Chapter 3
Traditional Growth Theories and the OSG Model

Fundamental progress in economics has to do with the restudy of basic ideas. It is through the reexamination as well as reinterpretation of basic ideas that we can fully appreciate the past and become aware of possibility of and hindrance to advancing. The Solow model is the starting point for almost all analyses of economic growth. It was the first neoclassical version of the Harrod-Domar growth model. It opened a new way to modeling economic growth. The previous chapter asserted that the only difference between the Solow model and the OSG model, so far, is description of consumer behavior. This chapter explains differences between the Solow model and the OSG model. We also examine differences in modeling consumer behavior between the other traditional approaches and the OSG model. Nevertheless, for illustrating the historical path of growth theory, it is proper for us to introduce the Harrod-Domar model - the most important formal (mathematical) growth model prior to the Solow growth model.¹

This chapter is organized as follows. Section 3.1 represents the Harrod-Domar model and examines its dynamic behavior. In Section 3.2, first we define the Solow model and provide dynamic properties of the model. Then, we examine some relationships between the Solow model and the OSG model. We demonstrate that the OSG model exhibits identical dynamic behavior with these revealed by the Solow model when we specify preference change of the OSG model. Section 3.3 introduces the life cycle hypothesis and discusses relationships between the life cycle hypothesis and our approach to consumer behavior. We show that the two approaches have similar implications for consumers' decision on saving and consumption. Section 3.4 discusses the permanent income hypothesis and studies relationships between the hypothesis and our approach to consumer behavior. It is concluded that the two approaches have similar implications for consumers' decision on saving and consumption. This is expectable because the life cycle hypothesis and the permanent income hypothesis have similar economic implications. Section 3.5 defines the standard Ramsey model and examines its dynamic properties. We also discuss limitations and some invalidities in modeling consumers' behavior in the Ramsey growth model. In Section 3.6, we show the connections between the Ramsey model and the OSG model by introducing some pattern of preference change. It is demonstrated that under the preference change, the two models reveal the same behavior pattern. Section 3.7 shows how poverty traps can be generated by the Solow model through making the saving rate an endogenous variable as well as by the OSG model through making the propensity to save an endogenous variable. Section 3.8 reasons our approach to consumer behaviors. In the appendix to this chapter, we deal with the golden rule of capital accumulation in the Solow model.

3.1 The Harrod-Domar Model

In the Harrod-Domar model, it is assumed that any change in the rate of investment per year I(t) will affect the aggregate demand and productivity of the economy. The demand effect of a change in I(t) operates through the multiplier process. An increase in I(t) will raise the rate of income flow per year Y(t) by a multiple of the increment in I(t).

In the literature of economic dynamics, there are various ways to allocating household income among saving and expenditures on different goods. One way in the neoclassical growth theory to avoiding this complex of the decision is to assume that all expenditure on consumption goods can be aggregated into that on a single consumer good and there are some simple rule relating saving to aggregate income. In the Harrod-Domar model, it is assumed that the agents regularly set aside some predictable portion of its output for the purpose of capital accumulation. Since there is only one good, no question of changes in relative price can arise, nor can any questions of capital composition. Let us denote ŝ constant fraction of the total output flow that is saved and set aside to be added to the capital stock.

For a predetermined ŝ, the multiplier is a = 1/ŝ. As I(t) is the only expenditure flow that influences the rate of income flow, we have Ẏ(t) = İ(t)/ŝ. The capacity effect of investment is reflected by the change in the rate of potential output the economy is capable of producing. The capacity-capital ratio is defined by ρ ≡ κ(t)/K(t), where κ(t) stands for capacity or potential output flow and ρ represents a (predetermined) constant capacity-capital ratio. The above equation implies that with a capital stock K(t) the economy is potentially capable of producing an annual product κ. Taking derivatives of κ(t) = ρK(t) with respect to t yields:

$\dot{\kappa }=\rho \dot{K}=\mathit{\rho }I\mathrm{.}$(3.1.1)

Here, equilibrium is defined as a situation, in which productive capacity is fully utilized, i.e., Y(t) = κ(t). If we start initially from equilibrium, the requirement means the balancing of the respective changes in capacity and in aggregate demand: Ẏ = $\dot{\kappa }$. The question is what kind of time path of investment I(t) will keep the economy in equilibrium at all times. To answer this question, from ρ = κ(t)/K(t), equation (3.1.1), and equation Ẏ = $\dot{\kappa }$, we have İ = ŝρI. Therefore, the required path is given by the following solution of the above differential equation I(t) = I(0)e^ρŝt, where I(0) is the initial rate of investment. This implies that to maintain the balance between capacity and demand over time, the rate of investment flow must grow precisely at the exponential rate of pŝ.

We now examine what will happen if the actual rate of growth of investment, denoted by r, differs from the required rate ps. According to the definition of r, we have I(t) = I(0)e^rt. Therefore, we have, according to Ẏ(t) = İ(t)/ŝ and equation (3.1.1):

$\dot{Y}\mathrm{=}\frac{\mathrm{1}}{\stackrel{\mathrm{ˆ}}{S}}\dot{I}\mathrm{=}\frac{\mathit{r}}{\stackrel{\mathrm{ˆ}}{S}}\mathrm{/}\mathrm{(}\mathrm{0}\mathrm{)}{\mathit{e}}^{rt},\dot{\kappa }\mathrm{=}\mathit{\rho }I\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\mathit{\rho }I\mathrm{(}\mathrm{0}\mathrm{)}{\mathit{e}}^{\mathit{r}t}\mathrm{.}$

The ratio between these two growth rates is Ẏ/$\dot{\mathit{\kappa }}$ = r/ρŝ. The ratio gives the relative magnitude of the demand-creating effect and the capacity-generating effect of investment at any time t, for the given actual growth rate. If r exceeds ρŝ, Ẏ > $\dot{\kappa }$. The demand effect will outstrip the capacity effect, resulting in a shortage of capacity. If r lags behind ρŝ, Ẏ < $\dot{\kappa }$. This will cause a surplus of capacity.

If investment actually grows at a faster rate than the required rate, the economy will be faced with a shortage rather than a surplus of capacity; if the actual growth rate of investment lags behind the required rate, the economy will encounter a capacity surplus rather than a shortage. This implies that if we allow the entrepreneurs to adjust the actual growth rate according to the prevailing capacity situation, they tend to make the 'wrong kind' of adjustment. For instance, in the case of r > ρŝ, the emergent capacity shortage will encourage an even faster rate of investment, which will actually intensify the discrepancy between the two rates of growth.

Once ρ and ŝ are determined, the only way to avoid both shortage and surplus of productive capacity is to guide the investment flow along the equilibrium path with a growth rate equaling ρŝ. Any deviation from such a razor's edge time path will bring about a persistent failure to satisfy the condition for full utilization of capacity. Nevertheless, the three elements - the saving rate, the growth rate of investment, and the capital-output ratio - are not only changeable in realty but are changed by different forces. The saving rate is determined by preference of consumers; the growth rate of investment by population growth and other conditions; the capital-output ratio by technology. A balance economic growth is simply a miraculous stroke of luck. It implies that any little departure from a steady state will be magnified indefinitely by a process that is determined by multiple independent forces. From the balance condition, r = ρŝ, we see that an economy doubles its growth rate simply by doubling the saving rate. These undesirable properties of the Harrod-Domar model explain why it was replaced by the Solow model irrespective of its pioneer position in growth theory.

We omit describing connections between the Harrod-Domar and the OSG model. As shown below, the OSG model is an extension of the Solow model; the Solow model is an extension of the Harrod-Domar model. There are detailed studies about relationships between the Solow model and Solow model,² which are also valid for our OSG model.

3.2 The Solow Model and the OSG Model

The Harrod and Domar were concerned with the question of when an economy is capable of steady growth at a constant rate. The answer from the Harrod-Domar model is that the national saving rate has to equal the product of the capital-output ratio and the rate of growth of the effective labor force. Along the balance path the economy grows with its stock of capital in balance with its supply of labor - there is neither labor shortage nor labor surplus on the balance path. Nevertheless, A tiny deviation from this condition will drive the economy gradually far away from the balance growth path. In his Nobel Lecture in 1987, Solow stated what he felt about the Harrod-Domar model:³ 'Growth theory, like much else in macroeconomics, was a product of the depression of the 1930s and of the war that finally ended it. So was I. Nevertheless, it seemed to me that the story told by these models felt wrong. An expedition from Mars arriving on Earth having read this literature would have expected to find only the wreckage of a capitalism that had shaken itself to pieces long ago. Economic history was indeed a record of fluctuations as well as of growth, but most business cycles seemed to be self-limiting. Sustained, though disturbed, growth was not a rarity.'

The standard neoclassical growth model characterized by the Solow model initiated a new course of development of economic growth theoiy by using the neoclassical production function and neoclassical production theory, still maintaining the traditional way in handling with consumer behavior in dynamic analysis. Solow recalled:⁴ 'That was the spirit in which I began tinkering with the theory of economic growth, trying to improve on the Harrod-Domar model. I cannot tell you why I thought first about replacing the constant capital-output (and labor-output) ratio by a richer and more realistic representation of technology. ... I know that it occurred to me early, as a natural-born macroeconomist, that even if technology itself is not so very flexible for each single good at a given time, aggregate factor-intensity must be much more variable because the economy can choose to focus on capital-intensive or labor-intensive or land-intensive goods.'

