of research next period.

Since it is technically laborious, we represent a simplified version of the model by Solow.¹¹ The model includes the R&D process and tries to formally explain Schumpeter's processes of creative destruction - successful R&D can make the technology invented by previous R&D unprofitable and the rents from successful innovation are temporary. Because of its nonlinearity and being in discrete time, the model may exhibit endogenous cyclical behavior. We now describe the model.

A period is the time between two successive innovations. The length of each period is random because of the stochastic nature of the innovation process, but the relationship between the amounts of research in two successive periods can be modeled as deterministic. The amount of research this period depends negatively upon the expected amount next period, through two effects. The first is that of creative destruction. The payoff from research this period is the prospect of monopoly rents next period. Those rents will last only until the next innovation occurs. The expectation of more research next period will discourage research this period. The second effect is that of a general equilibrium in the labor market. Workers can be used either in research or in manufacturing. To maintain labor market equilibrium, the expectation of more research next period must correspond to an expectation of higher demand for labor in research next period, which implies the expectation of a higher wage rate. Higher wages next period will reduce the monopoly rents that can be gained by exclusive knowledge of how to produce the best products. Thus the expectation of more research will discourage research this period.

Like most of models in the new growth theory, the model omits capital accumulation and assumes a constant employment. There are three classes of tradable objects: labor, a consumption good, and an intermediate good. There is a continuum of infinitely-lived individuals, with identical intertemporally additive preferences defined over lifetime consumption, and the constant rate of time preference. The marginal utility of consumption is equal to the rate of interest. There are three categories of labor: unskilled labor, which can be used only in producing the consumption good; (skilled) labor, which can be used either in research or in intermediate sector; and specialized labor, which can be used only in research. Each individual is endowed with a unit flow of labor. Only one final good is produced by the fixed quantity of unskilled labor and skilled labor x. The production function is F(t) - Af(x), where we omit expressing unskilled labor and f(x) is increasing (f′(x) > 0) and concave (f″(x) < 0). The variable A is a technological variable. Final good is used as numeriare.

Some skilled labor is devoted to R&D. When successful, the innovation is a new intermediate good that allows a higher value of A and thus makes the old intermediate good obsolete. Let t stand for the t th innovation (not time). For convenience of description, we consider that each successful innovation increases the final output producible with any x by a multiplicative factor γ, i.e., ^A_t+1^/A_t = γ.

Suppose n units of labor are devoted to R&D and innovations arrive according to a Poisson process with arrival rate λn. It should be noted that in the original model the specialized labor affects the arrival rate. Since the number of specialized labor is prefixed, we omit mentioning this type of labor. The probability of an innovation in a given short unit of time equals λn, and the probability of no innovation is equal to 1 - λn, and the probability of two or more innovations is equal to zero. The assumption of the Poisson process says that the probability of making an innovation of given size depends only on n, independent of past history of innovation. In fact, innovation is hard to model for anyone because innovations cannot be predetermined even in probability sense.

The innovating firm acquires a monopoly on the final production that is useful until the next innovation. Thus the t th innovation brings a negative externality through killing the rents of the firm that produced the (t - 1) st innovation and a positive externality through making the (t + 1) st innovation possible. A successful innovator is a monopoly of the intermediate good and is faced with a demand curve from the final-goods industry: Af'(x_t) = P_t, where P_t is the price of the intermediate good.

We introduce V_t and Π_t, to respectively stand for the expected discounted rents associated with the t th successful innovation and the constant flow of rent expected by the t th innovator during the profitable life of the innovation. Let ρ stand for the discount rate of the rent expected by the I th innovator. Then the Fisher equation tells that the interest on the value of innovation equals the current income plus the expected capital gain (which equals λn_t(-V_t) + (1 - λn_t)0). That is: ρV_t = Π_t, - λn_tV_t. This equation is solved as:

$V_{\iota }=\frac{{\mathrm{\Pi }}_{\iota }}{\mathit{\rho }+\mathit{\lambda }{\mathit{n}}_{\iota }}\mathrm{.}$(10.4.1)

The equation says that a large value of n_t reduces V_t. In other words, research, like capital investment, is discouraged by the prospect of future R&D. Free entry and risk neutrality in R&D guarantees that entry will occur until the cost of conducting R&D is equal to the expected value of the innovation:

${\mathit{w}}_t{\mathit{n}}_t=\mathit{\lambda }{{\mathit{n}}_t}_{}{\mathit{V}}_{t\mathrm{+}\mathrm{1}}+\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\lambda }{\mathit{n}}_t\mathrm{)}\cdot \mathrm{0}\Rightarrow {\mathit{w}}_t=\mathit{\lambda }{V}_{t+\mathrm{1}}\mathrm{.}$(10.4.2)

Labor market is cleared for every t:

$n_t+x_t=N,$(10.4.3)

where N is the constant volume of employment. Solow holds that equation (10.4.3) contains a major limitation of this model: 'one of the true risks of R&D is that economic conditions should be cyclically weak during the effective life of an innovation, so that it turns out to be unprofitable because sales of final product are poor'.

The intermediate good is produced using skilled labor alone. The production function is specified in such a way that the intermediate product is equal to the flow of skilled labor used in the intermediate sector. With the one-to-one technology for producing intermediate good, the monopolist maximizes:

$P_t{x}_t-w_t{x}_t=A_{t}f\prime \left(x_t\right)x_t-w_t{x}_t\mathrm{.}$(10.4.4)

Provided that marginal revenue falls with x_t, the optimal x_t is a decreasing function of w_t/A_t, and the best achievable value of (Π_t/A_t) falls as (w_t / A_t) rises. Denote this function by x, = ϕ̄(wt/At). From equation (10.4.3), n_t is an increasing function of w_t/A_t. We have:

${{\mathit{n}}_t}_{}=\mathit{N}\mathrm{-}{{\mathit{x}}_{}}_t=\mathit{N}\mathrm{-}\overline{\mathit{\varphi }}\left(\frac{{{\mathit{w}}_{}}_t}{{{\mathit{A}}_{}}_t}\right)\mathrm{.}$

Solving the above equation with w_t/A_t as the variable yields:

$\frac{{{\mathit{w}}_{}}_t}{{{\mathit{A}}_{}}_t}=\mathit{\varphi }\mathrm{(}{{\mathit{n}}_{}}_t\mathrm{)},$(10.4.5)

where ϕ(n_t) rises in n_t Now equations (10.4.1) and (10.4.2) imply:

${\mathit{w}}_t=\mathit{\lambda }{\mathit{V}}_{t+\mathrm{1}}=\frac{\mathit{\lambda }{\mathrm{\Pi }}_{t+1}}{\mathit{\rho }+\mathit{\lambda }{\mathit{n}}_{t+1}}\mathrm{.}$

Inserting equation (10.4.5) and A_t+1 = γA_t into the above equation yields:

$\frac{{{\mathit{w}}_{}}_t}{{{\mathit{A}}_{}}_t}=\frac{\mathit{\gamma }\mathit{\lambda }\mathrm{(}{\mathrm{\Pi }}_{t\mathrm{+}\mathrm{1}}\mathrm{/}{\mathit{A}}_{t\mathrm{+}\mathrm{1}}\mathrm{)}}{\mathit{\rho }+\mathit{\lambda }{\mathit{n}}_{t\mathrm{+}1}}\mathrm{.}$(10.4.6)

As (Π_t+1/A_t+1) is a decreasing function of (w_t+1/A_t+1) and (w_t+1/A_t+1) is an increasing function of n_t+1, (Π_t+1 A_t+1) falls as n_t+1 rises. Consequently, the right-hand side of equation (10.4.6) is a decreasing function of n_t+1, denoted by ψ (n_t+1). Equations (10.4.5) and (10.4.6) imply:

$\mathit{\varphi }\mathrm{(}{\mathit{n}}_t\mathrm{)}=\mathit{\psi }\mathrm{(}{\mathit{n}}_{t\mathrm{+}\mathrm{1}}\mathrm{)}\mathrm{.}$

This dynamic equation closes the model. As ψ(n_t+1) falls in n_t+i and ϕ(n_t) rises in n_t, we rewrite the above discrete mapping the following form:

$n_{t+1}=h\left(n_t\right),h\prime <0.$(10.4.7)

Equilibrium is a solution of:

${\mathit{n}}^{*}=\mathit{h}\mathrm{(}{\mathit{n}}^{*}\mathrm{)}\mathrm{.}$

In general, as shown in Figure 10.2, there will be a unique steady state. We know that n, tends to n* if |h′(n)| < 1 for all n and will converge locally if:

$|\mathit{h}\mathrm{\text{'}}\mathrm{(}{\mathit{n}}^{*}\mathrm{)}\mathrm{|}\mathrm{<}\mathrm{1}\mathrm{.}$

Once we determine n_t, we determine all the other variables in the system.

