CHAPTER 2
PRODUCTION AND THE THEORY OF THE FIRM
We will use the word production or productivity in a very general sense to denote all activities that satisfy consumer demand for goods and services. In the classic use of the term, production is associated with the use of natural resource inputs to produce finished products that are subsequently marketed to consumers through various service efforts. In the initial part of this book, we will be concerned with this classic interpretation. In the later part, we will also consider production of inherently information and knowledge intensive products, such as software. Production is therefore one of the basic components of microeconomic theory. A (classic) firm is a unit that uses natural resource inputs, capital, and labor to produce output goods or services for purchase by consumers. The economic problem faced by a production unit or a firm is that of determining the quantity of output to produce and the amounts of the various input factors to be used in the production process. This will depend, of course, on technological relationships between the prices or costs of input factors, or input supplies, and the price that can be obtained for the output quantity, which is a function of the demand for the product of the firm. This is a problem in resource allocation that involves
1. the technology of the production process or the production function,
2. the price, or equivalently costs, of the input resource quantities or input factors such as labor, natural resources, and capital, and
3. the price of and demand for the output quantity.
Figure 2.1 is a block diagram of the firm as an economic institution and it shows the role of the three elements above. To obtain an economic theory of the firm we will need to discuss each of these elements, which is the objective of this chapter.
Figure 2.1. Block Diagram of the Production Process.
We will use the term producer to denote a decision-making entity that converts commodity bundles or inputs, by means of production, into other commodity bundles or outputs. We will generally assume that the producer desires to maximize the profits of the firm. We will define profit to be the difference between the revenue that the firm obtains for the products it produces minus all of the costs of producing the products. The particular decision variables that are subject to adjustment by the producer depend, of course, on the type of production issue that is being considered. In the simplest problem, the decision involves setting the output level; in a slightly more complicated problem both the output level and the price may be set by the firm. This setting of the price by the firm violates assumptions concerning “perfect competition.” It introduces the need for imperfect competition—in this case monopoly—considerations.
We begin our discussion of production with the simplest case of a single-product firm. We will then briefly mention how the results of a single-product firm can be extended to include the multiproduct firm, where the same firm generally produces a number of goods.
We assume that our single-product firm produces an output from several inputs. We will use the symbol q to denote the (scalar) output quantity. We assume that xi is the ith input quantity to the production process. Thus xi represents the various input resources or input factors to the production process. These resources will generally be of three distinguishable types
1. raw materials or natural resources (traditionally often denoted by M),
2. capital, such as investment in production machinery (traditionally often denoted by K or C), and
3. labor (traditionally often denoted by L).
In many economic studies, particularly traditional ones, only an economic and technological valuation of these inputs is attempted. However, it is becoming increasingly recognized that psychological, social, and political valuations are of great importance. In our later chapters we will consider these to some extent, although a full development of a behavioral theory of the firm is, while very important, beyond the scope of this book.
The production function, traditionally denoted by f, is a specific mapping or technological relation between the input variables xi and the production process and the output quantity produced, denoted by q. The specific mapping chosen is that presumably unique number that represents the maximum output that can be produced for a given set of input quantities.
Example 2.1:
If we have two input factors x1 and x2 to the production process, then we say that the product output level or quantity of production q is, for the ith production technique Pii,
We define the following: the output product q is the number of suits of clothing manufactured; xi are the input factors, such that x1 is the number of hours of input labor to the manufacturing process and x2 the number of square meters of input fabric; and ai are the coefficients of production, such that a1 is the number of hours of labor needed to make a suit and a2 the number of square meters of fabric necessary to make a suit. Then we see that the output production level is the quantity x1/a1 or x2/a2, which is the smallest. We have for our specific production process or technique
This is a very restrictive production process. Normally it would be possible to trade off clothing for labor such that if labor costs rose, we could cut the amount of labor, say, by a less restrictive process, which could increase the amount of fabric used per suit. We might then obtain a production function like
Here we could maintain a constant amount of production by decreasing x1 and increasing x2. We would obtain this production function from a class of production techniques by finding the one that yields the maximum output for a given set of inputs. This would be of the form
We will generally assume a production function of the form
in which there are n production input factors defined by the n-vector
These factors are determined such that there is no wastage of the input production factors. We do this by defining the production function f such that the following two conditions are satisfied.
1. For any given fixed level of input factors, the technique of production that gives the maximum output production is selected.
2. At any given fixed output production level q, the consumption of each input factor is at a minimum.
We ensure condition 1 by the requirement that, for fixed x,
We accomplish this by selecting the best production technique or process. We ensure condition 2 by the further requirement that the output production must increase if we increase the resource input to or factors of production. Thus, we require that q1 ≥ q2, or in equivalent terminology f (x1) ≥ f (x2), whenever1 x1 ≥ x2.
The desirability of these nonwaste conditions is physically apparent. In most physical production processes it is possible that further technological progress may reduce the amount of input products required for a given output or improve production efficiency or reduce waste. Also, there may be constraints, such as those imposed by worker safety, that will not allow complete freedom in process selection. All of these can, should, and will be incorporated into our theory of production. All that we require by our nonwaste condition is that under the existing production constraints there be no waste production factors.
Example 2.2:
We return to our previous example, in which the production function for a particular technology was selected as
where, as before,
q = output production level
x1 = input factor of labor
x2 = input factor of fabric
a1 = labor coefficient of production
a2 = fabric coefficient of production
For this example, our nonwaste requirement says that we would be foolish to use more than a1q units of labor or a2q units of fabric, where q is determined from Equation 2.5 for a specific x1 and x2.
To expand on this discussion further, let us suppose that
a1 = 4 h of labor per suit
a2 = 3 m2 of fabric per suit.
We can now use Equation 2.5 to draw a production isoquant, or curve showing the values of x1 and x2 that produce a constant output q. We do this for several constant values of q and easily obtain the set of isoquants shown in Fig. 2.2. Every value of x1, x2 that lies on one side of a given isoquant must necessarily result in a higher output production, whereas every value of x1, x2 on the other side must result in a lower production. The line q = x1/4 = x2/3 defines the minimum input factors for this production technique. Clearly, we should operate on this line to satisfy the nonwaste conditions.
Figure 2.2. Isoquants for Example 2.2, with 
.
Two important axioms are assumed to be satisfied by a production function. The first axiom is that there is an economic region in which a decreased output cannot result from an increased input. This axiom is entirely equivalent to and follows from condition 2 of noninput factor waste. This leads to an important quantity called the marginal productivity, MP(x), which is the first partial derivative or gradient of the production function. We have
An important feature of each component of the marginal productivity or marginal return that follows from condition 2 is that it is positive, or surely nonnegative to be more precise, in the economic region. We can easily illustrate this by any of several procedures. One particular approach that is very insightful is to consider a graph of the output quantity produced for various values of x1 with all the other factor inputs held constant. We will use the symbol 
to denote all the components of the vector x except xi. We may write, using this definition,
Now suppose that we fix the value of the vector at some particular value of . By increasing xi to xi + Δxi, for Δxi positive, we necessarily do not decrease the output q and thus have
The definition of marginal productivity is equivalent to
Thus, we immediately see that each component of the marginal productivity is always nonnegative, and generally always positive, in the economic region.