Most neoclassical models are extensions and generalizations of the pioneering works of Solow and Swan in 1956.⁵ Actually, the Solow model is often called the Solow-Swan model because Swan's work is similar to Solow's seminal paper. In the Solow model, capital and labor are substitutes for one another with the result that the long-run growth path of the economy is one of full employment. The model shows that the razor-edge growth path of the Domar model is primarily a result of the particular production function assumption adopted therein and that the need for delicate balancing may not arise when the production function is taken on a different type.

Most aspects of the OSG model in Chapter 2 are similar to the Solow model. The variables, Y, K, N, w, r, in the Solow model have the same meanings as in the OSG model. The production process, marginal conditions, population growth are the same as in the OSG model. The Solow model assumes that the agents regularly set aside some fairly predictable portion ŝ of its output for the purpose of capital accumulation; hence:

By the way, it is worthwhile to introduce Solow's recent comments on the linear depreciation function of capital, δ_kK. This assumption makes depreciation independent of the details of the history of technological change and past gross investment, even though this is known to be empirically inaccurate. Solow introduces a way to relax this assumption.⁶

Define a non-increasing survivorship function ϕ(a), with ϕ(0) = 1 and ϕ(+∞)= 0. Here, ϕ(a) stands for the fraction of any investment that survives to age a. Then if I(t) (which is assumed to equal ŝF(t)) is gross investment:

$\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{{\displaystyle \int }}}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathit{F}\mathrm{(}t\mathrm{-}a\mathrm{)}\mathit{\varphi }\mathrm{(}\mathit{a}\mathrm{)}\mathit{d}a\mathrm{.}$

Now differentiation with respect to time and one integration by parts with respect to a leads to:

$\dot{\mathit{\kappa }}\mathrm{(}\mathit{f}\mathrm{)}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathit{F}\mathrm{(}t\mathrm{)}\mathrm{-}\stackrel{\mathrm{ˆ}}{\mathit{s}}{\int }_0^{\infty }F(t-a)d\left(a\right)da$

where:

$\mathit{d}\mathrm{(}\mathit{a}\mathrm{)}\mathrm{\equiv }\mathrm{-}\frac{\mathit{d}\mathit{\varphi }\mathrm{(}\mathit{a}\mathrm{)}}{\mathit{d}\mathit{o}}$

is the rate of depreciation at age a. The net investment at time t depends on the whole stream of gross investments. If we specify:

$\mathit{\varphi }\mathrm{(}\mathit{a}\mathrm{)}\mathrm{=}\exp \mathrm{(}\mathrm{-}{\mathit{d}}_{\mathit{k}}\mathit{a}\mathrm{)}\text{,}$

then:

$\dot{\mathit{K}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\hat{\mathit{s}}\mathit{F}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{-}{\mathit{d}}_{\mathit{k}}\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$

This is reduced to the Solow capital accumulation equation.

As the production function is neoclassical, one can reduce equation (3.2.1) into the following single differential equation:

$\dot{\mathit{k}}\mathrm{(}t\mathrm{)}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathit{f}\mathrm{(}\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{-}\left(\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}\right)k\mathrm{(}\mathit{t}\mathrm{)},$(3.2.2)

where k(t) and f(k(t)) are the same as k(t) and f(k(t)) defined in Chapter 2. Similarly to Figure 2.8, we illustrate dynamics of capital-labor ratio in Figure 3.1.

We see that the differential equation for per-worker-capital accumulation in the Solow model is mathematically identical to the capital accumulation equation in the OSG model defined by equation (2.4.6) in Section 2.4:

$\dot{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\lambda }\mathit{f}\mathrm{(}\mathit{k}\mathrm{)}\mathrm{-}\mathrm{(}{\mathit{\xi }}_{\mathit{k}}\mathrm{+}\mathit{n}\mathrm{)}\mathrm{t}\mathrm{(}t\mathrm{)}\mathrm{.}$

The Solow model and the OSG model have the same dynamic properties - the system has a unique stable equilibrium. But the OSG model holds that saving rate is time-dependent; the Solow model predetermines saving rate. As in the OSG model, the Solow model implies that changes in the level of technology, the saving rate, the rate of population, and the depreciation rate do not affect the long-run growth rates of per capita output, capital, and consumption, which are all equal to zero.

Historically, the Solow hypothesis is approximately valid for the U.S. economy, while it is hardly acceptable for many other economies. There is some evidence that over the last hundred and fifty years saving rates as percentage of the GNP for the U.S. economy have remained stable while those in many other countries have risen. The Solow growth model was constructed on the basis of the empirical verification for the U.S. economy. From the saving rate s̄ = λ - ξδk^β/A in the OSG model, we see that saving rate is time-dependent in general rather than constant. In particular, the saving rate is directly related to technological change. We see that a country endowed with wealth tends to have a low saving rate, while a country with high tech but less wealth tends to have a high saving rate. Table 2.1, which is referred to Maital and Maital,⁷ presents data on historical gross saving rates as a percentage of GNP for some industrial economies. Empirical studies show that the saving rates varied over time, except only for the U.S. economy. It was observed that national saving as a proportion of GNP or GDP tend to fall in contemporary industrial countries. It is observed that the United States and many other OECD countries implemented many tax reforms in the 1980s that were strongly pro-rich. Between 1984 and 1990, top individual tax rates fell by 22 percent in the United States, 12 percent in Japan. Since the rich save more of their income than the middle class or poor, and since it is the rich that mainly pay the highest tax rates, low tax rates for the rich should have led to higher personal saving rates. But it is observed that saving rate fell for every income during the 1980s. It is worthwhile to cite what Maital and Maital observed: 'Conventional economic theory has been largely unable to explain this decline in saving. Moreover, empirical evidence has long been inconsistent with the predominant models of consumption and saving the "permanent income" hypothesis and the life-cycle model. Therefore, lacking an empirically well-grounded theory of saving, conventional macroeconomics has little to say, either positive or normative, on a vital economic issue that touches on virtually every important policy question for this decade - intergenerational equity, growth, productivity, inflation, competitive advantage.'

Table 3.1 Historical Gross Saving Rates as a Percentage of GNP

Solow explicated:⁸ 'I know that even as a student I was drawn to the theory of production rather than to the formally almost identical theory of consumer choice. It seemed more down to earth.' The main difference between the OSG model and the Solow model is about consumer behavior. Solow and Swan introduced a consumption function, saying that a fixed fraction of income is consumed. Although this assumption greatly simplifies consumers' behavior and economists could have analyzed capital accumulation, it has become clear that this assumption has limitations in extending dynamic analysis to a dynamic economy with multiple consumption goods. Its analytical framework is mainly limited to one-sector model or multi-sector model with a single consumption good. The history of the neoclassical growth theory shows that this assumption has limitations to extend dynamic analysis to including preference changes and other important dynamic issues.

We now show that under certain circumstances the OSG model includes the Solow model as a special case. The OSG model endogenously determines saving and consumption. Its capital accumulation is given by equation (2.4.4):

$\dot{\mathit{\kappa }}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\lambda }\mathit{F}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathrm{(}\mathit{\xi }\mathrm{+}\mathit{\lambda }{\mathit{\delta }}_{\mathit{k}}\mathrm{)}\mathit{K}\mathrm{(}t\mathrm{)}\mathrm{.}$(3.2.3)

We explain under what conditions the OSG model has the same behavioral implications as the Solow model.

Remember that in the OSG model, the APS is given by equation (2.3.5):

$\overline{\mathit{s}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\frac{\mathit{\lambda }\mathit{F}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\lambda }\delta K\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{\delta }\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}}{\mathit{F}\mathrm{(}t\mathrm{)}}\mathrm{.}$

We are interested in when s̄(t) in the OSG model equals the predetermined saving rate ŝ in the Solow model, i.e.:

$\frac{\mathit{\lambda }\mathit{F}\mathrm{(}t\mathrm{)}\mathrm{+}\mathit{\lambda }\mathit{\delta }\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{\delta }\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}}{\mathit{F}\mathrm{(}t\mathrm{)}}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{.}$(3.2.4)

If the propensity to save λ is considered as an endogenous variable λ(t), the above equation holds if:

$\lambda \mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathit{\delta }\frac{\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}}{\mathit{F}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\delta }\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{+}\frac{\mathrm{(}\mathrm{1}\mathrm{-}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathit{\delta }}{\mathit{f}\mathrm{(}\mathit{k}\mathrm{(}t\mathrm{)}\mathrm{)}\mathrm{/}\mathit{k}\mathrm{(}t\mathrm{)}\mathrm{+}\mathit{\delta }}\mathrm{(}\mathrm{<}\mathrm{1}\mathrm{)}\mathrm{.}$(3.2.5)

As ξ(t) + λ(t) = 1, ξ(t) as function of k/f(k). If the propensity to save is related to ratio of capital per capita and output per capita as in equation (3.2.5), then s̄(t) in the OSG model is constant and is equal to the saving rate S in the Solow model. Inserting equation (3.2.5) into equation (3.2.3) yields:

$\dot{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathit{F}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathrm{(}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}{\mathit{\delta }}_{\mathit{k}}\mathrm{)}\mathit{K}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{(}\mathit{F}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\delta K\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{-}\mathit{K}\left(t\right)=\hat{s}\mathit{F}\mathrm{(}t\mathrm{)}\mathrm{-}{\mathit{\delta }}_{\mathit{k}}\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$(3.2.6)

We see that under equation (3.2.5) the evolution of capital in the OSG model is identical to that in the Solow model. By equation (2.3.5), we have APC + APS = 1. As s̄(t) = ŝ at any point of time, c̄(t) (= 1 - s̄(t)) in the OSG model is equal to the consumption rate ĉ(=(1 - ŝ)) in the Solow model. Summarizing the above discussions yields the following theorem.