We may conduct usual comparative statics analysis with respect different parameters in the system. Explanation about the model and its further implications is referred to Aghion and Howitt.¹²

10.5 Growth with Improvements in Quality of Products

The previous two sectors treated technological progress as an increase in the number of types of products. This section deals with types of technological changes that improve quality of the current products.¹³

Like the model in Section 10.2, there are N types of non-durable intermediate goods. For convenience of analysis, we fix N. It is assumed that when a product or technology is improved, the new good or method tends to displace the old one. As we assume that different qualities of a particular type of intermediate input are perfect substitutes, the discovery of a higher grade turns out to drive out completely the lower grades. The process is characterized by Schumpeter's creative destruction (as explained in the previous section) in the sense that successful researchers tend to eliminate the monopoly rentals of their predecessors. We consider that the duration of successful research is random, because it depends on the uncertain outcomes from the research efforts by competitors. There is no permanent monopoly position.

The production function for firm i is:

${\mathit{Y}}_i=\mathit{A}{L}_i^{1-\alpha }\underset{j=1}{\overset{N}{\mathrm{\Sigma }}}{X}_{ij}^{\alpha },\mathrm{0}\mathrm{<}\mathit{\alpha }\mathrm{<}\mathrm{1},$(10.5.1)

where L_i, is labor input and X_ij is the quality-adjusted amount employed of the j th type of intermediate good. The potential grades of each intermediate good are arranged along a quality ladder with rungs spaced proportionately at interval q > 1. We normalize so that each good begins at quality 1. The subsequent rungs are at the levels q, q², and so on. Thus, if sector j has experienced k_j improvements in quality, then the available grades in the sector are 1, q, ..., q^kj. Let X_ijk stand for the quantity used by the ith firm of the jth type of intermediate good of quality rung k, which corresponds to quality q^k. The overall input from a sector, X_ij, is the quality-weighted sum of the amounts used of each grade, q^kX_ijk. Thus, if κ_j, is the highest quality level available in sector j, then the quality-adjusted input from this sector is given by: X_ij; = $\sum _{k=0}^{k_j}{q}^k{X}_{ijk}$, which implies that the quality grades within a sector are perfect substitutes as input to production.

Assume that goods of quality 1 can be produced by anyone. The researcher responsible for each quality improvement in sector j retains a monopoly right to produce the jth intermediate good at that quality level. In particular, if the quality rungs k = 1,..., κ_j have been reached, then the kth innovator is the sole source of intermediate goods with the quality level q^k. The intermediate good entails a unit marginal cost of production (in terms of output, Y). Let the cost of production of intermediate goods be the same for all qualities q^k, k = 0, 1,..., κ_j. Thus, the latest innovator has an efficiency advantage over the prior innovators in the sector, but will be at a disadvantage relative to future innovators. In the situation that only the best existing quality of intermediate good j with quality level q^kj is available currently for the production (the other grades will not be used in equilibrium), by equation (10.5.1) and the definition of X_ij, the marginal product of X_ijk is:

$\frac{\mathrm{\partial }{\mathit{Y}}_i}{\mathrm{\partial }{\mathit{X}}_{ijk}}=\mathit{\alpha }A{L}_i^{1-\alpha }{\mathit{q}}^{\alpha k_j}{X}_{ijk}^{\alpha -1}\mathrm{.}$

If units of the leading-edge good are priced at ${\mathit{P}}_{j{\mathit{\kappa }}_j}$ and if no other quality grades of good j are available, then $\mathrm{\partial }{Y}_i\mathrm{/}\mathrm{\partial }{\mathit{X}}_{ijk}={\mathit{P}}_{j{\kappa }_j}$. Hence, the implied demand function from the aggregate of final- goods producers is:

${{\mathit{X}}_{ijk}}_{}={\left(\frac{{{\mathit{P}}_{j{k}_j}}_{}}{\mathit{\alpha }A{\mathit{q}}^{\mathit{\alpha }{k}_j}}\right)}^{1/(\alpha -1)}L\mathrm{.}\left(\mathrm{10.5.2}\right)$

As the leading-edge producer acts as a monopolist in this environment, the monopolistic pricing is given by: P_jki = P = 1/α.

The monopoly price is thus constant over time and across sectors. Substituting P = 1/α into equation (10.5.2) yields:

${\mathit{X}}_{ijk}={\left(\frac{\mathrm{1}}{{\mathit{\alpha }}^2\mathit{A}{\mathit{q}}^{\alpha {\kappa }_j}}\right)}^{1/(\alpha -1)}L\mathrm{.}$(10.5.3)

In the demand functions, only κ_j are changeable. As κ_j changes over time in each sector, X_ijk vary over time and across sectors.

We are now concerned with situations that goods from quality rungs below κ_j are also available for production in sector j. We neglect the possibility that the κ_j; th innovator was also the (κ_j - 1) th (and below) innovator. If the leading-edge producer charges the monopoly price and if this price is high enough, then the producer of the next lowest grade will produce to obtain non-negative profits. By X_ij = $\sum _{k=0}^{{\kappa }_j}{q}^k{X}_{ijk}$, each unit of the leading-edge good is equivalent to q > 1 units of the next best good; thus if the highest grade is priced at ${\mathit{P}}_{j{\kappa }_j}$, then the next grade good could be sold at most at the price ${\mathit{P}}_{j{\kappa }_j}\mathrm{/}\mathrm{q}$, the one below that at the price ${\mathit{P}}_{j{\kappa }_j}\mathrm{/}{\mathit{q}}^{\mathrm{2}}$ and so on. If ${\mathit{P}}_{j{\mathit{\kappa }}_j}\mathrm{/}\mathit{q}$ is less than the unit marginal cost of production, then the next best grade (and all of the lower quality grades) cannot survive. As ${\mathit{P}}_{j{\mathit{\kappa }}_j}=1/\alpha$, the next best grade and all of the lower quality grades are priced at most 1/αq, 1/αq². ... If 1/αq is less than one, then the next best producer cannot compete against the leader's monopoly price. Therefore, αq > 1 implies that monopoly pricing will prevail. If αq ≤ 1, then the providers of intermediate goods of a given type engage in Bertrand price competition. In this case, the quality leader employs a limit-pricing strategy ${\mathit{P}}_{{j\mathit{\kappa }}_j}=q$ so as to make it just barely unprofitable for the next best quality to be produced. Because the condition for limit pricing to prevail αq ≤ 1, ${\mathit{P}}_{j{\mathit{\kappa }}_j}=\mathit{q}\mathrm{\le }\mathrm{1}\mathrm{/}\mathit{\alpha }$ = monopoly pricing.

When ${\mathit{P}}_{j{\kappa }_j}=q$ prevails, the total quantity produced is:

${\mathit{X}}_{ijk}={\left(\frac{\mathit{q}}{\mathit{\alpha }\mathit{A}{\mathit{q}}^{\mathit{\alpha }{\kappa }_j}}\right)}^{1/(\alpha -1)}\mathit{L}\mathrm{.}$

This equation and equation (10.5.3) show that if αq < 1, the quantity produced under limit pricing is at least as large as the amount that would have been produced under monopoly. In the remainder of this section, we require αq > 1. That is, the monopoly prevails. The case of αq ≤ 1 can be similarly discussed. We thus can neglect any goods of less than leading-edge quality and rewrite equation (10.5.1) as:

${\mathit{Y}}_i=\mathit{A}{L}_i^{1-\alpha }\underset{j=1}{\overset{N}{\mathrm{\Sigma }}}{\mathit{q}}^{\mathit{\alpha }{\kappa }_j}{\mathit{X}}_{ij{\kappa }_j}^{\mathit{\alpha }}\mathrm{.}$

Substituting equation (10.5.3) into the above equation (where L_i is replaced by L) and then aggregating over the firms, we get:

$\mathit{Y}\mathrm{=}{\mathrm{A}}^{1/(\mathrm{1}-\mathit{\alpha }\mathrm{)}}{\mathit{\alpha }}^{2\alpha /\mathrm{1}/\mathrm{(}1-\alpha \mathrm{)}}\mathit{L}\underset{j=1}{\overset{N}{\mathrm{\Sigma }}}{\mathit{q}}^{\mathit{\alpha }{\kappa }_j/1/\mathrm{(}1-\alpha \mathrm{)}}\mathrm{.}$

As L and N are constant, the growth of Y is due to changes in κ_j. We may rewrite the above equation as:

$\mathit{Y}\mathrm{=}{\mathit{A}}^{1/(1-\alpha )}{\mathit{\alpha }}^{2\mathit{\alpha }/\mathrm{(}1-\alpha \mathrm{)}}\mathit{L}\mathit{Q}={\mathit{\alpha }}^{-\mathrm{2}}\mathit{X}\mathrm{,}$

where X is total quantity of intermediates produced and:

$\mathit{Q}\mathrm{\equiv }\stackrel{N}{\underset{j=1}{\mathrm{\Sigma }}}{\mathit{q}}^{\mathit{\alpha }{\kappa }_j/\mathrm{(}1-\mathit{\alpha }\mathrm{)}},\mathit{X}\mathrm{\equiv }{\mathit{A}}^{\mathrm{1}\mathrm{/}(1-\alpha )}{\mathit{\alpha }}^{2/\mathrm{(}1-\alpha \mathrm{)}}\mathit{L}\mathit{Q}\mathrm{.}$(10.5.4)