A quantity similar to the marginal productivity that is of importance is the average productivity or average return, defined by the expression
This average productivity, the total productivity with all the input factors but xi fixed divided by the input factor xi, must always be positive, regardless of whether we are in the economic region.
Our second production function axiom is that there exists a relevant economic region, denoted by Rp, for which the second derivative of the production function, or Hessian matrix, is negative definite. Thus, we have in this relevant economic region a matrix expression
that is negative definite.2 The production function q = f(x) is therefore strictly concave (for every nonnegative output q) in the relevant economic region.
In the relevant economic region for a production function, we can show that the main diagonal components of this Hessian matrix are negative, in that
for all i in the input factor set. This is physically a very significant result. It is a statement of the law of diminishing marginal productivity or the law of diminishing marginal returns. As we add more and more input xi to a process with all the other input factors, , fixed, we must eventually reach a point at which the marginal productivity for that input xi drops to zero and we reach the boundary of the relevant economic region.
We illustrate these concepts graphically in Fig. 2.3, which shows a typical production curve and the variation of production with a single input factor, as in Equation 2.7. Figure 2.3 also illustrates the marginal productivity MP1(x) and the average productivity AP1(x) for the hypothetical production function chosen. We note several interesting items concerning Fig. 2.3:
Figure 2.3. Typical Productivity (q), Marginal Productivity (MP), and Average Productivity (AP) Curves.
1. For small x1 factor inputs the marginal and average productivities increase with increasing factor input.
2. There is a maximum factor input 
, and if x1 is increased above this value, we move outside of the relevant economic region and production is decreased.
3. Above 
the marginal productivity is negative. This is another symptom of our being outside the relevant economic region.
4. For values of x1 between zero and 
the marginal productivity is greater than the average productivity. Further increases in x1 result in the marginal productivity becoming less than the average productivity. Often this will be undesirable. The value of x1 at which the maximum average productivity occurs is the value beyond which the marginal productivity is less than the average productivity.
5. There are three distinct productivity regions for this input factor x1:
a. a region in which the marginal productivity is greater than the average productivity,
b. a region in which the marginal productivity is positive but less than the average productivity, and
c. a region in which the marginal productivity is negative.
When all the factor inputs change by the same constant multiple, the concept of return to scale may be used to further characterize the production function. A production function is said to possess constant return to scale characteristics if an increase in the input factor x to ax produces a proportional output increase, that is, if
If a production function possesses constant return to scale characteristics, then a 50% increase in all the inputs will result in a 50% increase in output productivity.
A production function is said to possess decreasing return to scale characteristics if the output increases by a smaller proportion than does the input, that is, if
Similarly, a production function possesses increasing return to scale characteristics if changing all the inputs by a constant multiple a changes the output by more than this proportion, that is, if
Production functions may possess increasing returns to scale for some points in input factor space, constant returns to scale for other points in input factor space, and decreasing returns to scale for still other points in input factor space. Generally, for sufficiently large x, production of all realistic classical natural physical products will exhibit decreasing return to scale. In the very important case of goods strongly dependent on information, which we will examine in the later chapters, this is not the case and increasing returns to scale may well be present.
The case of constant return to scale is particularly important, as production functions with this property are mathematically convenient. Fortunately, a great many production functions do possess constant or nearly constant return to scale characteristics. Two things that will often prohibit constant return to scale are limited input factor supplies that cannot be greatly increased to accommodate greatly expanded outputs and the fact that some inputs only occur in integer units.
Example 2.3:
It is convenient to discuss return to scale in terms of the homogeneous production function of degree s for which
Here, a is greater than 1. When s > 1, we have increasing returns to scale; when s < 1, we have decreasing returns to scale; and when s = 1, we have the constant return to scale case. We may be tempted to conclude that a linear production function is the only one that has the constant return to scale property. It is the case that any linear function that has zero output for zero input
certainly does have constant return to scale, in that
However, the nonlinear production function
is also a homogeneous production function, in that
This is a linear homogeneous production function if b1 + b2 = 1. Thus we see that functions that are not linear may also possess constant return to scale.
Elasticity of production or production elasticity or output elasticity is a measure of the return to scale as a function of the input factor vector x. It is defined as the ratio of the marginal productivity to the average productivity with respect to a change in the ith input. This factor elasticity3 of production is written as
We can use the definitions of marginal and average productivity, Equations 2.6 and 2.8, such that the foregoing becomes
Factor elasticity has a number of interpretations. Perhaps the most interesting one is that it is the proportional variation in the output divided by the proportional variation in the factor input. Thus if xi increases by 2% and the output increases by 6%, the elasticity is 3. Of course, the definition we have used of elasticity, that is, the ratio of the marginal productivity to the average productivity, is important and has much physical meaning.
For the vector case, we can formally define the scalar elasticity as the inner or scalar product of the marginal productivity and the reciprocal average product vector defined by
We thus have for the elasticity
which becomes
Thus we see that the total elasticity is just the sum of the local factor input elasticities and given by
where εi(x) is defined by Equation 2.14.
Example 2.4:
We consider the linear production function of the form
The individual production factor elasticity is given by its fundamental definition, which is the ratio of the marginal productivity to the average productivity, as
and the total factor elasticity of production is
This is as we should expect, since the linear production function possesses constant return to scale and is a homogeneous function of degree 1.
For the homogeneous production function of degree bl + b2
we have for the factor elasticity
and we have for the total elasticity
The total elasticity is 1 if b1 + b2 = 1, and this is the requirement we had obtained in our previous example for constant return to scale.
The marginal rate of substitution or the rate of technical substitution of one input product for another is an important concept, as it allows insight into the alternate factor input possibilities that result in the same output. Along a specific isoquant of constant production qk, given by
we must have dqk = 0. We will constrain all factor inputs but xi and xj to be constant. Thus we have from the general relation along an isoquant
the relation
We rearrange this relation and obtain for the marginal rate of substitution along an isoquant
An interesting property of the marginal rate of substitution MRSij (x) is the fact that it is the ratio of the marginal productivities of factors i and j. This is of considerable value, as it allows us to determine the optimum economic production factor inputs.
We may also define an elasticity of substitution as the expression
This elasticity represents the proportional change in the ratio of the inputs i and j divided by the proportional change in the ratio of the marginal products or in the marginal rate of substitution. We will be able to show, using the results of the next section, that the elasticity of substitution is also the ratio of the change in the relative factor inputs to the associated change in their relative wages or prices. This is given by
Example 2.5:
For the linear production function
we obtain the marginal rate of substitution, Equation 2.20, from the marginal productivities MPi(x) = bi as
This relation tells us that to maintain a constant output production q we can substitute input products xi and xj according to the relationship
This relation is, of course, valid only for “small enough” changes. In Examples 2.3 and 2.4, we found that a linear production function has constant return to scale and a production elasticity of 1. In this example, we have found that the marginal rate of substitution MRSij (x) is bi/bj. Since this is a constant, the elasticity of substitution is infinite. Such a large elasticity means that even the smallest change in the rate of marginal substitution, which is constant everywhere, will produce an extraordinarily large change in the ratio of input products that determines the proportional mixes of the input factors.