Theorem 3.2.1

Let the production sectors be identical in the OSG model and the Solow model. If the saving rate ŝ in the Solow model and the propensity to save λ(t) in the OSG model satisfy equation (3.2.4), i.e.:

$\mathit{\lambda }\mathrm{(}t\mathrm{)}\mathrm{=}\hat{s}\mathrm{+}\frac{\mathrm{(}\mathrm{1}\mathrm{-}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathit{\delta }}{\mathit{f}\mathrm{(}\mathit{k}\mathrm{(}t\mathrm{)}\mathrm{)}\mathrm{/}\mathit{k}\mathrm{(}t\mathrm{)}\mathrm{+}\mathit{\delta }},$

then the OSG model is identical to the Solow model in terms of the saving rate (out of current income), the consumption rate, the interest rate, the wage rate, output, income, consumption, and saving.

The propensity to save λ(t) in this case is not constant. To see how it changes as economic conditions vary, we take derivative of equation (3.2.5) with respect to t:

$\dot{\lambda }\mathrm{=}\frac{\mathrm{(}\mathrm{1}\mathrm{-}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathit{\delta }\mathrm{(}\mathit{f}\mathrm{-}kf\prime \mathrm{)}}{\mathrm{(}f\mathrm{+}\delta k)}\dot{\mathit{k}}\mathrm{=}\frac{\mathrm{(}\mathrm{1}\mathrm{-}\hat{\mathit{s}}\mathrm{)}\mathit{\delta }w}{\mathrm{(}f\mathrm{+}\delta k)}\dot{\mathit{k}}\mathrm{,}$(3.2.7)

where we use w = f(k) - kf′(h) in equation (2.1.2). The propensity to save rises (falls) as wealth per capita rises (falls).

Corollary 3.2.1

If the propensity to save evolves according to equation (3.2.5), then the propensity to save rises (falls) as wealth per capita rises (falls).

By the way, as the Solow model does not provide rational mechanism for the household to make decision on consumption and saving, one may ask under what conditions one may claim that the household in the Solow model makes a rational decision. In a well-cited paper published in 1966, Phelps addressed this issue. Phelps identified a level of the saving rate that maximizes per capita consumption in the steady state. In Appendix A.3.1, we discuss the golden rule in the Solow model.

3.3 The Life Cycle Hypothesis and the Generalized Keynesian Consumption Function in the OSG Model

It is well observed that the saving rate is not constant in general. It is found to be related to different variables. Two well-known models of consumers' saving behavior are the so-called life cycle theory of consumption proposed by Modigliani and Brumberg in 1954 and Ando and Modigliani in 1963.⁹ The theories were developed to reconcile the disparate implications from data resources.

The life cycle hypothesis is to explain the empirical work on consumption functions.¹⁰ It has been observed that the relationship between consumption and current income would be nonproportional and the intercept of consumption function is not constant over time. As stated out by Modigliani,¹¹ 'The point of departure of the life cycle model is the hypothesis that consumption and saving decisions of households at each point of time reflect a more or less conscious attempt at achieving the preferred distribution of consumption over the life cycle, subject to the constraint imposed by the resources accruing to the household over its lifetime.' Consumption depends not just on current income but also on long-term expected earnings over their lifetime. The life cycle hypothesis accounts for the dependence of consumption and saving behavior on the individual age. Young people entering the labor force have relatively low incomes and possibly low or even negative saving rates. As they become elder, they earn more and tend to have higher saving rates. Retirement brings a fall income and may become a period of dissaving. The life cycle theory portrays a typical pattern of lifetime consuming and saving as in Figure 3.2. Consumption rises gradually over the life cycle. Income rises sharply over the early working years, peaks, and then falls. In early working years and the late stage of the life cycle the typical consumer undergoes dissaving; over the high-income middle period of life cycle the consumer experiences saving.

The general form of the aggregate consumption function implied by the life cycle hypothesis is:

$\mathit{C}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}{\mathit{b}}_1\mathit{w}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}{\mathit{b}}_{\mathrm{2}}\tilde{Y}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}{\mathit{b}}_3\mathit{K}\mathrm{(}t\mathrm{)},$(3.3.1)

where b₁ b₂, and b₃ are nonnegative parameters and:

C(t) = consumption at time t;

w(t) = the individual's labor income at time t;

Y(t) = the average labor income expected over the remaining years during which the individual plans to work; and K(t) = the value of presently held assets.

To use equation (3.3.1) to examine actual consumer behavior, it is necessary to make further assumptions about the way in which individuals form expectations concerning lifetime labor income. In Ando and Modigliani,¹² it is assumed:

$\tilde{\mathit{Y}}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\mathit{\beta }w\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\mathit{\beta }\mathrm{>}\mathrm{0.}$(3.3.2)

Individuals expect their future income flows through working by some proportion β of a change in current labor income. Under equation (3.3.2), equation (3.3.1) becomes:

$\mathit{C}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\mathrm{(}{\mathit{b}}_{\mathrm{1}}\mathrm{+}\mathit{\beta }{\mathit{b}}_{\mathrm{2}})\mathit{w}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}{\mathit{b}}_{\mathrm{3}}\mathit{K}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{.}$(3.3.3)

In the OSG model, consumption (in terms of per capita) is given by the generalized Keynesian consumption function:

$\mathit{C}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\xi }\stackrel{\mathrm{ˆ}}{\mathit{Y}}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\mathit{\xi }\mathit{w}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\xi }\mathrm{(}\mathit{r}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\delta }\mathrm{)}\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\mathit{N}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathrm{1}\mathrm{,}$(3.3.4)

We see that both equations (3.3.3) and (3.3.4) show that consumption is positively related to labor income and wealth.

The OSG model does not require that each household have at all times a definite vision of the family's future size and composition, the life expectancy, the entire lifetime profile of the income from birth to death, and even the future emergencies, opportunities and the like. It can be seen that our approach does not impose so strict information about the future, even though it can take any important information into consideration by changing preference parameters ξ(t) and λ(t). For instance, it is straightforward to take account of liquidity constraints for young people because our utility function can be age-dependent.

3.4 The Permanent Income Hypothesis and the Generalized Keynesian Consumption Function

According to Friedman,¹³ the consumption is proportional to permanent income Y^p(t) as follows:

$\mathrm{C}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\kappa }{\mathit{Y}}^p\mathrm{(}\mathit{t}\mathrm{)},$

where κ is the positive factor of proportionality. Permanent income is expected average long-term income from human and nonhuman wealth. In this theory, human capital is a significant variable in determining consumption - we will include this consideration when dealing with endogenous human capital in Chapter 7. According to this conception, we see that it is necessary to include wealth, human capital, working time, family structure, and institutional factors in determining consumption.

To implement the permanent income hypothesis, it is necessary to make some assumptions about how individuals form long-term expectations about income. Friedman assumed that individuals revise their initial estimate of permanent income from period to period in the following manner:

${\mathit{V}}^{\mathit{p}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}{\mathit{Y}}^p\mathrm{(}\mathit{t}\mathrm{-}\mathrm{1}\mathrm{)}\mathrm{+}\mathit{j}(Y\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}{\mathit{Y}}^p\mathrm{(}\mathit{t}\mathrm{-}1\mathrm{)}\mathrm{)}\mathrm{,}\mathrm{0}\mathrm{<}\mathit{j}\mathrm{<}\mathrm{1}\mathrm{,}$

which states that in each period individuals adjust their estimate of permanent income by a fraction j of the discrepancy between actual income in the current period and the prior period's estimate of permanent income.

Following Romer,¹⁴ we illustrate the essence of the permanent income hypothesis. Let us consider an individual who lives for T periods. The individual has initial wealth of Y₀ and labor incomes of Y₁, Y₂, ..., Y_T in the T periods of lifetime. The individual takes these as given. Under the assumption that the individual's discount rate of utility is zero, lifetime utility is equal to the sum of the level of utility over each period. That

$\mathit{U}\mathrm{=}\underset{t=1}{\overset{T}{\mathrm{\Sigma }}}\mathit{u}\mathrm{(}{\mathit{C}}_t\mathrm{)}\mathrm{,}\mathit{u}\prime \mathrm{(}\cdot \mathrm{)}\mathrm{>}\mathrm{0}\mathrm{,}u"\mathrm{(}\cdot \mathrm{)}\mathrm{<}\mathrm{0}\mathrm{,}$(3.4.1)

where u(C_t) stands for lifetime utility, u(Ct) is the instantaneous utility, and C_t is consumption in period t.