We now design a mechanism to determine κ_j, the key element for economic growth. The κ_j th innovator in sector j raises quality from $q^{{\mathit{\kappa }}_j-1}$ to $q^{{\mathit{\kappa }}_j}$. The flow of profit, ${\pi }_{j{k}_j}$, associated with quality rung κ_j equals (P-1)$X_{j{\kappa }_j}$. By equation (10.5.3), ${\pi }_{j{\kappa }_j}$ are given by:

${{\mathit{\pi }}_{}}_{j{\kappa }_j}=\mathit{L}{\mathit{A}}^{1/\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}\left(\frac{\mathrm{1}\mathrm{-}\alpha }{\mathit{\alpha }}\right){\alpha }^{2/\mathrm{(}\alpha -1\mathrm{)}}{\mathit{q}}^{{\kappa }_j\mathit{\alpha }/\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}\mathrm{.}$(10.5.5)

This profit is continued - over the interval $T_{j{\kappa }_j}$ = $t_{k_j+1}$ + – $t_{k_j}$ - from the time of the κj th quality improvement, $t_{{\kappa }_j}$, until the time of the next improvement by a competitor, $t_{{\kappa }_{j+1}}$. If the interest rate, r, is constant, then the present value evaluated at $t_{{\kappa }_j}$ of the profit from the κ_j th quality improvement is:

${{\mathit{V}}_{j{\kappa }_j}}_{}=\frac{{\mathit{\pi }}_{j{\kappa }_j}\lfloor 1-\exp (-r{T}_{j{\kappa }_j})\rfloor }{\mathit{r}}\mathrm{.}$

Since ${\pi }_{j{\mathit{\kappa }}_j}$ are known, we now determine $T_{j{\kappa }_j}$ to finally determine $V_{j{\kappa }_j}$.

Let $Z_{j{\kappa }_j}$ stand for the flow of resources in terms of Y expended by the aggregate of potential innovators in sector j when the highest quality-ladder number reached in that sector is κ_j. Assume that the probability, $p_{j{\kappa }_j}$, per unit of time of a successful 379 is related to $Z_{j{\kappa }_j}$, and κ_j as follows:

${\mathit{p}}_{j{\kappa }_j}={\mathit{Z}}_{j{\kappa }_j}\mathit{\varphi }\mathrm{(}{\mathit{\kappa }}_j\mathrm{)},$

where ϕ′(κ_j,)< 0 implies that as project becomes more complicated, the probability of success in research declines. Define G(τ) the cumulative probability density function for $T_{j{\kappa }_j}$, that is, the probability that $T_{j{\kappa }_j}$ ≤ τ. The change in G(τ) with respect to τ represents the probability per unit of time that the 379 occurs at τ. An 379 at τ implies that it had not occurred earlier, an outcome that has probability 1 - G(τ). According to the definitions: dG/dτ = [1 - G(τ)]$p_{j{\kappa }_j}$.

Assume that $p_{j{\kappa }_j}$ and $Z_{j{\kappa }_j}$ do not vary over time between innovators in a sector. As G(0)=0, we solve G(τ)= 1 - exp(- $p_{j{\kappa }_j}$ τ). The propensity density for $T_{j{\kappa }_j}$ is then given by:

$\mathit{g}\mathrm{(}\mathit{\tau }\mathrm{)}=G\prime \mathrm{(}\mathit{\tau }\mathrm{)}={\mathit{p}}_{j{\kappa }_j}\exp (-p_{j{\kappa }_j}\tau )\mathrm{.}$

By this equation, we can compute the expected present value evaluated at $t_{{\kappa }_j}$ of the profit from the κ_j th quality improvement:

$$\begin{gathered} \mathit{E}\mathrm{(}{\mathit{V}}_{j{\kappa }_j}\mathrm{)}=\frac{{\mathit{p}}_{j{\kappa }_j}{\mathit{\pi }}_{j{\kappa }_j}}{\mathit{r}}\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{{\displaystyle \int }}}\mathrm{(}1\mathrm{-}\exp \mathrm{(}\mathrm{-}\mathit{r}\mathit{\tau }\mathrm{)}\mathrm{)}\exp \mathrm{(}\mathrm{-}{\mathit{p}}_{j{\kappa }_j}\mathit{\tau }{\mathrm{)}}_{}d\tau = \\ \frac{\mathit{L}{\mathit{A}}^{1/(\alpha -1)}(\mathrm{l}\mathrm{/}\mathit{\alpha }\mathrm{-}\mathrm{1}){\alpha }^{2/(\alpha -1)}{\mathit{q}}^{{\mathit{\kappa }}_j\alpha /\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}}{\mathit{r}\mathrm{+}{\mathit{p}}_{j{\kappa }_j}}\mathrm{,} \end{gathered}$$(10.5.6)

where we use equation (10.5.5).

It is assumed that potential innovators care only about the expected value in equation (10.5.6). The expected reward per unit of time for pursuing the (κ_j + 1)th 379 is $p_{j{\kappa }_j}$ E($V_{j{\kappa }_j}$). Hence, the expected flow of net profit, ${\mathrm{\Pi }}_{j{\kappa }_j}$, from research currently at κ_j equals ${\mathit{p}}_{j{k}_j}$ $\mathit{E}\mathrm{(}{\mathit{V}}_{{j\mathit{\kappa }}_j\mathrm{+}\mathrm{1}}\mathrm{)}$–$s$. By equations (10.5.6) and:

$p_{j{\mathit{\kappa }}_j}={\mathit{Z}}_{j{\kappa }_j}\varphi \mathrm{(}{\mathit{\kappa }}_j\mathrm{)},$

we have:

$$\begin{gathered} {\mathrm{\Pi }}_{j{\mathit{\kappa }}_j}={\mathit{p}}_{j{\mathit{\kappa }}_j}\mathit{E}\mathrm{(}{\mathit{V}}_{j{\mathit{\kappa }}_j+\mathrm{1}}\mathrm{)}\mathrm{-}{\mathit{Z}}_{j{\kappa }_j}= \\ {\mathit{Z}}_{j{\mathit{\kappa }}_j}\mathrm{[}\frac{\varphi \left({\kappa }_j\right)L{A}^{1/(\alpha -1)}(\mathrm{1}\mathrm{/}\mathit{\alpha }\mathrm{-}\mathrm{1}){\alpha }^{\mathrm{2}/\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}{\mathit{q}}^{\left({\mathit{\kappa }}_j\mathrm{+}\mathrm{1}\right)\alpha /\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}}{\mathit{r}\mathrm{+}{\mathit{p}}_{j{\kappa }_j+1}}-\mathrm{1}\mathrm{]}\mathrm{.} \end{gathered}$$(10.5.7)

Free entry into the research business guarantees ${\mathrm{\Pi }}_{j{\kappa }_{j+1}}$ = 0. For $Z_{j{\kappa }_j}$ > 0, ${\mathrm{\Pi }}_{j{\kappa }_j}$. = 0 becomes:

$\mathit{r}+{\mathit{p}}_{j{\kappa }_j+1}=\varphi \left({\mathit{K}}_j\right)L{A}^{1/(\alpha -1)}(\frac{\mathrm{1}}{\mathit{\alpha }}\mathrm{-}1){\alpha }^{\mathrm{2}\mathrm{/}(\alpha -1)}{\mathit{q}}^{{(\kappa }_j+1)\alpha /(\alpha -1)}\mathrm{.}$(10.5.8)

There are two forces ϕ(κ_j) (meaning that 379s are increasingly difficult, with ϕ′ < 0) and q^(κ_j ^{+ 1)α/(α – 1)} (reflecting that the expected reward from an 379 is increasing in κ_j) in the equation. If the first force dominates, then more advanced sectors tend to grow relatively slower; and vice versa. If the two forces exactly offset, then all sectors will tend to grow at the same rate. To simplify the analysis, we specify ϕ(κ_j) as:

$\varphi \mathrm{(}{{\mathit{\kappa }}_{}}_j\mathrm{)}=\frac{q^{-({\kappa }_j+1)\alpha /(\alpha -1)}}{\mathit{\zeta }}\mathrm{,}\mathit{\zeta }\mathrm{>}\mathrm{0}\mathrm{.}$(10.5.9)

This formulas simplifies the free entry condition (10.5.8) as follows:

$\mathit{r}+\mathit{p}=\frac{\mathit{L}{\mathit{A}}^{1/\mathrm{(}\mathit{\alpha }-1\mathrm{)}}}{\mathit{\zeta }}(\frac{\mathrm{1}}{\mathit{\alpha }}\mathrm{-}\mathrm{1}){\alpha }^{2/\mathrm{(}\alpha -1\mathrm{)}},$

where p = $p_{j{\kappa }_{j+1}}$denotes that $p_{j{\kappa }_{j+1}}$ is invariant in κ_j. This equation gives the probability of an 379 per unit of time as:

$\mathit{p}=\frac{\mathit{L}{\mathit{A}}^{1/\mathrm{(}\mathit{\alpha }-1\mathrm{)}}}{\mathit{\zeta }}(\frac{\mathrm{1}}{\mathit{\alpha }}\mathrm{-}\mathrm{1}){\alpha }^{\mathrm{2}/\mathrm{(}\alpha -\mathrm{1}\mathrm{)}}-r\mathrm{.}$(10.5.10)

If r is constant, then p is also constant. If we substitute the above equation for r + p into equation (10.5.6), we find the market value of the κ_j, th 379 as:

$\mathit{E}\mathrm{(}{\mathit{V}}_{j{\kappa }_j}\mathrm{)}=\mathit{\zeta }{\mathit{q}}^{{\kappa }_j\mathit{\alpha }/(\alpha -1)}\mathrm{.}$

The aggregate market value of firms, denoted by V, is the sum of the above equations over N:

$\mathit{V}={\mathrm{\Sigma }}_{j=1}^N\mathit{\zeta }{\mathit{q}}^{{\kappa }_j\alpha /\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}=\mathit{\zeta }\mathit{Q}\mathrm{.}$

The amount of resources devoted to R&D in sector j is:

$Z_{j{\kappa }_j}=\frac{\mathit{p}}{\varphi \mathrm{(}{\mathit{\kappa }}_j\mathrm{)}}={\mathit{q}}^{\left({\mathit{\kappa }}_j\mathrm{+}1\right)\mathit{\alpha }/(\mathit{\alpha }-1)}\left[L{A}^{1/(\alpha -1)}\right(\frac{1}{\alpha }-1){\alpha }^{2/(\alpha -1)}-r\zeta ],$

where we use equations (10.5.9) and (10.5.10). The aggregate of R&D spending, denoted by Z, is:

$Z\mathrm{\equiv }\underset{j=1}{\overset{N}{\mathrm{\Sigma }}}{Z}_{j{\kappa }_j}=Q{q}^{\alpha /(\alpha -1)}\left[L{A}^{1/(\alpha -1)}\right(\frac{1}{\alpha }-1){\alpha }^{2/(\alpha -1)}-r\zeta ]\mathrm{.}$(10.5.11)

As Y, X, V, and Z are all constant multiples of Q, they have the same growth rate:

${\mathit{g}}_Y={\mathit{g}}_X={\mathit{g}}_V\mathrm{=}{\mathit{g}}_Z\mathrm{=}{\mathit{g}}_{\mathit{Q}}\mathrm{.}$(10.5.12)

In $\mathit{Q}={{\displaystyle \sum }}_{\mathit{j}}{\mathit{q}}^{\alpha {\mathit{\kappa }}_j/(1-\alpha )}$, term $q^{\alpha {\mathit{\kappa }}_j\mathrm{/}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\alpha }\mathrm{)}}$ does not change if no 379 occurs in sector j; but changes to q^α(κ_j ^{+ 1)/(1-α)} if an 379 occurs. The proportionate change in this term due to an 379 is q^{α/(1 - α).} Since p is the same for all the sectors, the expected proportionate change in Q per unit of time is:

$\mathit{E}\left(\frac{\mathrm{\Delta }\mathit{Q}}{\mathit{Q}}\right)=\mathit{p}({\mathit{q}}^{\mathit{\alpha }\mathrm{(}\mathrm{1}-\mathit{\alpha }\mathrm{)}}\mathrm{-}\mathrm{1})\mathrm{.}$

We assume that N is large enough to treat Q as differentiable, and thus we have g_Q ≈ E(ΔQ/Q) With this equation and equation (10.5.10), we have:

${\mathit{g}}_Q=\mathrm{\left[}\frac{\mathit{L}{\mathit{A}}^{\mathrm{1}/\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}}{\mathit{\zeta }}\right(\frac{\mathrm{1}}{\mathit{\alpha }}\mathrm{-}1){\alpha }^{2/(\alpha -1)}-r](q^{\alpha /(1-\alpha )}-1)\mathrm{.}$(10.5.13)

Hence, to determine the growth rates, we have to find conditions for determining the rate of interest. We now turn to behavior of households to close the model.

Each household maximizes:

$\mathit{U}=\stackrel{\infty }{\underset{0}{\underset{}{{\displaystyle \int }}}}\left(\frac{{{\mathit{c}}^{1-\theta }}^{}\mathrm{-}\mathrm{1}}{\mathrm{1}\mathrm{-}\mathit{\theta }}\right)e^{-\rho t}\mathit{d}\mathit{t}\mathrm{,}$

where c is consumption per person and population growth rate is zero. The key condition for this study from household optimization is g_c - (r - ρ)/θ, where C is the aggregate consumption.

The economy's overall resource constraint is C = Y - X - Z. Substituting equations (10.5.4) and (10.5.11) and Y = α^-2X into the above equation yields:

$C=\left[A^{1/(1-\alpha )}\right(\mathrm{1}\mathrm{-}{\mathit{\alpha }}^{\mathrm{2}}){\alpha }^{2\alpha /(1-\alpha )}L-p\zeta q^{\alpha /(1-\alpha )}]Q\mathrm{.}$

If p is constant, g ≡ g_c = g_Q. From g_c = (r - ρ)/θ and equation (10.5.113), we solve r and the growth rate g as:

$$\begin{gathered} \mathit{r}=\frac{\mathit{\rho }+\mathit{\theta }({\mathit{q}}^{\mathit{\alpha }/\mathrm{(}1-\alpha \mathrm{)}}\mathrm{-}\mathrm{1})\mathrm{[}\mathrm{(}\mathit{L}\mathrm{/}\mathit{\zeta }\mathrm{)}{\mathit{A}}^{\mathrm{1}/\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}(\mathrm{1}\mathrm{/}\mathit{\alpha }\mathrm{-}\mathrm{1}\left){\alpha }^{\mathrm{2}/\mathrm{(}\mathit{\alpha }-\mathrm{1}\mathrm{)}}\mathrm{\right]}}{1+\theta (q^{\alpha /(1-\alpha )}-1)}, \\ \mathit{g}=\frac{(q^{\alpha /(1-\alpha )}-1)\left[\right(L/\zeta )A^{1/(\alpha -1)}(1/\alpha -1){\alpha }^{2/(\alpha -1)}-\rho }{1+\theta (q^{\alpha /(1-\alpha )}-1)}\mathrm{.} \end{gathered}$$(10.5.14)

We assume that the parameters are such that g > 0 so that the free-entry condition holds with equality and r > g so that the transversality condition is satisfied. We have thus determined the rate of interest and the growth rate.

We have thus closed the model. The single state variable is now Q. Given an initial value Q(0), by equation (10.5.14) and:

$g=g_Y=g_X=g_V=g_Z=g_Q=g_C,$

we determine the growth rates as well as the variables at any point of time. Further analysis of the model is referred to Barro and Sala-i-Martin.¹⁴

10.6 Learning by Doing and Research

Zhang constructed an economic growth with learning by doing, education, and learning by leisure within the Solow framework.¹⁵ Zhang's model attempts not only to integrate traditional models of learning by doing and education within a single framework but also to introduce learning by consumption (with poor countries in the mind when constructing the model) into growth theory. As far as I know, no other formal growth model takes account of learning by leisure. Zhang's model emphasizes the interdependence between education and research activity, learning through production, and learning through consumption and leisure activity. In 1993, Young constructed a model with invention and bounded learning by doing within a framework of monopolistic competition. We now introduce the Young model. It should be noted that because of its technical difficulties, we only illustrate the main assumptions and conclusions without explanation in detail.

The Young model is a hybrid of models of invention - in which technical change is the outcome of costly and deliberate efforts aimed at the development of technologies - and models of learning by doing - in which technical change is the serendipitous byproduct of experience gained in the production of goods. Models of invention are generally concerned with factors that influence the incentive to consciously innovate, such as institutional structures, market size, and opportunity of extra profits; models of learning are generally concerned with factors that influence the incentives to produce different types of goods, such as the pattern of comparative advantage. The Young model integrates these two types of models, demonstrating that sustained technical change involves an interaction between deliberative invention and serendipitous learning. In this model, if market size is small, rate of time preferences is large, or invention is relatively costs, the profitability of inventive activity is too low for invention to be made. If market size is large, or invention is relatively inexpensive, profitability of inventive activity is high and invention takes place.

It is assumed, like the previous models in this chapter, that the invention of new products requires costly sacrifice of current consumption as resources are reallocated from the production of goods to research. To explain the incentives for costly research, we consider that a successful innovator will receive an infinitely lived patent to the product that he invents. By learning by doing we have that production experience generates new knowledge on how to produce goods more efficiently. The knowledge generated by learning by doing is not appropriated by any economic actor; the productivity gains generated through learning by doing spill over all sectors. It is assumed that the potential for learning-induced productivity improvement in each good is finite and bounded. This means that the measure of the society's cumulative learning experience will be bounded from above by the total number of goods society has invented. Young considers the case that newly invented goods are initially inferior to mature technologies that have attained their maximum productivity, even though they have the potential to give more utility per unit of factor input than the older goods.