For the homogeneous production function of degree b1 + b2,
we have for the marginal rate of substitution
which indicates that input factor substitutability does depend on the present factor input level.
The elasticity of substitution is obtained from Equation 2.21 as ε12 (x1, x2) = 1. The fact that the elasticity of substitution is 1 throughout the complete range of input factors is desirable, because it ensures reasonable substitutability of one input factor for another over a wide range of inputs. This particular production function is a special case of the Cobb–Douglas production function, which is frequently used in microeconomic analysis.
In this section, we have introduced the subject of production and the theory of the firm. A number of examples have been presented that deal with two frequently used types of production functions: the linear production function
and the Cobb–Douglas production function
There are many theoretical advantages to linear functions. Fortunately a function that is log linear, such as the Cobb–Douglas production function
has many of these advantages. For example, linear regression analysis may be used to estimate the bi coefficients in Equation 2.24 and we may use it to estimate the bi coefficients in Equation 2.22.
Figures 2.4 and 2.5, respectively, present curves of the linear and Cobb–Douglas production functions for the case of two factor inputs. Various characteristics of these two useful production functions are summarized in Table 2.1 for the general case.
Figure 2.4. Linear Production Function.
Figure 2.5. Cobb–Douglas Production Function.
TABLE 2.1. Properties of Linear and Cobb–Douglas Production Functions
A summary of the central results of this section, in the context of a profit maximization example, yields much that is of importance to our future efforts. We will suppose that there is a single factor input to production, perhaps labor. The production function is
We assume that the firm maximizes profits subject to constraints involving the production function and market realities concerning marketing products and purchasing inputs. The firm is assumed to be too small to affect the wages w of labor or the price of its product—it is a “price taker in all markets.”
We now examine some issues concerning profit of a firm. For simplicity, we will initially assume that profit is not a function of time or, equivalently, that we are considering a fixed and known time interval and all data are for that known time interval. Thus, the profit of the firm is given by the expression
Here the assumed variable is the amount of labor to use in the production process. To maximize profit we must have
and
The first-order condition is a necessary condition for optimality; it represents both the output supply function and the input factor–demand function. This second order condition is also very important, as it represents the sufficient condition for a maximum. From it we see that the marginal product (of labor in this case) must be decreasing at the optimum production level for maximum profit. This is the same as the requirement that the production function be concave in the region of maximum profit.
Example 2.6:
We can see the optimality requirements graphically and physically from examination of the first-order condition Equation 2.27, and a typical profit. This curve, given by Equation 2.26, is illustrated in Fig. 2.6 for the production function
Figure 2.6. Revenue and Cost Curves for Example 2.6.
with w = 45, p = 1.
We obtain two possible values of labor, L = 1 and 10, which cause the marginal profit dΠ/dL to be zero. But L = 1 is a point of minimum, and negative (Π = −21.75), profit. At L = 10 we obtain the true maximum profit of Π = 525.
The major conclusion that we obtain from this last example is that the production function must be concave at the (local) optimum where profit-maximizing behavior occurs. We also know that the maximum profit is nonnegative, since the firm can always obtain zero profit by doing nothing at all. We use these observations to obtain some interesting relations concerning sensitivities. From Equation 2.26 we obtain
Using the wage–price ratio obtained from Equation 2.27, we have
We can rewrite this relation as the equivalent sensitivity expression
Thus, we see that the “sensitivity” or “elasticity” of the output with respect to the input is less than or equal to 1. Again this requires nonincreasing return to scale. We can show, in general, that a profit-maximizing firm will only produce a finite nonzero amount under conditions of nonincreasing returns to scale.
Often it is more meaningful to interpret these results in terms of the average and marginal costs of production. The cost of production for our simple labor-only production function is C = cost = wL. Thus, the average and marginal costs per unit of goods produced are
Here we use the fact, from Equation 2.27, that the marginal labor productivity is just the ratio of the wages to the prices. We divide the marginal costs by the average costs and use Equation 2.30 to obtain the important result that AC = average costs ≤ MC = marginal costs, or that average costs cannot be greater than marginal costs. At the factor input that leads to profit optimality, the marginal cost is at least as great as the average cost. If this were not the case, we could change the operating point and this would have led to lower average costs and hence greater profits. We will now examine these results in greater detail and with greater generality.
Example 2.7:
The concept of a production possibility frontier is often used in representing trade-offs between produced items that result from constrained inputs to production. The production frontier allows us to represent trade-offs between goods (and services and investments) that compete for scarce resources and may be of assistance in determining an operating point that allocates resources to production in an efficient, effective, and equitable way. We will need this concept in our discussions of welfare, or normative, economics in Chapter 5.
Suppose that we have a very simple economy that produces food and clothing according to the production functions
where LF and LC are the labor devoted to production of food and clothing. For simplicity, we assume that the price of food and the price of clothing are each 1. We also assume that the wages for labor are the same in the two markets. We may obtain the conditions for maximum profit of the two firms from the profit equations
The optimum operating conditions and associated profits are obtained from dΠi/dLi = 0 as
The optimization conditions assume a potentially unlimited supply of potential labor. If the wages wL are between 0 and 1, there will be a profit to both food and clothing producers. For labor wages in excess of 1 the profit in food production is negative and so no food will be produced. It is interesting to note that the maximum amount of labor available LM may well be less than that demanded by the two firms, which is 
If we have for the wages of labor wL = 0.1, then we obtain, for the unrestricted labor case, the results F = 2.30, LF = 9, ΠF = 1.4, LC = 25, ΠC = 2.5, and C = 5. It turns out here that if labor is restricted but there are more than 34 units available, the firms will not use as much labor as is available and, assuming perfect competition, each firm can maximize its profit equation as given by Equations 2.35 and 2.36. If there are less than 34 units of labor available, then some sort of rationing of labor will be needed, since we do not allow a firm to raise its wage rate to attract more labor. Firm F will not want any more than 9 units of labor, and firm C will not need any more than 25.
If LM = 9, then we see in this case that FM will vary from 2.30 to 0 as CM varies from 0 to 3.0. The profits of the two firms will vary from ΠF = 1.4 to 0, and from ΠC = 0 to 2.5, as FM varies from 2.30 to 0 and LFM varies from 9 to 0. If the object were to maximize total profit Π = ΠF + ΠC, then the best operating point on the production frontier is LFM = 0, LCM = 9, FM = 0, CM = 3.0, ΠF = 0, ΠC = 2.5. But society may well not desire a situation in which all labor is devoted to clothing manufacture and none to food production. We see that there may be concerns other than profit maximization that are introduced into even a simple economic model. We will examine a number of these later as we explore such subjects as normative economics.