It is assumed that the individual can save or borrow at an exogenous interest rate (set to zero for simplicity) subject only to the constraint that any outstanding debt must be repaid at the end of the life. The individual's budget constraint is thus given by:

$\underset{t=1}{\overset{T}{\mathrm{\Sigma }}}{\mathit{C}}_t\mathrm{\le }\underset{t=0}{\overset{T}{\mathrm{\Sigma }}}{\mathit{Y}}_t\mathrm{.}$(3.4.2)

The individual maximizes equation (3.3.1) subject to equation (3.3.2). The problem is a standard optimization with an inequality constraint. Define the Lagrangian for the problem:

$\mathrm{Z}\mathrm{=}\underset{t=1}{\overset{T}{\mathrm{\Sigma }}}\mathit{u}\mathrm{(}{\mathit{C}}_t\mathrm{)}\mathrm{+}\mathit{\lambda }(\underset{t=0}{\overset{T}{\mathrm{\Sigma }}}{{\mathit{Y}}_t}_{}\mathrm{-}\underset{t=1}{\overset{T}{\mathrm{\Sigma }}}{\mathit{C}}_t)\mathrm{.}$(3.4.3)

The first-order condition is:

$$\begin{gathered} \frac{\mathrm{\partial }\mathrm{Z}}{\mathrm{\partial }{\mathit{C}}_t}\mathrm{=}\mathrm{0}\Rightarrow {\mathit{u}}^{}\prime \mathrm{(}{\mathit{C}}_t\mathrm{)}\mathrm{=}\mathit{\lambda }\mathrm{,}\mathit{t}\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\text{..., }}\mathit{T}\mathrm{,} \\ \frac{\mathrm{\partial }\mathrm{Z}}{\mathrm{\partial }\mathit{\lambda }}\mathrm{=}\mathrm{0}\Rightarrow \underset{t=0}{\overset{T}{\mathrm{\Sigma }}}{\mathit{Y}}_t\mathrm{=}\underset{t=1}{\overset{T}{\mathrm{\Sigma }}}{\mathit{C}}_t\mathrm{.} \end{gathered}$$

The first T equations guarantee that the marginal utility of consumption is equal in every period. Since the level of consumption uniquely determines its marginal utility, the level of consumption must be equal in every period, i.e., C₁ = C₂ =... = C_T. From the last equation in equation (3.4.4), this means:

${{\mathit{C}}_t}_{}\mathrm{=}\frac{\mathrm{1}}{\mathit{T}}\underset{t=0}{\overset{T}{\mathrm{\Sigma }}}{\mathit{Y}}_t\mathrm{.}$(3.4.5)

The rational individual would divide the lifetime resources equally among each period of life. It is straightforward to show that the above solutions satisfy the second-order conditions.

Different from the assumption on saving behavior in the Solow model, this model predicts that the individual's consumption in a given period is related not only to income in that period, but also the incomes over the rest periods of the entire lifetime. Friedman dubbed the right-hand side of equation (3.3.5) the permanent income. The difference between current and permanent income is transitory income. The above equation means that consumption is determined by permanent income. This result implies that a temporary rise in income may have little impact on consumption - an important difference between Solow's assumption of predetermined saving rate. Consider the effect of a windfall gain of amount ΔY₁in the first period of life. The windfall raises current income by ΔY₁; but permanent income by only ΔY₁/T. If the individual's lifetime is long, the impact on current consumption is small. Saving is determined not only by the permanent income but also by the current income. When current income is less (more) than permanent income, saving is negative (positive). The individual uses saving and borrowing to smooth the path of consumption. We also see that if a windfall is large, the individual saves most of it for future consumption. The above conclusions are the key ideas of the lifecycle/permanent-income hypothesis.

We define the individual's saving in period t as the difference between income and consumption:

${\mathit{S}}_t\mathrm{=}{{\mathit{Y}}_t}_{}\mathrm{-}{\mathit{C}}_{\mathit{t}}\mathrm{=}{\mathit{C}}_t\mathrm{=}{\mathit{Y}}_t\mathrm{-}\frac{\mathrm{1}}{\mathit{T}}\underset{\tau =0}{\overset{T}{\mathrm{\Sigma }}}{Y}_{\tau }\mathrm{.}$(3.4.4)

According to this formulation, the average propensity to save from the current income is:

$\mathit{A}\mathit{P}\mathit{S}\mathrm{=}\frac{{{\mathit{S}}_{}}_t}{{\mathit{Y}}_t}\mathrm{=}\mathrm{1}\mathrm{-}\frac{1}{T}\frac{{\mathrm{\Sigma }}_{\tau =0}^T{\mathit{Y}}_{\tau }}{{{\mathit{Y}}_t}_{}}\mathrm{.}$

Comparing the APS in the permanent income hypothesis and the APS, s̄(t), in the OSG model, we conclude that the basic mechanism of determining consumption is similar - if the life time average income has a similar impact on consumption to the impact by the current assets.

3.5 The Ramsey Growth Model

There are two main modeling frameworks in the neoclassical growth theory, the Solow model and the Ramsey growth model, with capital accumulation. The main difference between the two approaches is related to consumers' behavior. The Solow model introduces a plausible consumption function with some empirical support. The other modeling strategy is to imagine the economy to be populated by a single immortal representative household that optimizes its consumption plans over infinite time in the sort of institutional environment that will translate its wishes into actual resources allocation at any point of time. We now introduce this framework initiated by Ramsey in 1928, which was proposed and developed as a story about centralized economic planning. The Ramsey model was refined by Cass in 1965 and Koopmans in 1965.¹⁵

We assume a one-sector economy in which 1 unit of output can be used to generate 1 unit of household consumption, or 1 unit of additional capital. Each household consists of one or more adults who are employed in the competitive labor market and receive wages for providing labor services. A household is imagined as an immortal extended family. The households receive interest income on assets, purchase goods for consumption and save by accumulating additional assets. Each household maximizes utility and incorporates a budget constraint over an infinite horizon. Denote C(t) the total consumption at time t and c(t) ≡ C(t)/N(t) is consumption per worker. It is assumed that the labor market clears at any point of time and each adult supplies 1 unit of labor services per unit of time. Households take the net rate of return r(t) on assets and the wage rate w(t) paid per unit of labor services as given in the competitive markets.

Most aspects of the Ramsey model are similar to the Solow model. The variables, Y(t), K(t), N(t), k(t), w(t), r(t), in the Ramsey model have the same meanings as in the Solow and the OSG model. The production process, marginal conditions, population growth are the same as in the OSG model. Different from the Solow model which assumes the agents regularly set aside some fairly predictable portion of its output for the purpose of capital accumulation, the Ramsey model endogenously determines saving and consumption. The firms maximize the present value of profits. Since firms rent capital and labor services and has no adjustment costs, there are no intertemporal elements in the firm's optimal behavior. Maximizing the profit with K and N as variables yields the same conditions as in equation (2.1.2).

We now describe households' behavior. The extended family is assumed to grow at an exogenously given rate n. Let the number of adults at time 0 be unity, the family size at time t is N(t)=e^nt. Each member supplies one unit of labor per unit time, without disutility. The household's preferences are expressed by an instantaneous utility function u(c(t)), where c(t) is the flow of consumption per person, and a discount rate for utility, denoted by ρ. For simplicity, specify u(c) as:

$\mathit{u}\mathrm{(}\mathit{c}\mathrm{)}\mathrm{=}\frac{\mathit{c}\mathrm{(}\mathrm{t}{\mathrm{)}}^{1-\theta }\mathrm{-}\mathrm{1}}{\mathrm{1}\mathrm{-}\mathit{\theta }},\theta \mathrm{>}\mathrm{0}\mathrm{,}$(3.5.1)

where θ is a parameter, u′ > 0, u″ < 0, and u satisfies the Inada conditions: u′ → ∞ as c → 0 and u′ → 0 as c → ∞.

Assume that each household maximizes utility U as given by:

$\mathit{U}\mathrm{=}\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{{\displaystyle \int }}}\mathit{u}\mathrm{(}\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}{\mathit{e}}^{nt}{\mathit{e}}^{-\rho t}\mathit{d}\mathit{t},\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{\ge }\mathrm{0}\mathrm{,}\mathit{t}\mathrm{\ge }\mathrm{0.}$(3.5.2)

The household makes the decision subject to a lifetime budget constraint. We denote the net assets per household by k(t), which is measured in units of consumables. The total income at each point of time is equal to w + rk. The flow budget constraint for the household is:

$\dot{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{w}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{r}\mathrm{(}\mathit{t}\mathrm{)}\mathit{k}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{-}\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{n}\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{n}\mathit{k}\mathrm{(}t\mathrm{)}\mathrm{.}$(3.5.3)

The equation means that the change rate of assets per person is equal to per capita income minus per capita consumption and the term, nk. It is assumed that the credit market imposes a constraint of borrowing, the present value of assets must be asymptotically nonnegative, that is:

$\underset{t\to \infty }{\lim }\mathrm{\left[}\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}\exp \right\{-\underset{\mathrm{0}}{\overset{}{{\displaystyle \stackrel{t}{\int }}}}\mathrm{(}\mathit{\rho }\mathrm{-}\mathit{n}\mathrm{)}\mathit{d}\mathit{v}\mathrm{\left\}}\mathrm{\right]}\mathrm{\ge }\mathrm{0.}$(3.5.4)

The present-value Hamiltonian is given by:

$J\mathrm{=}\mathit{u}\mathrm{(}\mathit{c}\mathrm{)}{\mathit{e}}^{-(\rho -n)t}\mathrm{+}\overline{\mathit{\lambda }}\mathrm{(}\mathit{w}\mathrm{+}\mathit{r}\mathit{k}\mathrm{-}\mathit{c}\mathrm{-}\mathit{n}\mathit{k}\mathrm{)},$(3.5.5)

where λ̄ is the present-value shadow price of income.