We now describe the Young model. Like in the Romer model, assume that all the goods that have been or will be invented can be arranged along the real line in order of increasing technical sophistication. Let [0, A(t)] denote a subset of this real line that a society can produce at time t. Here, A(t) stand for the most sophisticated good that the society is currently able to produce. Labor is the sole factor of production, and the function a(τ, t) is the amount of labor necessary to produce one unit of good i at time t. This function is defined only on the domain [0, A(t)]. It is assumed:

$$\begin{gathered} a\mathrm{(}\mathit{\tau }\mathrm{,}\mathrm{t}\mathrm{)}=\overline{\mathit{a}}{\mathit{e}}^{-\tau }\mathrm{\forall }\mathit{\tau }\mathrm{\in }\mathrm{[}\mathrm{0},\mathit{T}\mathrm{(}t\mathrm{)}\mathrm{]}\mathrm{,} \\ \mathit{a}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}=\overline{\mathit{a}}{\mathit{e}}^{-T\mathrm{(}t\mathrm{)}}{\mathit{e}}^{\tau -T\left(t\right)}\mathrm{\forall }\mathit{\tau }\mathrm{\in }\mathrm{[}\mathit{T}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\mathit{A}\mathrm{(}t\mathrm{)}\mathrm{]}, \end{gathered}$$

where T(t) evolves according to the learning-by-doing equation:

$\dot{\mathit{T}}\mathrm{(}\mathit{t}\mathrm{)}={\displaystyle \underset{T\left(t\right)}{\overset{A\left(t\right)}{\int }}}\mathrm{\Psi }\mathit{N}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}\mathit{d}\mathit{\tau }\mathrm{,}$(10.6.1)

in which ψ is the rate of learning by doing and N(τ, t) is the number of workers producing good τ at time t. The formation that the unit labor requirement curve a(τ,t) is upward sloping beyond T(t) means that new technologies may initially be inferior to mature technologies. The above specification is interpreted as that the economy experiences bounded learning by doing with symmetric spillovers across goods. Leaning is bounded as the amount of labor required to produce good x cannot fall below āe^-τ. The formation for Ṫ(t) essentially implies that there are spillovers in learning across goods, with technical improvements that originate in any particular industry τ having applications in other industries. Once an industry has reached its lower bound, āe^-τ, it has nothing to learn. If T(t) reaches A(t), then Ṫ(t) is equal to zero.

Firms are free to enter the process of invention. They finance their research and development efforts by selling shares in capital market. A firm will distribute any profits to its shareholders. After a good is invented, the firm that owns the patent will engage in monopolistic competition with all other patent holders. New goods are invented through the creative efforts of entrepreneurs, which acquire an infinitely lived patent on each good they invent. The rate of invention is simply given by:

$\dot{\mathit{A}}\mathrm{(}\mathit{t}\mathrm{)}=\frac{{\mathit{N}}_R\mathrm{(}t\mathrm{)}}{{{\mathit{a}}_{}}_R}\mathrm{,}$(10.6.2)

where N_A(t) is the amount of labor devoted to research and 1 / a_R is the measure of creativity of workers engaged in research.

Let labor be the numeraire - the flow of each consumer's labor income is equal to one. All prices and values are denominated in units of labor. There are N consumers, each of whom inelastically supplies one unit of labor at all times. For each consumer, consumer's instantaneous consumer expenditure, E(t), is:

$\mathit{E}\mathrm{(}t\mathrm{)}=\underset{\mathrm{0}}{\overset{A\left(t\right)}{{\displaystyle \int }}}\mathit{p}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}\mathit{C}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}\mathit{d}\mathit{\tau },$(10.6.3)

where p(τ, t) and C(τ, t) are goods prices and individual consumption along τ ∈ [0, A(t)].

Consumers maximize the present value of the logarithm of a time separable function form:

$\mathit{M}ax{\mathit{U}}^{*}\mathrm{=}\underset{}{\overset{\mathrm{\infty }}{{\displaystyle \underset{t}{\int }}}}{\mathit{e}}^{-\mathit{\rho }(v-t)}\log \mathrm{\left[}\mathit{U}\mathrm{\right\{}\mathit{C}\mathrm{(};\mathit{v}\mathrm{)}\mathrm{\left\}}\mathrm{\right]}\mathit{d}v,$

subject to the intertemporal budget constraint:

$\stackrel{\infty }{{\displaystyle \underset{t}{\int }}}{\mathit{e}}^{-\mathit{R}\left(v\right)+R\left(t\right)}\mathit{E}\mathrm{(}\mathit{v}\mathrm{)}\mathit{d}\mathit{v}=\mathit{S}\mathrm{(}\mathit{t}\mathrm{)}+\underset{t}{\overset{\infty }{{\displaystyle \int }}}{\mathit{e}}^{-\mathit{R}\left(v\right)+R\left(t\right)}\mathit{w}\mathrm{(}\mathit{v}\mathrm{)}\mathit{d}\mathit{v}\mathrm{,}$(10.6.4)

where R(t) is the cumulative interest factor up to time t, and w(t) and S(t) are the nominal wage and individual assets at time t. The utility function U{C(.,v)} is given by:

$\mathit{U}\mathrm{\left\{}C(\cdot ,v)\mathrm{\right\}}=\underset{0}{\overset{A\left(t\right)}{\int }}{C}^{*}\left(t\right)g\left(\frac{\mathit{C}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}}{{\mathit{C}}^{*}\mathrm{(}\mathit{t}\mathrm{)}}\right)\mathit{d}\mathit{\tau }\mathrm{,}$

where g(·) is strictly concave and continuously differentiable with g(0) = 0 and g′(0) < ∞, and C*(t) ≡ ${{\displaystyle \int }}_0^{\mathit{A}\mathrm{(}t)}$ C(τ,t)dτ.

For simplicity of analysis, we specify g(·) as follows: g(x) = x - x²/2. The concavity of g(·) and g′(0) < ∞ imply a strong but not unbounded preference for variety. Changes in p(τ, t) will lead to changes in the set of goods consumed.

Given the time separability of the consumer's utility function, we can break down the consumer's optimal consumption and expenditure program into a two-stage analysis. In stage 1, we maximize instantaneous utility subject to instantaneous expenditure. In stage 2, with U{C(.,v)} defined as a function of E(t) and p(.,t), we maximize total intertemporal utility subject to the intertemporal budget constraint. In what follows we often suppress the notion denoting the implicit dependence of the variables on the time.

Introduce f(τ) ≡ C*(τ,t)/C*(τ,t). With respect to the maximization of the instantaneous utility function, the consumer's problem is to allocate this consumption density across goods and then adjust the consumption scaling factor C* on the basis of the desired level of instantaneous expenditure. That is:

$$\begin{gathered} \mathrm{Max}\mathit{U}\mathrm{\{}\mathit{C}\mathrm{(}\mathrm{\cdot }\mathrm{)}\mathrm{\}}\mathrm{=}\underset{0}{\overset{A}{{\displaystyle \int }}}{\mathit{C}}^{*}\mathit{g}\mathrm{(}f\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{)}\mathit{d}\mathit{\tau }\text{subject to} \\ \mathrm{1}=\underset{0}{\overset{A}{{\displaystyle \int }}}\mathit{f}\mathrm{(}\mathit{s}\mathrm{)}\mathit{d}\mathit{s}\mathrm{,}\mathit{E}=\underset{0}{\overset{A}{{\displaystyle \int }}}p\mathrm{(}\mathit{s}\mathrm{)}\mathit{f}\mathrm{(}\mathit{s}\mathrm{)}{\mathit{C}}^{*}\mathit{d}\mathit{s}\mathrm{.} \end{gathered}$$(10.6.5)

The problem has a solution. The optimal f(τ) is give by:

$\mathit{f}\mathrm{(}\mathit{\tau }\mathrm{)}=\mathit{\lambda }\mathrm{[}\mathit{p}\mathrm{(}\mathit{Z}\mathrm{)}\mathrm{-}\mathit{p}\mathrm{(}\tau \mathrm{)}\mathrm{]},$(10.6.6)

where λ is the marginal utility derived from an additional unit of expenditure E at the consumer optimum and Z denotes some limit good such that the consumer consumes all goods that are cheaper than Z and consumes no goods that are more expensive than Z. Good Z is determined by 1 = ${{\displaystyle \int }}_{\mathrm{0}}^{\mathit{A}}\mathit{f}\mathrm{(}\mathit{s}\mathrm{)}$ ds with the above formation of f(τ). The budget constraint in equations (10.6.5) then determines C*.