2.3 MULTIPRODUCT FIRMS AND MULTIPRODUCT PRODUCTION FUNCTIONS
Extensions of our production function concepts from the case of the single-product firm are relatively straightforward conceptually. We now briefly consider the case where there are a number of goods and services produced from the input factors. We will let qj denote the output level of good or service j produced. In general, it will turn out that we will need output good or service 
as an input to the production process to produce good or service qj. Also, there is no fundamental reason why good or service is not needed in the process of producing qj. Thus we see that our generalization to the multiproduct case results in the need for production functions of the form
for i = 1, 2,..., m. Using vector notation, we can write this as
or in an equivalent form
It will sometimes be convenient to define an augmented vector z consisting of q and x,
With this notion, we have for the generalized vector production function of a multiproduct firm
In Chapter 4, concerning supply–demand equilibrium and input–output analysis, we will need to consider the multiproduct firm and will therefore delay further discussion of this topic at this point.
2.4 CLASSIC THEORY OF THE FIRM
There are four fundamental economic concerns of a firm and the microeconomic decision problem for a firm involves each of these:
1. What total amount of factor inputs xi should be purchased?
2. What is the best allocation of the factor purchase resources among the input factors?
3. What is the best level of total output production?
4. What is the best allocation of the input factors to the various output production components?
For the single-product firm, three basic economic problems emerge from these considerations.
1. We wish to minimize the total cost or value of the input factors or input resources used in the production process to produce a given output product.
2. We wish to maximize the profit of the firm.
3. We wish to maximize the profit of the firm subject to resource constraints on input factors.
Often small-scale projects resolve the first problem. There is no fixed total budget constraint; however, we wish to turn out a product that has the lowest production cost. If we wish to optimize the production of soft drink containers, we might thus wish to solve problem 1. Problem 1 is equivalent to the problem of maximizing output production subject to a budget constraint on the input factors. Problem 2 is generally spoken of as the problem of the firm in the long term. When input resource constraints, such as those due to physical plant size or input factor availability, are present, we generally speak of the resulting problem as a problem of the firm in the short term. This is problem 3.
In all cases we wish to maximize the output q, or the profit Π, generally through minimizing production costs C by choosing appropriate inputs to the production process. The production function is assumed to be known, as is the output product price p. The input prices, or wages per unit of factor input, and denoted by the vector
are also assumed to be known. The revenue R to the firm is a product of the output level and the output price and is
The production cost PC is equal to the total price or payments for all factor inputs and is
The fixed cost FC is generally assumed to be independent of production costs.4 The total cost C is the sum of the fixed cost and the production cost and is given by the expression
The profit of the firm, Π, is the difference between the total revenue and the total cost and is
We are now in a position to state the classic microeconomic optimization problem of the firm in a relatively general form.
The resource allocation problem for each firm is that of minimizing its C(x), given by Equation 2.44, subject to the equality constraint q = f(x). In other words, we minimize the production costs associated with producing a fixed output quantity. We assume that all factor inputs will be nonnegative and thus do not impose an inequality constraint on the factor inputs. This general problem may be resolved by using the well-known Lagrange multiplier method of optimization theory. We adjoin the general equality constraint to be satisfied,
to the cost function to be minimized,
by means of a vector Lagrange multiplier λ to obtain
This function J′ is often called the Lagrangian and is denoted by 
. We note that we have added a special form of nothing to the original cost function J, since if Equation 2.46 is satisfied, J′ will surely become equal to J of Equation 2.47. Our procedure will be to minimize J′ of Equation 2.48 and then to determine λ such that Equation 2.46 is satisfied.
Use of basic calculus leads to the necessary requirement to minimize J′ in Equation 2.48. We obtain
and these are the two vector equations that must be solved to determine the solution to the original problem. Table 2.2 summarizes the results of optimization, with an equality constraint.
TABLE 2.2. Basic Maximization (Minimization) Problem with Equality Constraints
For our particular resource allocation economic problem, Equations 2.49 and 2.50 become
since the Lagrangian for this problem is, in the single-product firm case,
Example 2.8:
We desire to ensure a given output level from a productive unit that has a Cobb–Douglas production function
so as to minimize the total cost of production
We adjoin Equation 2.53 to Equation 2.54 by means of a Lagrange multiplier to obtain
We use the basic calculus necessary conditions and obtain
By combining Equations 2.56 and 2.57 we obtain
We will obtain a similar relation often in this chapter and will give it a very special and useful interpretation later. We may combine Equations 2.59 and 2.58 to obtain the desired solutions for the optimum factor inputs 
and 
. There are
Example 2.9:
An interesting general resource allocation problem that has an analytic solution consists of minimizing the input factor costs
subject to the constraint of a given output level q from the productive unit, where the production function is assumed to be a general quadratic plus linear function of input factors
with B a symmetric negative definite matrix and a > 0. For this production function, the marginal productivity is given by Equation 2.6 as
Since the marginal productivity is always positive in the economic region and since use of negative amounts of the input factors to production is not meaningful, we see that the economic region is given by 0 ≤ x ≤−B−1a. The elasticity of production is, from Equation 2.16,
We see that the elasticity is 1 for low values of x and decreases as x increases. At the end point x = −B−1a of the economic region, the elasticity is zero as it should be, since the marginal productivity is zero at the boundary of the economic region.
To obtain the best resource allocation we adjoin Equation 2.61 to Equation 2.60, using a Lagrange multiplier λ to obtain
The necessary conditions for minimum cost of the input factors are, from Equations 2.49 and 2.50, given by the expressions
Solution of these two equations yields the optimum Lagrange multiplier 
. From Equation 2.65 we obtain
and from Equation 2.66, using this value of 
, we have
Thus, we see that the optimum Lagrange multiplier is given by
The optimum factor input is then
The minimum cost of the input factor resources is, from Equation 2.60,
This results in the value of the minimum production cost, which is a function of the quantity output q. We will use this result in a later example in this section to maximize the profit of the firm expressed as a function of the production output q.
We can show5 that the problem of minimizing the input factor costs subject to a fixed production output is equivalent to that of maximizing production q subject to the constraint of a fixed budget, Cf. To resolve this equivalent problem we adjoin the constraint to the cost function and obtain
To obtain the minimum input factor costs, we set
Then we solve for the resulting optimum Lagrange multiplier and factor inputs. From Equation 2.73 we obtain
and by premultiplying this by wT and using Equation 2.74, we see that
For this particular example the optimum factor input increases linearly with increases in the budget for factor inputs. However, is not linear with respect to changes in w.
We should be careful to ensure that the results obtained and the models used for an optimization procedure are physically meaningful. In this particular example, for instance, the unconstrained maximum production output occurs at
and yields
The cost of this input factor is
In this example, we should never pay any more for the factor inputs than this. If we do, we are forcing a wasteful production system to accommodate the input factor cost greater than Cm(x). The cost of the factor input vector of Equation 2.77 is Cf. If Cf < Cm(x), we should use 
of Equation 2.77 as the optimum factor input. If Cf > Cm(x), we should use 
of Equation 2.78 and accept the fact that we do not need all of the budget for the factor inputs to the production process.
A reason for our dilemma here is that we really should have maximized output production q subject to the inequality constraint that the factor input cost is less than or equal to some budget amount, wTx ≤ Cf. We will now extend our treatment of the theory of the firm to include inequality constraints. This will allow us to consider problems, such as this, in a more realistic manner.