The first-order conditions are:

$$\begin{gathered} \frac{\overline{o}J}{\overline{o}\mathit{c}}\mathrm{=}\mathrm{0}\Rightarrow \overline{\mathit{\lambda }}\mathrm{=}\mathit{u}\prime {\mathit{e}}^{-\mathrm{(}\mathit{\rho }-\mathit{n}\mathrm{)}t} \\ \frac{\mathit{d}\lambda }{\mathit{d}\mathit{t}}\mathrm{=}\mathrm{-}\frac{\mathrm{\partial }J}{\mathrm{\partial }\mathit{k}}\Rightarrow \frac{\mathit{d}\mathit{\lambda }}{\mathit{d}\mathit{t}}\mathrm{=}\mathrm{-}\mathrm{(}\mathit{\rho }\mathrm{-}\mathit{n}\mathrm{)}\overline{\mathit{\lambda }}\mathrm{.} \end{gathered}$$(3.5.6)

The transversality condition is given by lim_t→∞ [λ(t)k(t)] = 0. By equation (3.5.6), we can derive:

$\mathit{r}\mathrm{=}\mathit{\rho }\mathrm{-}\frac{u"\mathit{c}}{{\mathit{u}\prime }^{}}\left(\frac{1}{c}\frac{\mathit{d}\mathit{c}}{\mathit{d}\mathit{t}}\right)\mathrm{.}$(3.5.7)

This equation says that households choose consumption so as to equate the rate of return r to the rate of time preference ρ plus the rate of decrease of the marginal utility of consumption u′ due to growing per capita consumption c. Inserting equation (3.5.1) into equation (3.5.7) yields:

$\dot{\mathit{c}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\frac{\mathit{r}\mathrm{(}t\mathrm{)}\mathrm{-}\mathit{\rho }}{\theta }\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\frac{{\mathit{f}\prime }^{}\mathrm{(}\mathit{k}\mathrm{)}\mathrm{-}\mathit{\rho }}{\theta }\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}$(3.5.8)

The trajectoiy of the economy is determined by equations (3.5.3) and (3.5.8). The phase diagram in c(t) and k(t) is as shown in Figure 3.3. Along the vertical line defined by:

$\mathit{f}\mathrm{\prime }\mathrm{(}\mathit{k}*\mathrm{)}\mathrm{=}\mathit{\rho },$

the change rate of the consumption per capita is equal to zero, i.e., ċ(t) = 0. The consumption per capita increases to the left of the curve and falls to the right. Along the locus defined by:

$\mathit{c}\mathrm{(}t\mathrm{)}\mathrm{=}\mathit{f}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{-}\mathit{n}\mathit{k}\mathrm{(}t\mathrm{)},$

the change rate of the capital-labor ratio equals zero. The capital-labor ratio falls above the curve and increases below it. With the requirement ρ > n (without which the utility becomes unbounded along feasible paths), the intersection of the two curves determines a unique steady state, (k*, c*).

Local stability of the c-k system is determined by the characteristic roots of the following matrix of the coefficients of k(t) and c(t) equations linearized around the equilibrium point:

$\mathit{J}\mathrm{\equiv }\left[\begin{array}{cc}\frac{\mathrm{\partial }\dot{\mathit{k}}}{\mathrm{\partial }\mathit{k}}& \frac{\mathrm{\partial }\dot{\mathit{k}}}{\mathrm{\partial }c}\\ \frac{\mathrm{\partial }\dot{c}}{\mathrm{\partial }k}& \frac{\mathrm{\partial }\dot{c}}{\mathrm{\partial }c}\end{array}\right]\mathrm{=}\left[\begin{array}{cc}f\prime -n& -1\\ \frac{f"c}{\theta }& 0\end{array}\right]$(3.5.9)

We have:

$$\begin{gathered} \mathrm{t}\mathrm{r}\mathrm{(}\mathit{J}\mathrm{)}\mathrm{=}\mathit{f}\prime \mathrm{-}\mathit{n}\mathrm{=}\mathit{\rho }\mathrm{-}\mathit{n}\mathrm{>}\mathrm{0}, \\ \mathrm{\left|}\mathit{J}\mathrm{\right|}\mathrm{=}\frac{\mathit{f}"\mathit{c}}{\mathit{\theta }}\mathrm{<}\mathrm{0.} \end{gathered}$$(3.5.10)

Therefore, the characteristic equation is:

${\mathit{\varphi }}^2\mathrm{-}\mathrm{t}\mathrm{r}\mathrm{(}J\mathrm{)}\mathit{\varphi }\mathrm{+}\mathrm{\left|}\mathit{J}\mathrm{\right|}\mathrm{=}\mathrm{0.}$

Inserting equation (3.5.10) into the above equation yields:

${\mathit{\varphi }}^2\mathrm{-}\mathrm{(}\mathit{\rho }\mathrm{-}\mathit{n}\mathrm{)}\varphi \mathrm{+}\frac{{\mathit{f}"}^{}\mathit{c}}{\mathit{\theta }}\mathrm{=}\mathrm{0}\Rightarrow {\mathit{\varphi }}_{1,2}\mathrm{=}\frac{\mathrm{(}\mathit{\rho }\mathrm{-}\mathit{n}\mathrm{)}\mathrm{\pm }\sqrt{{\mathrm{(}\mathit{p}\mathrm{-}\mathit{n}\mathrm{)}}^2\mathrm{-}\mathrm{4}\mathit{f}"\mathit{c}\mathrm{/}\mathit{\theta }}}{\mathrm{2}}$(3.5.11)

The characteristic roots are real and opposite in sign. The equilibrium point is a saddle point.

Equation (3.5.8) implies that the difference between r and ρ determines whether households choose a pattern of per capita consumption that rises, stays constant or falls over time. That is, the optimizing household has increasing, stationary, or decreasing consumption according as the current real interest rate exceeds, equals, or falls short of the utility discount rate. According to this result, consumption always falls if the interest rate is low and the utility discount rate is high. 'There is, however, strong empirical evidence that for most of us, the marginal rate of time preference exceeds the rate of interest, often by a large margin ... Kurz found empirically measured subjective interest rates as high as 60 percent.'¹⁶

The Ramsey model is controlled by a system of two differential equations. Together with the initial conditions and the transversality condition, this system determines the path of the two variables. At stationary state, the per capita variables, k, c and y( ≡ Y/N), grow at the rate 0, and the level variables, K, C and Y, grow at the rate, n. It can be shown that the system has a unique steady state. Since the two eigenvalues have the opposite signs, the system is locally saddle-path stable.¹⁷

Since we accept the OSG model in modeling consumer behavior, it is necessary to discuss advantages and disadvantages of the two approaches.

With regard to the Ramsey approach, Solow pointed out:¹⁸ 'These formulations all allocate current output between consumption and investment according to a more or less mechanical rule. The rule usually has an economic interpretation, and possibly some robust empirical validity, but it lacks "microfoundations". The current fashion is to derive the consumption-investment decision from the decentralized behavior of intertemporal-utility-maximizing households and perfectly competitive profit maximizing firms. This is not without cost. The economy has to be populated by a fixed number of identical immortal households, each endowed with perfect foresight over the infinite future. No market imperfections can be allowed on the side of firms.' The Ramsey modeling framework is analytically far more complicated than the OSG model - it is partially due to this analytical complex that growth theory is not integrated. Any concrete problem results in a complicated analytical problem beyond being thoroughly analyzed. For instance, the OSG model is one-dimensional, while the Ramsey model is two-dimensional. From operational point of view it is more convenient to use the OSG formulation. The equilibrium of the Ramsey is a saddle point for the two-dimensional differential equations. If the initial values of the two variables are located anywhere but on the saddle path, the resulting trajectory is nonoptimal for the household (or else ultimately infeasible). The appropriate path for this economy is along the saddle path, which leads asymptotically to the equilibrium point.