Each firm maximizes its current profits:

$\underset{\mathit{p}\mathrm{(}\tau )}{\max }\mathit{\pi }\mathrm{(}\mathit{\tau }\mathrm{)}=\mathit{N}\mathit{C}\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{[}\mathit{p}\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{-}\mathit{a}\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{]}=\mathit{N}{\mathit{C}}^{*}\mathit{f}\mathrm{(}\mathit{\tau }\mathrm{)}[p\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{-}\mathit{a}\mathrm{(}\mathit{\tau }\mathrm{)}\mathrm{]}\mathrm{.}$

Using f(τ)= λ[p(Z) – p(τ)], we find the equilibrium price of each good:

$\mathit{p}\mathrm{(}\mathit{\tau }\mathrm{)}=\frac{\mathit{p}\mathrm{(}\mathit{Z}\mathrm{)}\mathrm{+}\mathit{a}\mathrm{(}\mathit{\tau }\mathrm{)}}{\mathrm{2}}=\frac{a\left(Z\right)+a\left(\tau \right)}{2},$(10.6.7)

where we also use a(Z) = p(Z). Given the pattern of consumer expenditure determined by equations (10.6.6) and (10.6.7), we can calculate the consumer price of a unit of instantaneous utility P_U and the actual labor cost of producing this unit of utility MC_U:

$$\begin{gathered} {\mathit{P}}_{\mathit{U}}=\overline{\mathit{a}}{\mathit{e}}^{-T}\mathrm{\left[}\frac{(\mathit{\tau }\mathrm{+}\mathit{\eta }\mathrm{-}\mathrm{1})e^{\tau }\mathrm{+}\mathrm{2}\mathrm{-}{\mathit{e}}^{\mathit{\eta }}}{\mathrm{2}}\mathrm{\right]}\mathrm{,} \\ \mathit{M}{\mathit{C}}_U=\overline{a}{{\mathit{e}}^{-T}}^{}\mathrm{\left[}\frac{\{2+{\mathit{e}}^{\mathrm{2}\tau }\mathrm{-}\mathrm{4}{\mathit{e}}^{\tau }\mathrm{+}\mathrm{2}{\mathit{e}}^{\tau }{\mathit{e}}^{\mathit{\eta }}\mathrm{-}{\mathit{e}}^{\mathrm{2}\mathit{\eta }}\}\left\{\right(\mathit{\tau }\mathrm{+}\mathit{\eta }\mathrm{-}1)e^{\tau }\mathrm{+}\mathrm{2}\mathrm{-}{\mathit{e}}^{\mathit{\eta }}\mathrm{\}}}{\mathrm{2}\mathrm{+}\mathrm{(}\mathit{\tau }\mathrm{+}\mathit{\eta }\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{2}{\mathit{e}}^{\mathrm{2}\mathit{\tau }}\mathrm{-}{\mathit{e}}^{\mathrm{2}\mathit{\eta }}}\mathrm{\right]}, \end{gathered}$$(10.6.8)

where η ≡ A - T. From the above equations, we directly calculate the markup for this monopolistically competitive setting as P_U / MC_U = Φ (A - T), where Φ is a function decreasing in A - T. By equations (10.6.8) we can explicitly express Φ (A - T). The distribution of the labor N_M between learning and non-learning industries, denoted respectively by N_M1 and N_M2, is determined by η (= A - T):

${\mathit{N}}_{\mathit{M}\mathrm{1}}={\mathit{N}}_{\mathit{M}}{\mathrm{\Phi }}^{*}\mathrm{(}\mathrm{A}\mathrm{-}\mathit{T}\mathrm{)}\mathrm{,}{\mathit{N}}_{\mathit{M}\mathrm{2}}={\mathit{N}}_{\mathit{M}}\mathrm{[}1\mathrm{-}{\mathrm{\Phi }}^{*}\mathrm{(}\mathrm{A}\mathrm{-}\mathit{T}\mathrm{)}\mathrm{]}\mathrm{,}$

where:

${\mathrm{\Phi }}^{*}\mathrm{(}\mathit{\eta }\mathrm{)}\mathrm{\equiv }\frac{\mathrm{2}{\mathit{e}}^{\mathit{\eta }+\tau }\mathrm{-}\mathrm{2}{\mathit{e}}^{\tau }\mathrm{-}\mathrm{2}{\mathit{e}}^{\mathit{\eta }}+\mathrm{1}}{\mathrm{2}+{\mathit{e}}^{2\tau }\mathrm{-}\mathrm{4}{\mathit{e}}^{\tau }\mathrm{+}\mathrm{2}{\mathit{e}}^{\mathit{\eta }+\tau }\mathrm{-}\mathrm{2}{\mathit{e}}^{2\mathit{\eta }}}\mathrm{\le }\frac{\mathrm{1}}{\mathrm{2}}\mathrm{,}{\mathit{h}}^{\prime }\mathrm{(}\mathit{\eta }\mathrm{)}\mathrm{>}\mathrm{0}\mathrm{.}$

Finally, we discuss market equilibrium. Let N_M and N_R stand for the demand for labor in manufacturing and research. Labor market equilibrium is guaranteed by: N_M + N_R = N.

The amount of labor in final goods production is equal to aggregate consumer expenditure divided by the price of a unit of utility times amount of labor required to produce a unit of utility, i.e.:

${{\mathit{N}}_{}}_M=\frac{\mathit{E}\mathit{N}}{{\mathit{P}}_U}\mathit{M}{\mathit{C}}_U=\frac{\mathit{E}\mathit{N}}{\mathrm{\Phi }\mathrm{(}\mathit{A}\mathrm{-}\mathit{T})}$(10.6.8)

Equation (10.6.8), together with N_M + N_R = N, determines the labor distribution N_M and N_R.

We have determined the equilibrium price and labor distribution at any point of time. We now consider the consumer's optimization problem in stage 2. The problem is to make an expenditure plan, E(t), that maximizes:

${\mathit{U}}^{*}\mathrm{=}{\displaystyle \underset{t}{\overset{\infty }{\int }}}{\mathit{e}}^{-\mathit{\rho }\mathrm{(}v-t\mathrm{)}}\mathrm{\left\{}\log \mathrm{\right[}\mathit{E}\mathrm{(}\mathit{v}\mathrm{)}\mathrm{]}\mathrm{-}\log \mathrm{[}{\mathit{P}}_U\mathrm{(}\mathit{v}\mathrm{)}\left]\right\}dv,$

subject to the intertemporal budget constraint (10.6.4). This leads to the following familiar maximal condition:

$\frac{\dot{\mathit{E}}\mathrm{(}t\mathrm{)}}{\mathit{E}\mathrm{(}t\mathrm{)}}=\dot{\mathit{R}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{\rho }\mathrm{.}$(10.6.9)

The above equation determines behavior of the consumer. We now describe behavior of firms. Let V(τ,t) stand for the asset market value of holding the patent to good τ at time t, i.e.:

$V\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}=\underset{t}{\overset{\infty }{{\displaystyle \int }}}{\mathit{e}}^{-R\left(v\right)+\mathit{R}\left(t\right)}\mathit{\pi }\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{v}\mathrm{)}\mathit{d}\mathit{v},$

where π(τ, t) denotes the profits of firm τ at time t. Differentiation of the above equation with respect to t yields:

$\dot{\mathit{R}}\mathrm{(}t\mathrm{)}=\frac{\dot{\mathit{V}}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}}{\mathit{V}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}}\mathrm{+}\frac{\mathit{\pi }\mathrm{(}\mathit{\tau }\mathrm{,}t\mathrm{)}}{\mathit{V}\mathrm{(}\mathit{\tau }\mathrm{,}\mathit{t}\mathrm{)}}\mathrm{,}\mathrm{\forall }\mathit{\tau }\mathrm{.}$(10.6.10)

This equation states that in asset market equilibrium the return to holding the patent to any good τ (profits plus patent value appreciation) must equal the risk-free rate of return. Free entry into the inventive process implies that the present discounted value of the profits of firm A(t) will be less than or equal to the cost of invention:

$a_{\mathit{R}}\mathrm{\ge }\mathit{V}\mathrm{(}\mathrm{A}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}t\mathrm{)}={\displaystyle \underset{t}{\overset{\infty }{\int }}}{\mathit{e}}^{-\mathit{R}\mathrm{(}\mathit{v}\mathrm{)}\mathrm{+}\mathit{R}\left(t\right)}\mathit{\pi }\mathrm{(}A\mathrm{(}\mathit{t}\mathrm{)},\mathit{v}\mathrm{)}\mathit{d}\mathit{v}\mathrm{(}=\mathit{i}\mathit{f}\dot{\mathit{A}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{>}\mathrm{0}\mathrm{)}\mathrm{.}$(10.6.11)

Assuming that Ȧ(t) > 0, we take derivatives of equation (10.6.11) with respect to time yields:

$\dot{\mathit{R}}\mathrm{(}t\mathrm{)}=\frac{\mathit{\pi }\mathrm{(}\mathit{A}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\mathit{t}\mathrm{)}\mathrm{-}\dot{\mathit{A}}\mathrm{(}\mathit{t}\mathrm{)}{\mathit{V}}_1\mathrm{(}\mathit{A}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}t\mathrm{)}}{{{\mathit{a}}_{}}_R},$(10.6.12)

where:

$V_1\mathrm{(}\mathit{A}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\mathit{t}\mathrm{)}\mathrm{\equiv }\underset{t}{\overset{\infty }{\int }}{\mathit{e}}^{-\mathit{R}\mathrm{(}\mathit{\nu }\mathrm{)}\mathrm{+}\mathit{R}\left(t\right)}\frac{\mathrm{\partial }\pi }{\mathrm{\partial }\mathit{A}}\mathit{d}\mathit{v}\mathrm{.}$

At any time, the value of each firm depends on its position along the real line. Firms τ ≤ Z(t) have a value of zero since, as demand has moved to the right, they will never again earn any profits. Free entry into the inventive process yields V(A(t),t) = a_R.