The nonbudget-constrained problem of the firm, which is the classic long-term problem of the firm, consists of maximizing profits as given by
subject to the constraint that we can have only nonnegative input factors, or
We accomplish this maximization by choosing the input factor vector x. This is a classic problem in nonlinear programming and there are a number of nonlinear programming algorithms that could resolve this problem for a specified production function.
One approach is the basic one of expanding the function to be extremized in a Taylor’s series. We assume that there is a local maximum of Equation 2.80 at 
. Then for any small change Δx we must have
If we assume that Π is differentiable in x, we can expand the right-hand side of Equation 2.45 in a Taylor’s series about Δx = 0 to obtain the relation
We drop all terms here of order higher than the first in Δx. When we compare the foregoing two equations, we see that we must have
Suppose that we are on the boundary where 
. On the boundary, we must have Δx > 0, since x ≥ 0 is a constraint (Eq. 2.81). The quantity Δx is otherwise arbitrary and so the foregoing is equivalent to
If we assume that we are not at the inequality boundary where some xi = 0, then we must require that 
be zero for an arbitrary Δxi. Thus for all xi solutions not on the boundary, where xi = 0, we require, to ensure that Equation 2.84 holds, that
For solutions on the boundary we cannot have arbitrary Δx, since we require that 
. We can combine this with the foregoing requirement and obtain, as an equivalent requirement to Equation 2.84,
We can sum this requirement over all i and obtain the necessary condition, sometimes called the complementary slackness conditions,
Table 2.3 represents the basic nonlinear programming problem and the necessary conditions for a solution, which we have obtained as Equations 2.81, 2.85, and 2.88. We may replace these three conditions, for 
, by the requirement that
TABLE 2.3. Basic Constrained Nonlinear Optimization Problem
For 
, we have the requirement
Equations 2.89 and 2.90 clearly indicate the complementary slackness condition in that either xi or ∂Π(x)/∂xi is zero.
Example 2.10:
As a simple example of profit optimization, we assume that the production function for a firm is of the form
and that the total costs are given by
Here, we see that the profit is, for p = 1, given by the expression
If we assume that the optimum is not on the boundary, then solution of Equation 2.89 for this example yields
which can be combined to give 
. We therefore have to solve the nonlinear algebraic equation 
. The solution of this equation represents our solution for this example and is 
and 
. We obtain a reasonable solution for this example, since the example is one in which the marginal productivity becomes negative for sufficiently large x. Thus, there is a relevant economic region and the optimum solution lies in this region. Often the production function is one, such as the linear or Cobb–Douglas production function, in which the marginal productivity is always positive and this unconstrained optimization is not meaningful.
To verify whether or not a solution obtained from the necessary conditions of our optimization procedures is actually a correct and meaningful one, we need to examine the sufficiency conditions that we can obtain from the second variation term in Equation 2.83. We obtain
This is negative definite at the obtained values for 
and 
, and so we do have a meaningful problem and solution.
We can draw some useful general conclusions from the problem of unconstrained profit maximization in which we desire to maximize6
Use of the necessary conditions of Table 2.2 leads to the requirements that for all i
and
with
Usually we require presence of some of all the input factor components xi, and so Equation 2.94 is routinely satisfied. Equation 2.93 will then be satisfied by the requirement of Equation 2.92 that
We have just obtained the important result that the price multiplied by the marginal productivity, pMP (x), is the wage or value for each input. In scalar component form, this relation states that
An important consequence of this result is that the optimum combination of input factors is such that the ratio of their marginal products is the ratio of their wages or values. We have from the foregoing discussion
This is the important result that we refer to in Equation 2.59 of Example 2.8.
Example 2.11:
We consider the quadratic production function where a > 0 and B is a negative definite symmetric matrix,
and find the optimum factor input vector to maximize the profit of the firm (in the long run). This profit is, for the production function of Equation 2.97,
We set
and obtain
as the optimum input factor vector. For this example we will assume that ap ≥ w, that is, ajp > wj ∀ i. See Example 2.13 for a discussion of the case where not all aip > wi. We can draw several important conclusions from this result. If the price w of the input factors is zero or if the price of the output product is infinite, then we will produce at the maximum value given by x = −B−1a. For any finite p and nonzero w the factor input is less than the value that yields the maximum output.
The optimum output 
to maximize profit is, from Equations 2.97 and 2.100,
The optimum value of the factor input becomes zero for w = ap, and for any w > ap the optimum factor input is zero. This simply illustrates the fact that the marginal return, or marginal revenue, MR(x) = pMP(x) = pa + pBx, reaches its maximum value at x = 0, and if this is less than the wage w, we should not produce anything at all.
These results may be obtained in another way that yields considerable insight into the optimization involved. If we could find the minimum (production) cost C(q) for a productive firm, we could maximize the profit as a function of output product. Since revenue R is equal to pq and since profit is revenue minus cost, we have
and obtain
and so we see that, for maximum profit, the price of the output product is equal to the marginal cost of the input factors
In Example 2.9 we found the minimum input factor cost for a productive firm as Equation 2.71, which is
Use of Equation 2.104 leads to a relationship for the price
which we may use to obtain for the production function
This is identical to Equation 2.101, as it should be. Equations 2.101 and 2.106 are very important results here: they indicate the impact of the relationship between wage and price on production and form the supply curve for this example.
The classic economic problem of the firm in the short run is to maximize profit
subject to a constraint on input factor availability,
Let us develop a set of conditions with which to accomplish this optimization. We convert the inequality constraint to an equality constraint by introducing a nonnegative slack variable s of appropriate dimension:
Our problem now becomes one of maximizing J = Π(x) subject to the equality constraint
and the inequality constraints
The basic optimization procedure we have just derived is applicable. We add a special form of nothing, Equation 2.110 multiplied by a Lagrange multiplier vector λ such that the resultant product is a scalar, to the cost function of Equation 2.107 to obtain the Lagrangian
Now when Equation 2.110 is satisfied, the cost function of Equation 2.113 is precisely the same as that of Equation 2.107. We will maximize Equation 2.113 and then adjust the Lagrange multiplier such that Equation 2.110 is valid. This is a standard approach in optimization theory, generally called the calculus of variations, and can easily be shown to be a valid one.
Table 2.4 illustrates the necessary conditions for an optimum easily obtained by this suggested procedure. It is obtained by using Table 2.3 with the cost function of Equation 2.113 and the inequality constraints of Equations 2.111 and 2.112. We obtain, considering x, s, and λ as variables,
TABLE 2.4. Nonlinear Optimization with Constraints: The Kuhn–Tucker Conditions for Optimality
Using b − C(x) to replace s leads immediately to Table 2.4. In Table 2.4 the equivalent Lagrangian, which does not contain s, has been defined. We could show that these conditions are both necessary and sufficient if the objective function is concave and the constraint function is convex, but to do this would take us far afield of our principle goal here, the study and engineering of economic systems. Unless we can demonstrate this concavity–convexity property or otherwise show that a locally optimal solution is also a globally optimal solution, we cannot claim sufficiency. The algorithms describing the conditions for optimality in Table 2.4 are known as the Kuhn–Tucker conditions and are discussed in most operations research texts; they are very useful theoretical constructs.