The form of utility formulation in the Ramsey optimal growth theory is given by ${\int }_0^{\infty }$ U[C(t)]e^-ρt. The specified form means that the household's utility at time 0 is a weighted sum of all future flows of utility. The parameter, ρ (≥ 0), is defined as the rate of time preference. A positive value of ρ means that utilities are valued less the later they are received. There are two assumptions involved in the Ramsey model. The first is that utility is additional over time. Although we may add capital over time, it is unrealistic to add utility over infinite time. Intuitively it is not reasonable to add happiness over time. It is well known in utility theory that when we use utility function to describe consumer behavior an arbitrary increasing transformation of the function would result in identical maximization of the consumer at each point of time. Obviously, the above formulation will not result in an identical behavior if U is subjected to arbitrarily different increasing transformations at different time. The second implication of the above formation is that the parameter ρ is meaningless if utility is not additional over times. It is obvious that the OSG model avoids these two issues. The OSG model is capable of taking account of social and cultural factors, which affect saving behavior by changing preference parameters.

It should be remarked that Ramsey interpreted the agent as a social planner, rather than a household. The planner chose consumption and saving for the current and future generations. Ramsey assumed ρ = 0 and considered ρ > 0 'ethically indefensible'. If ρ = 0, by equation (3.5.8) consumption per capita always grows if the interest rate is positive irrespective of whether wealth grows or falls.

The OSG model determines consumption as follows:

$\mathit{C}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\xi }\mathrm{(}\mathit{F}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\mathit{\delta }\mathit{K}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{)}\mathrm{.}$

Dividing the two sides of this equation by N(t) changes the generalized Keynesian consumption in per capita terms:

$\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\xi }\mathrm{(}\mathit{f}\mathrm{(}\mathit{k}\mathrm{)}\mathrm{+}\mathit{\delta }\mathit{k}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{)}\mathrm{.}$(3.5.12)

Taking the derivatives of equation (3.5.12) with respect to t yields:

$\dot{\mathit{c}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathrm{(}\mathit{r}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\mathit{\delta })\xi \dot{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$(3.5.13)

The change rate of consumption is related to the rate of interest; but the direction is not affected by the rate of interest. The rational household has increasing, stationary, or decreasing consumption according as the wealth rises, is stationary, or falls. The consumer adapts consumption level not according to the difference between the interest rate and discount rate for utility as the Ramsey model predicts. According to the OSG model theory, a Japanese consumer would consume more, irrespective of low interest, if his wealth increases; he would consume less, irrespective of high interest rate, when his wealth falls.

3.6 The Ramsey Model and the OSG Model

The Ramsey model has played the key role in describing consumers' behavior in growth theory. Unlike the Solow model that assumes that a consumer saves a fixed ratio of his income for future consumption, the Ramsey model considers that allocation of income between consumption and saving is determined by maximization of utility over lifetime. The OSG model takes a way between the two approaches. It should be mentioned that production side is identical in the three models. We have shown that the OSG model can generate the behavior of the Solow model by assuming a dynamics of the preference. It is reasonable to ask whether the OSG model can generate the same behavior as the Ramsey model under certain types of preference changes. We now show that the OSG model generates the same behavior as the Ramsey model.

The dynamical behavior of the Ramsey model is controlled by equations (3.5.3) and (3.5.8). If we find some equation of preference change in the OSG model to generate the same behavior as equation (3.5.8), then the two systems should exhibit the same behavior in terms of consumption, capital accumulation and incomes, even though they are built on different assumptions.

We now consider consumption of the OSG model. According to equation (3.5.12), with λ(t) as an endogenous variable and λ + ξ = 1, the consumption per capita in the OSG model is given by:

$c\mathrm{(}t\mathrm{)}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\left[\mathit{f}\mathrm{(}\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{+}\mathit{\delta }\mathit{k}\mathrm{(}t\mathrm{)}\right]\mathrm{.}$

Differentiation of this equation with respect to time yields:

$\frac{\dot{\mathit{c}}\mathrm{(}t\mathrm{)}}{c\mathrm{(}\mathit{t}\mathrm{)}}\mathrm{=}\frac{f\prime +\delta }{f\left(k\right(\delta \left)\right)}\mathrm{+}\dot{\mathit{k}}\mathrm{(}\mathit{t}{\mathrm{)}}^{}-\frac{\dot{\mathit{\lambda }}\mathrm{(}\mathit{t}\mathrm{)}}{\mathrm{1}\mathrm{-}\mathit{\lambda }\mathrm{(}\mathrm{t}\mathrm{)}}\mathrm{.}$(3.6.1)

For equations (3.6.1) and (3.5.8) to be equal, it is sufficient for λ(t) to evolve according to:

$\dot{\mathit{\lambda }}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\frac{\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\delta }}{\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\delta k\mathrm{(}\mathit{t}\mathrm{)}}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}\dot{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\frac{\mathit{f}\prime \mathrm{(}t\mathrm{)}\mathrm{-}\mathit{\rho }}{\mathit{\theta }}\mathit{\xi }\mathrm{(}t\mathrm{)}\mathrm{.}$(3.6.2)

The propensity λ to own wealth tends to rise (fall) when k rises (falls); it tends to rise (fall) when r < (>)ρ. We may interpret that the direction of change in λ is influenced by the direction of change in wealth as well as whether the rate of return of wealth is larger or smaller than the rate of time preference. If the wealth is increasing and the rate of time preference is larger than the rate of return, then the propensity to save will definitely rise. If the wealth is falling and the rate of time preference is smaller than the rate of return, the propensity tends to fall. In the other cases, the propensity may either increase or decrease.

Under equation (3.6.2), the consumption per capita in the OSG model evolves m the same way as in the Ramsey model. We now examine the fundamental equation (2.4.6), i.e.:

$\dot{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\lambda }\mathit{f}\mathrm{(}\mathit{k}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{)}\mathrm{-}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\lambda }\mathit{\delta }\mathrm{+}\mathit{n})\mathrm{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$

By c = (l - λ)(f + δk), the above equation can be rewritten as:

$\dot{\mathit{k}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{f}\mathrm{(}\mathit{k}\mathrm{(}t\mathrm{)}\mathrm{)}\mathrm{-}\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\left({\mathit{\delta }}_{\mathit{k}}\mathrm{+}\mathit{n}\right)k\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$(3.6.3)

Equation (3.6.3) for the OSG model is the same as equation (3.5.3) for the Ramsey model. Because the dynamics of the Ramsey model are controlled by equations (3.5.3) and (3.5.8), we conclude that the OSG model under equation (3.6.2) generates the same dynamics of k(t) and c(t) as the Ramsey model does.

Theorem 3.6.1

Let the production sectors be identical in the OSG model and the Ramsey model. If the propensity to save λ(t) evolves according to equation (3.6.2), then the OSG model generates the same dynamics of capital-labor ratio k(t) and per-capita consumption c(t) as the Ramsey model does.

This example illustrates how the Ramsey model is related to the OSG model. We can similarly examine relationships between the two approaches when utility functions are taken on other forms.

3.7 Poverty Traps Generated in the Solow Model

In the Solow model, the saving rate is fixed. We have many some directions to relax this assumption. In fact, as Solow pointed out, if one is only concerned with an endogenous saving rate,¹⁹ it is easy to generalize the traditional model. We may consider saving rate ŝ as a function of capital per capita ŝ(k). We may cite a special interpretation of endogenous saving rate from Solow. Suppose ŝ(k) is zero for low levels of k(t), and thereafter rises steeply toward the standard value ŝ. When people live on a survival level, they have almost nothing to save. For a low level of k(t) (which also means low levels of c(t) and y(t)), it is reasonable to assume the saving rate to be equal to zero. The modified phase for stationary states has two non-zero steady-state values of k(t), the larger of which is the same as in the Solow model. The smaller steady state is a sort of low-equilibrium trap.

Now, we illustrate an alternative way of making saving rates an endogenous variable. In a article on transitional dynamics in the neoclassical framework, King and Rebelo use a utility function in which there is a subsistence level of per capita consumption and the elasticity of intertemporal substitution varies over time.²⁰ The model has two equilibria - a Solow-type and an unstable steady state at the level of the capital stock comparable with subsistence consumption. In fact, the OSG model also has two equilibria - a Solow-type and the unstable origin (which corresponds to the unstable steady state in the King-Rebelo model). King and Rebelo assume that the saving rate is hump-shaped as a function of capital-labor ratio. This occurs because the elasticity of substitution rises as output increases. At low levels, saving rate rises as output increases. As the economy becomes mature, the saving rate falls. The model shows that even though the rate of interest falls as output increases, the growth rate of output does not diminish throughout the transition but has a hump-shaped path.