We have thus closed the model. Consumers' behavior is given by equations (10.6.6) and (10.6.9); firms' behavior is determined by equations (10.6.7) and (10.6.12). With the labor market conditions and equations (10.6.1), (10.6.2), and (10.6.4), we can determine all the variables in the system. We are concerned with whether the system has a meaningful a steady state.

In a steady state, E and A - T are constant. We thus also have Ṙ = ρ and Ȧ = Ṫ. With a constant-steady state of expenditure and an exponentially declining price P_U of a unit of U{C(·)} the proportional rate, denoted by g, of growth of U{C(·)} is equal to the equilibrium rate of learning and invention.

First, we are concerned with an extreme case of the model - a stagnant steady state with zero growth. We have A = T and Ȧ = Ṫ =0, with all the firms in (Z, T] earning an infinitely lived stream of constant profits. This is equilibrium if and only if the present discounted value of the flow of profits to firm T is less than or equal to the cost of invention; otherwise entrepreneurs would find it profitable to invent products to the right of T. For stagnation to occur, it is necessary and sufficient to have a_R ≥ N/ρ. We see that if the aggregate market (measured by N) is small enough, the cost of invention a_R is high, or the steady-state rate of interest ρ large enough, the economy will stagnate because firms would not find it profitable to invent.

If a_R < N / ρ, the steady-state growth rate is positive, with A - T > 0. In this case, firms find it profitable to invent products. If A - T > η* (see Young's paper for the definition of η*), Ṫ = g = ψN_M / 2. With A = g = N_R / a_R and N_M + N_R = N, we solve:

$\mathit{g}=\frac{\mathit{\psi }\mathit{N}}{\mathrm{2}+\mathit{\psi }{a}_{\mathit{R}}}\mathrm{.}$(10.6.13)

We conclude that the growth rate is independent of the incentives for 379 along equilibrium path, though the rate is influenced by the cost of invention and the learning efficiency.

The third type of steady state is when:

${\mathit{\eta }}^{*}\mathrm{>}\mathrm{A}\mathrm{-}\mathit{T}\mathrm{>}\mathrm{0}\mathrm{.}$

In equilibrium, N_R = ga_R. Labor market equilibrium condition becomes:

$N_M+g{a}_R=N\mathrm{.}$(10.6.14)

In the steady state, the rate of learning must equal the rate of invention. This relation can be expressed as:

$\mathit{\psi }{\mathrm{\Phi }}^{*}\mathrm{(}\mathit{A}\mathrm{-}\mathit{T}\mathrm{)}{\mathit{N}}_M=\mathit{g}\mathrm{.}$(10.6.15)

Moreover, free entry into invention defines a factor market equilibrium relation, which says that the return from an additional unit of labor to invention, V((A)(t),t)/ a_R, must be les than or equal to the real return to labor in manufacturing, i.e.:

$\mathrm{1}\mathrm{\ge }\frac{V\mathrm{(}\mathit{A}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}t\mathrm{)}}{a_{\mathit{R}}}\mathrm{.}$(10.6.16)

It can be shown that V(A(t), t) is dependent on the steady state size of the final goods market (N_M), lifetime profits per unit of market size that dependent on A - T, ρ and g:

$\mathit{V}\mathrm{(}\mathit{A}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\mathit{t}\mathrm{)}={\mathit{N}}_{\mathit{M}}\mathit{T}\mathrm{(}\mathit{A}\mathrm{-}\mathit{T}\mathrm{,}\mathit{\rho }\mathrm{,}\mathit{g}\mathrm{)}\mathrm{.}$(10.6.17)

Equations (10.6.14)-( 10.6.17) determine an equilibrium growth rate, g. It can be checked that the growth rate is affected by the parameters as follows:

$\frac{\mathrm{\partial }\mathit{g}}{\mathrm{\partial }\mathit{N}}\mathrm{>}\mathrm{0}\mathrm{,}\frac{\mathrm{\partial }\mathit{g}}{\mathrm{\partial }{a}_R}\mathrm{<}\mathrm{0},\frac{\mathrm{\partial }\mathit{g}}{\mathrm{\partial }\mathit{\psi }}\mathrm{>}\mathrm{0}\mathrm{,}\frac{\mathrm{\partial }\mathit{g}}{\mathrm{\partial }\mathit{\rho }}\mathrm{<}\mathrm{0}\mathrm{.}$(10.6.18)

Further interpretations of the comparative static analysis refer to Young.¹⁶

In the Young model, the dynamic force of growth is through Ȧ = N_R/a_R. Nevertheless, without such an extremely optimistic assumption about dynamics of technological change, one may not be able to obtain unlimited growth. This simplified dynamic force may explain why the Young model generates the three types of equilibrium growth that may not be different from those identified by the AK model. Moreover, it does not appear proper that consumers' behavior is so complicated and knowledge growth is so simple.

10.7 The New Growth Theory and Linearized Knowledge Growth

As Solow commented,¹⁷ the endogenousness of the growth rate in Aghion and Howitt's model discussed in Section 10.4 is merely assumed. This makes the model less attractive than it appears. Solow remarks, 'Their [Aghion and Howitt] ambition is to make a model that gets close to our intuition about the endogenous generation of new technology. Even so, it is still pretty far from anything that feels like a description of real research, academic or industrial. In one way this paper - and the whole literature [of the New Growth Theory] may be too ambitious.' Evenson and Westphal also point out:¹⁸ 'To date, endogenous growth theory has achieved few robust policy generalizations. Moreover, development economists who grew up arguing about the merits of Rosenstein-Rodan's "big push" and debating balanced versus unbalanced growth are prone to find much that is not really new in endogenous growth theory. The vocabulary is new, but many of the insights that are today considered novel were the staple of development economics in the 1950s and 1960s. Indeed, as is relatively well known, the basic insights on which much endogenous growth theory is built are present in Adam Smith's discussion of pin making technology.' These comments are acceptable if we are only limited to the level of 'rough insights' or conceptual discussion; but it may not be right to conclude that Smith's discussion of pin making technology already includes the basic insights simply because many insights into complexity of economic evolution could not be obtained in the 1950s and 1960s, not to mention in Smith's time.

A main concern of the new growth theory is to generate possibility of an unlimited per-capita consumption growth c(t). The new growth theory want a steady state that nothing will change essentially but living standard. This reverie is different from what the neoclassical growth theory wants. In most of the neoclassical growth models, the national economy or aggregated consumption level C(t) grows at a predetermined growth rate, denoted by n, in steady state. Since it is possible to find an unlimited growth path even without population growth, it is reasonable to see that an exogenous growth rate of the population well accepted in the neoclassical growth theory disappears in the new growth theory. Table 10.1 summarizes some differences in steady state between the two approaches.

Table 10.1 The Steady States in the Neoclassical and New Growth Theories

In the neoclassical growth theory, the national economic growth rate is exogenously determined by the population growth rate. In the new growth theory, national economic growth rates are 'endogenous' in the sense that they are determined as functions of other parameters in steady state. One may ask why the neoclassical growth theory had failed to identify such a situation of unlimited economic growth without population growth.

We show how the new growth theories generate endogenous economic growth. One may wonder how miracles of unlimited growth are possible in the new growth theory. In fact, we have already observed the mechanism of unlimited growth in terms of per-capita consumption or income in the AK model in Section 8.1. If we assume that human capital grows in proportion to capital stock, it is possible to generate unlimited growth. It is commonly perceived that it is monopolistic competition that generates endogenous economic growth rates. In fact, like the AK model, it is due to linearized growth of knowledge that makes endogenous growth rates possible.

For the Lucas model, this key mechanism for endogenous growth is given by equation (8.2.4):

$\dot{\mathit{H}}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{H}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\kappa }\mathrm{(}\mathrm{1}\mathrm{-}\mathit{T}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}{\mathit{T}}_{\mathit{h}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{.}$

The growth rate of human capital is a linear function in the time input for human capital accumulation:

${\mathit{g}}_{\mathit{h}}\mathrm{(}t\mathrm{)}=\mathit{\kappa }\mathrm{(}1\mathrm{-}\mathit{T}\mathrm{(}t\mathrm{)}\mathrm{-}{\mathit{T}}_{\mathit{h}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{.}$

If we consider the above equation as a technology for producing human capital, it has two inputs, human capital and time. The production has diminishing returns in neither of the two inputs.