Although the Kuhn–Tucker conditions of Table 2.4 may provide necessary and sufficient conditions for optimality, they provide little if any clues concerning how to seek the optimum. There are a number of methods, usually involving various gradient-type algorithms, discussed in nonlinear programming texts in operations research that can be used to determine numerical solutions to the Kuhn–Tucker conditions or the basic nonlinear programming problem from which these conditions result.
Example 2.12:
We assume a specific production function of the Cobb–Douglas type,
that has increasing return to scale, as we can easily show. The factor prices are w1 and w2 and the output product price is p. We desire to maximize profit and this is given by
subject to a budget constraint given by
We adjoin the equality constraint of Equation 2.116 to the cost function of Equation 2.115 by means of a Lagrange multiplier to obtain the Lagrangian
We set
and obtain
By dividing Equation 2.118 by Equation 2.119, we obtain
This relation is just a special case of Equation 2.96, which states that the ratio of marginal products is just the ratio of their input values.
When we combine Equations 2.120 and 2.121, we obtain a solution for the optimum input factors and the solution to the example is complete. It is a straightforward matter for us to show that the solution to the specific problem posed here is equivalent, in that the resource inputs for a given output are equal, to that of maximizing the output production q subject to the budget equality constraint.
Example 2.13:
We again consider the quadratic production function, where a > 0 and B is symmetric negative definite, as given by the expression
For this production function the profit, excluding fixed costs, is given by
As we have seen, the problem of the firm in the long run is solved if we set the marginal profit equal to zero to obtain
We should purchase the optimum input factor vector, which is determined from this relation as
as long as x ≥ 0. If some components of obtained from Equation 2.125 are negative, we set those values of 
equal to zero and modify the profit equation (Eq. 2.123) and the optimum factor input of Equation 2.125 accordingly. To illustrate this we assume that the initial solution of Equation. 2.125 leads to
where 
and 
. We can rewrite Equation 2.123 in terms of x1 and x2 as
We set purchases of factor x2 equal to zero and Equation 2.127 becomes, with x2 = 0,
and we see that the optimum input factor vector is
If we are concerned with the problem of the firm in the short run, we should maximize profits subject to the constraint of limited input resources, perhaps constraints that are due to budget limits or resource shortages. We suppose here that this constraint is, for d > 0 and C symmetric positive semidefinite,
If some of the components of obtained by unconstrained minimization are negative, they will be set equal to zero for both the constrained and the unconstrained optima. Thus we will minimize
subject to a constraint of the form
Use of Table 2.3 leads to the conditions for optimality of x1, where we assume that we have already satisfied x1 ≥ 0. These conditions are
There will generally be a number of rows of Equation 2.136 that will be satisfied owing to the inequality of Equation 2.134, and a number that will be satisfied owing to the equality of Equation 2.133. In other words, some of the inequality constraints in Equation 2.132 will become equality constraints and the rest of the inequality constraints in Equation 2.132 are really not needed, since satisfaction of other inequality constraints routinely ensures their satisfaction. For example, the inequality constraints x1 < 2 and x1 + x2 < 3 are ineffective if we also have x1 < 1 and x1 + x2 < 2. Unfortunately the Kuhn–Tucker conditions do not generally give any clues as to which is the set of truly needed constraints from the total set of inequality constraints of Equation 2.132. As we have mentioned, there are numerical algorithms from nonlinear programming that accomplish this. The result of using these algorithms is that we replace Equation 2.132 by a reduced set of equality constraints
In terms of these, the necessary conditions for optimality are given by
where λ is the Lagrange multiplier for the minimally sized effective constraints. From Equations 2.138 and 2.139 we obtain
as the optimum factor input.
To illustrate this further, let us assume a specific case with three input factors and
First we determine the unconstrained optimum input factors. Since
we have from Equation 2.125
which results in q = 6.25 and Π(x) = 12.5. Even though w3 > pa3, we do not obtain a negative x3 factor in our solution. This is because of the enhancing value x3 contributes to the output product q owing to the cross-product terms 0.5x1x3 + x2x3 that appear in the quadratic production equation. If the B matrix were, say,
then these enhancing features disappear and x3 is not nearly so desirable in influencing production. With this new B matrix and the same a, w, and p, we obtain from Equation 2.125
We also obtain q = 0.905 and Π(x) = 15.12. We see that we desire a negative input factor 3 with this new B. This is because of this new B matrix and the fact that a is not greater than w/p. Component x3 is negative in that a3 = 2 and w3/p = 3. Thus we find it desirable to sell x3 from the point of view of profit, and this is not allowed in our problem formulation. To allow this, would be, in effect, to allow the use of an invalid production model.
So we set x3 = 0 and then reformulate the problem as a two-input factor problem where
The optimum unconstrained factor input is now 
and
This results in an output q = 2.375 and a profit 
. As we should expect, our profit is reduced by forcing 
. But the “profit” obtained with a negative x3 is not realistic owing to the use of an invalid model.
Now suppose that we introduce the budget constraint
and a constraint, imposed, perhaps, by rationing or some other resource availability constraint,
Examination of the constraint of Equation 2.144 indicates that it is satisfied by the optimum solution of Equation 2.143. However, the constraint of Equation 2.145 is not satisfied by the solution of Equation 2.143. Thus we use Equation 2.140, where
to obtain
Because of the input resource constraint, we cannot use a budget that is anywhere near that of Equation 2.144 and the productivity and profit of the firm are reduced somewhat. With this constraint we spend 
of the 20 units available. We have q = 1.958 and Π(x) = 10.417. Thus we see that we are able to determine the factor inputs, subject to an inequality constraint, such that the profit is maximized.
Technological and market constraints affect the firm. All production schemes are not possible and, as we have seen, one of the problems of the firm is to determine production schemes that are technically feasible. Market conditions preclude decisions by firms only concerning their production quantity. Other agents and mechanisms influence the determination of production quantities. However, firms will certainly have some control over the prices to be charged for goods. If a firm is “large enough,” it can unilaterally set the prices it charges for products, or the wages that it will pay for factor inputs to production. We will examine some characteristics associated with monopoly and monopsony, the names given to these activities, in this section.
A purely monopolistic firm is one that is able to set the price of its product in accordance with an estimate of the demand for the product. The monopolist, in effect, has information that a firm in “perfect competition” does not have, and thus the monopolist should always be able to obtain greater profits than firms that accept prices as givens, if we assume that the monopolist’s demand estimate is correct. We will present some precise discussions of the implications of perfect competition in Chapter 4 and, especially, Chapter 6.
We will assume that efficient factor inputs are known7 so that we can write the cost of production in terms of the production level as C(q). We can write an expression for the revenue to the firm as
such that the profit of the firm is given by
The first-order necessary conditions for profit maximization are obtained as
This general relation, that a profit-maximizing firm will set marginal revenue equal to marginal cost, is a very important one. Ideally, it would be “nice” to obtain a marginal revenue that is much higher than marginal cost, but this is completely unrealistic. By using the revenue relation (Eq. 2.146), we obtain
where ε(q) is the price elasticity of demand, or factor elasticity,
which is easily shown to be negative. By using Equation 2.149, we obtain from Equation 2.148
From this relation we see that the price a monopolist will charge for a product is greater than that which would be justified by just setting the price equal to the marginal cost. Generally this will be higher than the price charged by a price-taking firm with the same minimum production cost function. Generally also, the quantity produced by a monopolistic firm will be less than that produced by a price-taking firm with the same minimum production cost function.