The dynamic processes described by the King-Rebelo model can be also found in the older development literature on poverty traps.²¹ We now illustrate this approach based on Ros.²² The model starts with a consumption function:

$\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathrm{(}{\mathit{y}}_{\mathrm{0}}\mathrm{-}{\mathit{\delta }}_{\mathit{k}}{\mathit{k}}_{\mathrm{0}}\mathrm{)}\mathrm{+}\overline{\mathit{\xi }}\mathrm{(}\mathit{y}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}{\mathit{c}}_{\mathrm{0}}\mathrm{)},$

where c(t) and y(t) are respectively per capita consumption and per capita income at time t, y₀ is subsistence income per capita, k₀ is the capital-labor ratio consistent with a subsistence level of income, δ_k the depreciation rate of capital, and ξ̄ (0 < ξ̄ < 1) is the propensity to consume out of nonsubsistence income. When income per worker is at the subsistence level, saving is equal to the depreciation of capital stock, i.e.:

$y_0-c={\delta }_k{k}_0\mathrm{.}$

The corresponding saving rate is:

$\hat{s}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\frac{\mathit{y}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathit{c}\mathrm{(}t\mathrm{)}}{\mathit{y}\mathrm{(}\mathit{t}\mathrm{)}}\mathrm{=}{\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}\mathrm{-}\frac{\mathit{\delta }*}{\mathit{y}\mathrm{(}\mathit{t}\mathrm{)}},$(3.7.1)

where:

${\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}\mathrm{\equiv }\mathrm{1}\mathrm{-}\overline{\mathit{\xi }}\mathrm{>}\mathrm{0}\mathrm{,}\mathit{\delta }\mathrm{\cdot }\mathrm{\equiv }{\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}{\mathit{y}}_{\mathrm{0}}\mathrm{-}{\mathit{\delta }}_{\mathit{k}}{\mathit{k}}_{\mathrm{0}},$

where ŝ₀ is the propensity to save out of nonsubsistence income. According to the definitions, the parameter δ* may be either positive or negative. If ŝ₀y₀ > δ_kk₀, δ* is positive, implying that as income rises, the saving rate rises. If ŝ₀y₀ < δ_kk₀, δ* is negative, implying that as income rises, the saving rate falls. The saving rate is a nonlinear function of the level of income per capita. The saving rate rises with income per capita if the marginal propensity to consume out of nonsubsistence income is less than the average propensity to consume out of subsistence income. Otherwise, the saving rate tends to fall as income rises above the subsistence level. Substituting equation (3.7.1) into the fundamental equation (3.2.2) in the Solow model yields:

$\dot{\mathit{k}}\mathrm{=}{\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}\mathit{f}\mathrm{(}\mathit{k}\mathrm{)}\mathrm{-}\left(\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}\right)k\mathrm{-}\mathit{\delta }*\mathrm{.}$(3.7.2)

We call the above model the generalized Solow model with poverty traps. We see that if δ* = 0, the model has the same structure as the Solow model. There are two steady states - an unstable origin and a stable equilibrium. Unlike the Solow model - where there is a unique value of capital-labor ratio, this model can yield two equilibria as shown in Figure 3.4 when δ* > 0.

In the case of δ* > 0, the equilibrium at the high-k level is similar to the steady state in the Solow model. The other is at the subsistence level of income. This is often referred to as poverty trap, but it is a fortune one in the sense that it is unstable. The economic system will not stay there long. Poverty may not be persistent under proper disturbances from, for instance, foreign aids and trade. This trap occurs because at low levels of capital-labor ratio income per capita is scarcely sufficient and savings fall below depreciation. If an economy is in this type of poverty, it is easy to start rapid development because once capital-labor ratio is higher than the level k₀, it will grow fast towards the stable equilibrium. Once it reaches at the high level of living standard, it is trapped because this is a stable state. As observed by Solow,²³ this model is actually not interesting for explaining poverty trap when it has two equilibria. We will show that a steady state with low level of capital-labor ratio is stable; while that with a higher level of capital-labor ratio is unstable. Here, it should be remarked that according to the traditional interpretations possibility of poverty persistence in this model is not due to internal economic mechanism but international trade. If other economies offer higher profits than poor economies, then poor countries cannot obtain capital sufficient to start rapid growth. To understand the world as a whole, it is necessary to extend our analytical framework to multiple countries and multiple regions.

The model has only one equilibrium - an unstable poverty when δ* < 0, i.e., ŝ₀y₀ < δ_kk₀. This situation occurs that the saving rate is too low or the subsistence income per capita is too low. If an economy is characterized by political and social instabilities, then it tends to have a low saving rate. Under such circumstances, the nation can hardly make any progress in economic development.

As an exercise, we now show that existence of multiple equilibria can also be generated by the OSG model, we now consider the propensity to own wealth as an endogenous variable. For the saving rate given by equation (3.7.1) to equal the saving rate given by equation (2.3.5), it is sufficient for the propensity to own wealth λ(t) to be taken on the following form:

$\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathrm{1}\mathrm{-}\frac{\overline{\mathit{\xi }}\mathit{y}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\delta }*}{\mathit{y}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\gamma k\mathrm{(}\mathit{t}\mathrm{)}}\mathrm{.}$(3.7.3)

Here, we use λ(t) + ξ(t) = 1. We assume δ* > 0. The propensity to hold wealth λ(t) increases as capital-labor ratio rises. The impact in change of income on λ(t) is given by

$\frac{\mathrm{\partial }\mathit{\lambda }\mathrm{(}\mathrm{t}\mathrm{)}}{\mathrm{\partial }\mathit{y}\mathrm{(}\mathit{t}\mathrm{)}}\mathrm{=}\frac{\delta *-\overline{\xi }\delta k\left(t\right)}{\left(y\right(t)+\delta k(t){)}^2}\mathrm{.}$

If δ* < 0, an increase in the current income reduces the propensity to own wealth. If δ* > 0, when the economy has low capital-labor ratio, an increase in the current income raises the propensity to own wealth. When capital-labor ratio becomes high, an increase in the current income reduces the propensity to own wealth. Under equation (3.7.3), s̄(t) = ŝ(t) for any values of k(t) and y(t). Inserting equation (3.7.3) into equation (2.4.6) yields:

$\dot{\mathit{k}}\mathrm{=}(y+\delta k)\lambda \mathrm{-}\left(\mathrm{1}\mathrm{+}\mathit{n}\right)k\mathrm{=}{\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}\mathit{f}\mathrm{(}\mathit{k}\mathrm{)}\mathrm{-}\left(\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}\right)k\mathrm{-}\mathit{\delta }*,$(3.7.4)

where we apply the definitions of ŝ₀ and δ. We see that under equation (3.7.3), the fundamental equation (3.7.4) for the OSG model is identical to equation (3.7.2) for the generalized Solow model with poverty traps. Consequently, the OSG model generates the same behavior as in Figures 3.4 and 3.5.

Theorem 3.7.1

Let the production sectors be identical in the OSG model and the generalized Solow model with poverty traps. If the saving rate ŝ in equation (3.7.1) and the propensity to save λ(t) in the OSG model satisfy equation (3.7.3), then the OSG model is identical to the generalized Solow model with poverty traps.

3.8 On the Utility Function in the OSG Model

This chapter examined relations of the OSG model with some of traditional growth models. We circumscribed our discussion to one-sector economies. Our comparison has to be narrow and partial because of the complexity of the literature. As concluding remarks on relations between the OSG model and consumer theory, we provide some insights into utility functions accepted in the OSG model. We consider the preference structure rather than given utility functions as the starting point of our analysis of consumer behavior. We show that it is possible to specify manifested forms of consumers' preference structures by utility functions.²⁴

We assume that at any point of time the consumer has preferences over alternative bundles of commodities which can be divided into goods, services, and time distribution of the consumer. The behavioral rule consists of maximization of these preferences under budgets restrictions of finance, or time, or human capital, or energy.

A commodity is characterized by its location, date at which it is available and its price. At each point of time, the consumer is faced with a commodity bundle consisting of (finite) real numbers:

$\mathrm{\left\{}{\mathit{x}}_{\mathit{j}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{\right\}},\mathit{j}\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{.}\mathrm{.}\mathrm{,}\mathit{m}\mathrm{,}$

indicating the quality of each commodity. The commodity space consists of commodity bundles. Here, we omit issues related to spatial location. Let us denote the price of commodity j by p_j(t). For simplicity, we omit time index of x and p except in some circumstances. Both commodity vector x, and price vector p, can be represented by points in Euclidean space R^m, i.e., x ∈ R^m and p ∈ R^m. The value of the commodity bundle at any point of time is given by:

$\mathit{p}\mathrm{(}t\mathrm{)}\mathit{x}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\underset{j}{\mathrm{\Sigma }}{\mathit{p}}_j\mathrm{(}\mathit{t}\mathrm{)}{\mathit{x}}_j\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{.}$

Obviously, some bundles of the commodity space may be excluded as consumption possibilities. The consumption set, denoted by X, consists of all consumption bundles, which are possible. Let us assume that the consumer's choices are restricted by the fact that the value of his consumption should not exceed his income w(t), at each point of time. The budget set:

$\mathit{\beta }\mathrm{(}\mathit{p}\mathrm{,}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\mathrm{\equiv }\mathrm{\{}\mathit{x}\mathrm{\in }\mathit{X}\mathrm{|}\mathit{p}\mathit{x}\mathrm{\le }\mathit{w}\mathrm{\}},$

is the set of possible consumption bundles whose value does not exceed the income.

The consumer has tastes and desires. They are important in analyzing why the consumer chooses a bundle from the consumption set. Mathematically, we represent the preference structure by the consumer's preference relation, ⪰_t, at each point of time which is a binary relation on X. For any two bundles x(t) and y(t), x ∈ X and y ∈ X, ⪰ y means that x is at least as good as y at time t. Before discussing the relation between the preference relation and utility functions, we introduce the following axioms.

Axiom 1 (Reflexibility)

For all x ∈ X, x ⪰ x, i.e., any bundle is as good as itself.