For the Romer model, the internal force is generated by: A(t) = kH_AA(t), where κ and H_A are fixed before steady state is determined. Similarly to the Uzawa-Lucas model with education, the growth rate of knowledge is also a linear function of the input factor: gA(t) = κH_A. For the Grossman-Flelpman model revised here, the miracle comes from dn/dt = l / a. In the Aghion-Howitt model, permanent percapita consumption growth is possible because, like the Romer model: A_t+1 / A_t γ. Irrespective of the internal structural differences in detail, the common feature of these models in the new growth theory is to replace one linear growth mechanism with another linear growth mechanism. In other words, these models, like the AK model, have assumed linearized growth in guaranteeing the vision of linear growth.

By the way, we show another way to generate permanent growth by using some linear production function. If we relax the assumption of diminishing returns to capital in the OSG model, the economy can sustain permanent growth. Following Jones and Manuelli,¹⁹ we replace the neoclassical production function with the following production function: F = K^αN^β + vK, v > 0, or in per-capita terms:

$\mathit{f}\mathrm{(}\mathit{k}\mathrm{)}={\mathit{k}}^{\alpha }\mathrm{+}\mathit{v}\mathit{k}\mathrm{.}$

This production is homogenous of degree one and has positive marginal products. It has diminishing marginal productivity because of f"(k) < 0. In contrast to the neoclassical growth function (which satisfies the Inada conditions, saying that the slope of the function is large near the origin and small at the other end), the slope of this function is positive, always decreases, but has a positive lower bound. It diminishes toward some large number. The neoclassical production function f(k) = k^α1 and the production function f(k) = k^a + vk are illustrated in Figure 10.3.

Suppose that consumers choose goods and saving exactly as in the OSG model in Chapter 2. Hence, corresponding to the fundamental equation (2.4.6), the dynamics with the new production function are controlled by:

$\dot{\mathit{k}}\mathrm{(}t\mathrm{)}=\lambda k\mathrm{(}\mathit{t}{\mathrm{)}}^{\mathit{\alpha }}\mathrm{-}\mathrm{(}{\mathit{\xi }}_{\mathit{k}}\mathrm{+}\mathit{n}\mathrm{-}\mathit{\lambda }\mathit{v}\mathrm{\rangle }k\mathrm{(}t\mathrm{)}\mathrm{.}$

Hence, the growth rate of k(t) is given by:

${\mathit{g}}_{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}=\frac{\dot{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}}{\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}}=\lambda k\mathrm{(}t{\mathrm{)}}^{-\mathit{\beta }}\mathrm{-}\mathrm{(}{\mathit{\xi }}_{\mathit{k}}\mathrm{+}\mathit{n}\mathrm{-}\mathit{\lambda }\mathit{v}\mathrm{)}\mathrm{.}$

If λv > ξ_k + n, the rate of growth of k(t) tends to λv - (ξ_k + n). Since c(t) eventually grows at the same rate as k(t), we conclude that the adopted production function generates permanent growth in per-capita consumption as well as in per-capita income.

Like in Section 2.8, we may also use the CES production function:

$\mathit{F}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{A}{\left\{\right(a(b\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}{\mathrm{)}}^{\mathit{\sigma }}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}a\mathrm{)}{\mathrm{[}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{b}\mathrm{)}\mathit{N}\mathrm{(}t\mathrm{)}]}^{\sigma }\}}^{1/\sigma },$

where 0 < a < 1, 0 < < 1 and σ < 1. The production function exhibits constant to scale for all values of σ. Dividing the two sides of the above equation by N(t) yields:

$f\mathrm{(}\mathit{k}\mathrm{)}=A{\mathrm{(}a{\mathit{b}}^{\mathit{\sigma }}{\mathit{k}}^{\mathit{\sigma }}\mathrm{+}\mathit{B})}^{1/\sigma },$

where B ≡ (1 - a)(1 - b)^σ. The marginal and average products of capital are:

$f\prime \left(k\right)=\mathit{A}\mathit{a}{\mathit{b}}^{\mathit{\sigma }}{\left(\mathit{a}{\mathit{b}}^{\mathit{\sigma }}\mathrm{+}\mathit{B}{\mathit{k}}^{-\mathit{\sigma }}\right)}^{(1-\sigma )/\sigma },\frac{f\mathrm{(}\mathit{k}\mathrm{)}}{\mathit{k}}=\mathit{A}{\left(\mathit{a}{\mathit{b}}^{\mathit{\sigma }}\mathrm{+}\mathit{B}{\mathit{k}}^{-\sigma }\right)}^{1/\sigma }\mathrm{.}$

Both f′(k) and f(k)/ k are positive and diminishing in k(t) for all values of σ. As the rest of the OSG model is the same as before, we can represent all the variables in terms of k(t). Here, we are interested in the dynamics of the growth rate of per-capita capital. The dynamics with the new production function are controlled by:

$\dot{k}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{\lambda }\mathit{A}\mathrm{(}\mathit{a}{\mathit{b}}^{\mathit{\sigma }}{\mathit{k}}^{\mathit{\sigma }}\mathrm{+}\mathit{B}{\mathrm{)}}^{1/\mathit{\sigma }}\mathrm{-}\mathrm{(}\mathit{\xi }\mathrm{+}\mathit{\lambda }{\mathit{\delta }}_{\mathit{k}}\mathrm{+}\mathit{n}\mathrm{-}\mathit{\lambda }\mathit{v}\mathrm{)}\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$

The growth rate of capital-labor ratio is:

$g_k\left(t\right)=\lambda A{(a{b}^{\sigma }+\mathit{B}{\mathit{k}}^{-\mathit{\sigma }})}^{1/\sigma }\mathrm{-}(\mathit{\xi }+\mathit{\lambda }{\mathit{\delta }}_{\mathit{k}}+\mathit{n}\mathrm{-}\mathit{\lambda }v)\mathrm{.}$

As in Section 2.8, the system can generate a permanent growth path.

Jones observes that growth has been relatively constant in the face of huge increases in the volume of resources devoted to R&D in advanced economies.²⁰ Young points out that these theories shed little light on the remarkable growth rates of newly industrialized countries in East Asia.²¹ It can be seen that the sustainable rapid growth of China over the last two decades cannot be explained by the growth models based on monopolistic competition. This observation is actually based on the models in the previous two chapters. Different from the models reviewed in this chapter in which labor is the only input to 379 and 379 is the only factor affecting long-term growth, the models in the previous chapters are built on the assumptions that human capital and knowledge creation are related not only by labor inputs, but also capital inputs and the current knowledge stocks as well as concrete ways of human activities. As pointed out by Aghion and Howitt,²² if capital accumulation and population are introduced into the growth models reviewed in this chapter, the new growth theory becomes broadly consistent with the empirical observations that have been adduced to refute them.

It is important to note that the models in the new growth theory discussed here do not provide any sophisticated analysis of stability. In the recent literature of economic growth with endogenous knowledge, the mathematical analysis has become much more refined, even though the main behavioral assumptions about companies and consumers are fundamentally the same as discussed in this chapter. It is interesting to cite from Paul Krugman's comments about the new growth theory:

It takes a long time to digest the 'funny assumptions', not to mention to follow the mathematical and computing skills in those works, in order to understand what are happening in the contemporary mainstream(s) of growth theory. Since assumptions in each model are 'funny' in its own way, the models in the current literature cannot be closely connected to each other but grow in a geometric progression.²⁴

¹ Shell (1973).

² Romer (1990).

³ Dixit and Stiglitz (1977), Judd (1985), and Ethier (1982).

⁴ Grossman and Helpman (1991, Chapter 3). See also Judd (1985).

⁵ Further discussions about human capital and human capital accumulation are referred to Grossman and Helpman (1991).

⁶ Following Dixit and Stiglitz (1977).

⁷ Ethier (1982).

8 Barro and Sala-i-Martin (1995, Section 6.2). The model is developed under the influence of Spence (1976) and Grossman and Helpman (1991).

⁹ Barro and Sala-i-Martin (1995, Section 6.2).

¹⁰ Aghion and Howitt (1992).

¹¹ Solow (2000).

¹² Aghion and Howitt (1992).

¹³ This section is based on Barro and Sala-i-Martin (1995, Chapter 7) and Grossman and Helpman (1991).

¹⁴ Barro and Sala-i-Martin (1995, Chapter 7).

¹⁵ Zhang (1990).

¹⁶ Young (1993).

¹⁷ Solow (2000).

¹⁸ Evenson and Westphal (1995: 220).

¹⁹ Jones and Manuelli (1990).

²⁰ Jones (1995).

²¹ Young (1995).

²² Aghion and Howitt (1998: 6).

²³ Fujita and Krugman (2004: 143).

²⁴ I have endeavored to integrate economic theories and develop economics to a new height about 16 years ago when I was a graduate student of civil engineering at Kyoto University, having first applied nonlinear science to economics (Zhang, 1991) to 'modernize' the vision and methodology of economic analysis, and then having developed the analytical framework with time and space for analyzing various economic issues with the vision of nonlinear science. This book is a part of the plain ambition. As mentioned in Zhang (2003a: preface), 'The proper evaluation of the book should be done by perusing and digesting it as a part of the edifice, rather than by scanning it as a diamond'. It will take quite some years for the approach to become stabilized.