Example 2.14:
As a simple illustration of the computations for a monopolist firm, and for comparison with an equivalent price-taking firm, we consider the production cost function
With this production cost, we have for average and marginal costs
For a price-taking firm we obtain the optimum production level by setting price equal to marginal cost and we have as a result
This is the supply function for the pure price-taking firm. There is no equivalent of a supply curve for a monopolistic firm. This is obviously so, since the monopolist will adjust production as a function of the available information base, which includes knowledge of the consumer demand curve. We assume that the monopolist’s estimate of demand is
From Equation 2.5.5, we have for the price elasticity of demand
We see that this is negative, since we must have a > bq for the demand to be positive. The marginal revenue is obtained as
Thus, we have for the operating condition for maximum profit where marginal revenue equals marginal cost
We substitute this value into Equation 2.156 and obtain for the price charged by the monopolist
If we assume that the pure price-taking firm faces the same demand curve as the monopolist, we have, from Equations 2.155 and 2.156, for the price and quantity produced, the relations
The profits of the two types of firms are obtained as
As we see, the profit of the monopolistic firm is always at least as great as that of the price-taking firm, assuming only that b > 0.
Figure 2.7 illustrates the average cost, marginal cost, marginal revenue, and profit curves for this example for the particular case where a = 4 and b = 2. Of particular interest in Fig. 2.7 is the intersection of the various curves that determine the production level, price, and profit received. We easily obtain qM = 2/3, pM = 8/3, ΠM = 1/3, qPT = 1, pPT = 2, ΠPT = 0.
Figure 2.7. Graphic Solutions for Example 2.14.
Example 2.15:
We consider a monopolist with a unit return to scale Cobb–Douglas production function of the form
The wages for labor and capital are wL and wK. We will first find the efficient cost of production relation. To do this, we maximize production subject to a constraint on production cost C. The Lagrangian associated with this maximization is given by
We obtain from setting ∂Λ/∂L = ∂Λ/∂K = ∂Λ/∂λ = 0, and then eliminating the Lagrange multiplier, the relations
Using these results in Equation 2.162 leads to the efficient production relation
This indicates that the efficient quantity produced is a linear function of the production cost.
Let us assume that the monopolist knows that there exists a linear demand function q = a − bp. We are interested in determining the quantity that the monopolist will produce and the resulting profit. The monopolist’s operating conditions are determined by equating marginal revenue to marginal cost. The revenue to the monopolist is given by
The operating costs are
Here, we obtain for the marginal revenue
This relation may be solved for the quantity produced as
The difference between revenue and operating costs is the profit for the monopolist and this and the optimum sales price are obtained as
It is of interest to examine conditions that could reduce the profit of the monopolist to the point where operation would become impossible. This would occur when the revenue equals the production cost, or when the average revenue equals the average production cost. We obtain for these results
This could be achieved by government action, such as through a price-freeze action to regulate the monopolist.
Some numerical results are of interest here. If we have α = 0.5, wL = wK = 0.1, a = 10, and b = 1, then we obtain the following for the monopolist’s operating conditions where there is no price fixing: β = 5, q = 4.9, p = 5.1, and Π = 24.01. With price fixing at p = 0.20, we get q = 9.8 and Π = 0. Generally, the monopolist operates by controlling prices. Price fixing by government action defeats a fundamental goal of a monopolist.
As we have seen in Example 2.14, under monopoly a productive unit will generally cut the output price of a product to sell more of the product. Thus, we have
We must modify our previous price-taking relationships to incorporate this variable-price structure. For example, the revenue of the firm now becomes
We now obtain marginal revenue, as we have just seen, as
We see that the marginal revenue now depends on the output quantity of the productive unit. The marginal revenue under monopoly is less than that obtained for the unit output price Þ(q) of the productive unit.
Monopsony is a term sometimes used in microeconomics. The monopsonist controls the price of input factors by adjusting input prices or wages as a function of the input factor purchases according to
Since a classic productive unit can generally only purchase more of a factor by paying a higher price for it, we require
The cost of the ith input factor is the factor production cost and is given by
The marginal cost of the ith factor input is the derivative of the cost of this factor with respect to the factor input level. This is
We see that this marginal cost exceeds the price of the input factor.
Under the imperfect competition of monopoly and monopsony, the unconstrained profit maximization problem is that of maximizing
subject to the production function equality constraint
We could insert q from Equation 2.171 into Equation 2.170 to eliminate the need for a Lagrange multiplier, but we will not do this here to give a special interpretation to the Lagrange multiplier for this problem.
To solve this problem, we adjoin Equation 2.171 to Equation 2.170 by means of a Lagrange multiplier and obtain the Lagrangian
The necessary conditions for an extremum are given by the expressions
Equation 2.173 leads to the interesting and useful result that the optimum Lagrange multiplier is, for this particular problem formulation, just the marginal revenue
Equation 2.174 leads to the conclusion that the product of the marginal revenue (λ) and the marginal factor productivity ∂f/∂xi is just the marginal cost of that factor. It is convenient to denote the factor marginal revenue product as this product, such that we have
If there are a few productive units and if these few units mutually agree to cooperate such as to jointly price their outputs, we have what is known as an oligopoly. For two productive units (duopoly) we have the outputs q1 = f1(x1) and q2 = f2(x2). The firms cooperate in setting prices such that p1 = Þ1(q1, q2) and p2 = Þ2(q1, q2). If two firms cooperate in such a way, as they could if they were the only buyers of a given factor input, to set prices and wages for a factor input according to their needs for the input factors 
, then we have what is known as an oligopsony.
Problems involving monopoly, monopsony, oligopoly, and oligopsony can be formulated as optimization problems such that the economic optimization methods we have formulated in this section are applicable. We shall not do this here, since determination of the necessary conditions is a relatively straightforward task for optimization problems of the sort we have been discussing. Unfortunately closed-form solutions to the problems we have been discussing in this chapter are difficult or impossible to obtain except in very special circumstances for relatively simple production functions, and numerical methods are generally essential. Problems 23 to 34 at the end of this chapter expand considerably on the concepts presented here concerning monopoly, monopsony, oligopoly, and oligopsony and provide for some interesting comparisons with firms in perfect competition.
In this chapter we have examined some elementary concepts from production theory or the theory of the firm. We have introduced the concept of a production function and a number of economic concepts concerning production that will be very useful to us later. We have briefly examined the fundamental or classic problem of the firm and used some elementary optimization concepts to illustrate how, in some very simple cases, profit maximization can be obtained.