Axiom 2 (Transitivity)

For any three bundles, x, y, z in X such that x ⪰ y and y ⪰ z it is true that x ⪰ z.

Axiom 3 (Completeness)

For any two bundles x and y in X, x ⪰ y or y ⪰ z.

Axiom 4 (Continuity)

For every x ∈ X the upper contour set {y ∈ X | x ⪰ y} and the lower contour set {y ∈ X | x ⪰ y} are closed relative to X.

A preference relation ⪰ which satisfies the first three axioms is a complete preordering on X and is called a preference order. A bundle x is said to be strictly preferred to a bundle y, i.e., x ≻ y iff x ⪰ y and not y ⪰ x. A bundle x is said to be indifferent to a bundle y, i.e., x ∼ y iff x ⪰ y and y ⪰ x. The indifference relation defines an equivalent relation on X, i.e., ∼ is reflexible, symmetric, and transitive. We always assume that X includes at least two bundles x′ and x″ such that x′ ≻ y″. In order to solve the problem of the representability of a preference relation by a numerical function, we introduce the concept of utility function.

Definition 3.9.1

Let X denote a set and ⪰_t a binary relation on X at time t. Then a function u from X into real R is a representation of ⪰, i.e., a utility function for the preference relation ⪰, if, for any two points x and y, u_t(x) ≥ u_t(y) iff x ર_t y at point of time t.

It seems that Pareto was the first to recognize that arbitrary increasing transformation of a given function would result in identical maximization of a consumer. From the above definition we see that for any utility function u_t and any increasing transformation f : R → R the function v_t = f ○ u_t is also a utility function for the same preference relation ⪰. The following theorem is referred to Debreu or Rader.²⁵

Theorem 3.9.1

Let X denote a topological space with a countable base of open sets and ⪰ a continuous preference order defined on X, i.e., a preference relation that satisfies Axioms 1-4. Then there exists a continuous function u.

The above theorem shows that under certain conditions the concepts of utility and of the underlying preferences may be used interchangeably to determine demand at any point of time. The above theorem is referred to any point of time. It does not involve how to calculate the future. When applying to our cases, we assume that a consumer is to choose consumption C(t) and saving S(t) with the disposable personal income Ŷ(t) at each point of time t.

Appendix

A.3.1 The Golden Rule of Capital Accumulation in the Solow Model

We now introduce Phelps' golden rule in the Solow model.²⁶ Let us consider the Solow model defined in Section 3.2. We know that for the given production function f(k) and given values of n and δ_k, there is a unique equilibrium value of per capita capital k* (> 0) for each value of the exogenous saving rate, ŝ, given by:

$\frac{\mathit{f}\mathrm{(}{\mathit{k}}^{*}\mathrm{)}}{{\mathit{k}}^{*}}\mathrm{=}\frac{\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}}{\stackrel{\mathrm{ˆ}}{\mathit{s}}}\mathrm{.}$(3.A.1.1)

We now consider k* a function of the saving rate, k*(ŝ). We know that a rise in the saving rate increases the steady-state level of per capita capital. The steady-state per capita consumption is:

${\mathit{c}}^{*}\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathit{f}\left({\mathit{k}}^{*}\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\right)\mathrm{.}$

From equation (3.A.1.1), we rewrite the above equation:

${\mathit{c}}^{*}\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathrm{=}\mathit{f}\mathrm{(}{\mathit{k}}^{*}\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathrm{)}\mathrm{-}\left(\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}\right)k^{*}\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathrm{.}$

To maximize c*(ŝ) with the saving rate as decision variable, we calculate:

$\frac{\mathit{d}\mathit{c}*}{d\hat{s}}\mathrm{=}\mathrm{\lceil }\mathit{f}\prime \mathrm{(}\mathit{k}*\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathrm{)}\mathrm{-}\mathrm{(}\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}\mathrm{)}]\frac{ds*}{d\hat{s}}\mathrm{.}$(3.A.1.2)

The first-order condition is:

${\mathit{f}\prime }^{}\mathrm{(}\mathit{k}*\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\mathrm{)}\mathrm{=}\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}\mathrm{.}$

This equation has a unique solution k*. We denote the value of k* that satisfies the first-order condition by k_g. Differentiation of equation (3.A.1.2) with respect to ŝ gives the following second-order condition:

$\frac{{\mathit{d}}^{\mathrm{2}}{\mathit{c}}^{*}}{{\mathit{d}}^{\mathrm{2}}\stackrel{\mathrm{ˆ}}{\mathit{s}}}\mathrm{=}{\mathit{f}"}^{}\mathrm{(}{\mathit{k}}^{*}\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}{\left(\frac{{\mathit{d}\mathit{k}}^{*}}{\mathit{d}\stackrel{\mathrm{ˆ}}{\mathit{s}}}\right)}^2\mathrm{<}\mathrm{0}\mathrm{,}\text{at }{\mathit{k}}^{*}\mathrm{=}{\mathit{k}}_{\mathit{g}}\mathrm{.}$

We conclude that at k_g, the per capita consumption is maximal. Let ŝ_g denote the level of the saving rate that corresponds to k_g. From the above discussions, we see that the equation: c* = f - (n + δ_k)k*, implies that the quantity c* increases in ŝ for low levels of ŝ and decreases in ŝ for high levels of values of ŝ; it achieves maximum at ŝ = ŝ_g . The condition:

${\mathit{f}}^{}\prime \left({\mathit{k}}^{*}\mathrm{(}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\right)\mathrm{=}\mathit{n}\mathrm{+}{\mathit{\delta }}_{\mathit{k}}\mathrm{,}$

is called the golden rule of capital accumulation. Figure 3.A.1 depicts the golden rule.

Figure 3.A.2 depicts the implications of the golden rule. The figure illustrates three possible saving rates, ŝ₁, ŝ_g, and ŝ₂ with ŝ_l< ŝ_g < ŝ₂. Per capita consumption, c, in each case equals the vertical distance between the production function, f(k), and the corresponding saving, ŝ_if(k), i = 1, g, 2. For each s_i the steady-state value of per capita capital is determined at the intersection between ŝ_if(k) and (n + δ_k)k line. The steady-state per capita consumption, c_g, is maximized when k_g. The first-order condition for maximization is satisfied because the tangent to the production function at this point parallels the (n + δ_kk line. Corresponding to ŝ₁, and ŝ₂, we have two equilibria k₁* and k₂*. As shown in the figure, we have: k₁* < k_g < k₂*.

Since the Solow model does not have an index for measuring welfare of consumers, the question of whether some saving rates are better than others remains to be examined. We now show that if ŝ > ŝ_g, then the economy is over saving in the sense that per capita consumption could be raised at 'all points in time' by lowering the saving rate. To show this, let us consider an economy at steady state k₂* with a saving rate ŝ₂ (> ŝ_g). Evidently, c₂* < c_g. Imagine that, starting from the steady state, we reduce the saving rate from ŝ₂ to ŝ_g. As seen from Figure 3.A.2, per capita consumption, c, initially increases and then falls during the transition toward its new steady-state value, c_g. As c₂* < c_g, per capita consumption exceeds its previous value c₂* at any time during the transition.

We thus conclude that if ŝ > ŝ_g, then the economy is over saving. It can be seen from Figure 3.A.2 that if ŝ < ŝ_g, raising the saving rate would reduce per capita consumption currently and during part of the transition period, but finally increase per capita consumption. We thus conclude that the outcome is ambiguous if consumers don't specify how they discount the future.

¹ The model revised here follows Chiang's description (1984: 465-469). The original articles refer to Domar (1946) and Harrod (1948).

² A recent comparison is given by Solow (2000).

³ Solow (2000: x).

⁴ Solow (2000: xi).

⁵ Solow (1956) and Swan (1956).

⁶ Solow (1999).

⁷ Maital and Maital (1994).

⁸ Solow (2000: x).

⁹ Modigliani and Brumberg (1954) and Ando and Modigliani (1963). See also Modigliani (1966). A related theory called the permanent income hypothesis is referred to Friedman (1956, 1957).

¹⁰ This section is referred to based on Froyen (1999:282-286).

¹¹ Modigliani (1966).

¹² Ando and Modigliani (1963).

¹³ Friedman (1957).

¹⁴ Romer (1996).

¹⁵ Ramsey (1928), Cass (1965) and Koopmans (1965). For extensions of the model, see Cass and Shell (1976), Brock and Scheinkman (1976), Araujo and Scheinkman (1977), Magill (1977), and Dolmas (1996), Benhabib and Nishimura (1979), Zhang (1988, 1989).

¹⁶ Quote from Maital and Maital (1994).

¹⁷ We will not provide a complete analysis of the model. Refer to Takayama (1985) and Romer (1996) in detail.

¹⁸ Solow (1999: 646).

¹⁹ Solow (1999).

²⁰ King and Rebelo (1993).

²¹ See Nelson (1956) and Leibenstein (1957).

²² Ros (2000: 60-62).

²³ Solow (1999).

²⁴ Our following discussion on the relationship between the preference structure and utility function is referred to Barten and Bohm (1982).

²⁵ Debreu (1959) and Rader (1963).