There are many topics concerning production that we have not discussed. Perhaps the most important one is that we presently have no way of determining how much output a firm should produce to accommodate purchases of this output. This is related, of course, to the demand for the product of the firm, a subject known as the theory of the household or the theory of consumer. Also, we did not examine a number of factors, such as advertising, capital costs, taxes, and the effects of negative externalities such as pollution, or the effects of social and regulatory inputs to the production process. Chapter 3 will examine the theory of the household and later chapters will discuss some of the other important inputs to the production determination process.
PROBLEMS
1. Show that the marginal productivity and average productivity are equal at the point of maximum average productivity. Is there any physical significance to this point?
2. What do the curves of Fig. 2.3 become for
a. a linear production function,
b. a Cobb–Douglas production, and
c. the production function 
.
3. Repeat Examples 2.3 through 2.5 using the production function 
.
4. Reconstruct Table 2.1 for the production function of Problem 3.
5. Investigate the behavior and important properties of the following production functions, using Table 2.1 as a guide:
6. Is it reasonable that a production function demonstrate diminishing average productivity? Is a law of diminishing average productivity implied by the law of diminishing marginal productivity? You may find it helpful to use the following production functions in support of your arguments. Examine the marginal and average productivities for input x2.
7. Prove or disprove the statement that if marginal productivity is everywhere a monotonically decreasing function of an input factor, then average productivity is everywhere a monotonically decreasing function of that input factor. Consider a specific example to demonstrate your result.
8. Prove or disprove the statement that if average productivity is everywhere a monotonically decreasing function of an input factor, then marginal productivity is everywhere a monotonically decreasing function of that input factor. Consider a specific example to demonstrate your result.
9. Find the necessary conditions to maximize the output of a productive unit subject to an equality constraint in total cost.
10. What are the requirements on the Cobb–Douglas production function of Equation 2.23 such that it be (a) a homogeneous production function and (b) a linear homogeneous production function?
11. Suppose that the manufacture of a number of products, q, is specified by the Cobb–Douglas production function 
where x1 represents the labor used in person-hours, x2 the number of hours of machine time, and x3 the aggregated amount of raw material in kilograms. Labor costs $200 per 40 h, machine use costs $300 per 40-h work week, and raw materials cost $50 per kilogram.
a. What are the optimal proportions of labor, machines, and raw materials in the production process?
b. What will be the effect of a 20% increase in labor costs on the production process?
c. What will be the effect of a 100% increase in raw material costs on the production process?
12. Repeat Problem 11 for the production function 
.
13. Repeat Problem 11 for the linear production function q = b1x1 + b2x2 + b3x3.
14. Investigate the behavior of the production function 
. Under what conditions will this become a linear production function? Under what conditions will this become a Cobb–Douglas production function?
15. A manufacturer produces a quantity q of a product using two inputs x1 and x2. The prices of the input and output quantities are fixed. Rationing is introduced and the manufacturer may purchase no more than x2 max units of x2. Without rationing, the manufacturer would like to produce using more than x2 max.
a. What are the conditions for profit maximization under rationing?
b. What are the marginal productivities of x1 and x2?
c. What is the relationship between x1 and x2?
d. The rate of technical substitution (RTS) or the marginal rate of substitution (MRS) or the marginal rate of technical substitution (MRTS) is defined as the ratio of marginal productivities at any point on the production curve. What is the RTS for this problem both with and without rationing?
16. We wish to allocate a scarce resource x1 between two production processes such as to maximize their total returns. Thus we wish to maximize ql + q2 where q1 = f1(x1,1) and q2 = f2(x1,2), and where we have the constraint x1,1 + x1,2 = x1 = fixed = b. Inputs to the production process other than x1 may be ignored for simplicity.
a. What are the conditions for optimality?
b. If f1(x1,1) = 25 − (1 − x1,1)2 and f2(x1,2) = 50 − (2 − x1,2)2, what are the optimal inputs?
c. Under what conditions will the marginal revenue be greater than the marginal cost? What is the physical significance of this?
17. For a particular monopolistic situation, marginal revenue is a linear function and marginal cost a quadratic function of production output such that MR = A + Bq and MC = C + Dq + Eq2. We will assume that the fixed cost of the product, FC, is known, as are A, B, C, D, and E.
a. What are reasonable signs for A, B, C, D, and E?
b. Find the revenue R, the production cost PC, the total cost C, and the profit Π of the firm.
c. What are optimality conditions for profit maximization?
d. Suppose that the government introduces a sales tax ST that is a linear function of the number of units sold and given by ST = Tq. What is the optimum value of T that produces maximum tax revenue? The tax is paid by the manufacturer.
18. Show that the problems of maximizing output production with a constraint on input costs and minimizing input factor costs subject to a constraint on total output are equivalent problems in the sense that the Lagrangian and the optimum solutions are the same for both problems.
19. The output of a firm is given by q = bTx − 0.5xTAx, where A is a positive definite matrix. Find the values of x that result in the following:
a. maximum output subject to the budget constraint xTw ≤ B;
b. minimum input factor cost C(x) = wTx for a fixed output q = qf;
c. maximum profit;
d. maximum profit subject to a budget constraint of the form xTw ≤ B; and
e. maximum profit subject to the budget constraint of d and rationing of the form xT1 ≤ k, where 1 is an appropriately dimensioned vector of all ones.
20. Obtain numerical results for Problem 19 for the case and illustrate your results as a function of B and k where
21. Repeat Problem 19 for the case where A is negative definite. Show that you are now minimizing output and profit!
22. Repeat Example 2.9 for the case where a = 0 and B is positive definite.
23. The price equation of a monopolist is Þ(q) = l00 − 5q. The optimum production cost of the monopolist is given by PC = 25q.
a. What is the profit equation for the monopolist?
b. What is the optimum production and price that maximize profit?
c. What is the profit that would result if the monopolist were to determine production by setting the price Þ(q) equal to the marginal production cost? This is the result obtained under perfect competition conditions. How do these two profits compare?
24. Suppose that a monopsonist sells products at a price p = 4. The production function for the monopsonist is q = 10x − 0.5x2. The wage the monopsonist pays for the input factors x is Ω(x) = 20 + 3x.
a. What is the profit equation for the monopsonist?
b. What is the optimum production and factor input that maximize profit?
c. What is the profit that would result if the monopsonist were to determine production by setting marginal revenue ∂pf(x)/∂x equal to the wage for the factor input w? This is the result obtained under perfect competition conditions. How do these two profits compare?
25. Suppose that the production function for a monopolist–monopsonist firm is q = aTx. The price of the product sold decreases as the quantity increases according to Þ(q) = c − dq and the wages increase as the input factor increases according to the relation Ω(x) = e + Fx.
a. What is the profit equation for the monopolist–monopsonist firm?
b. What is the optimum factor input x, wage Ω, price Þ, and quantity q?
c. What is the profit that would result if the firm were to act as a firm in perfect competition and set the product of the price Þ(x) and the marginal productivity equal to the wages Ω(x)? How does this profit compare to the profit under imperfect competition obtained in part b?
26. A discriminating monopolist will sell the product of the firm in two or more markets at different prices. Purchasers are assumed to be unable to purchase the product of the firm in one commodity market and sell it in another. The revenue to the firm for the two-market case is R = R1(q1) + R2(q2) where R1(q1) = Þ1(q1) q1 and R2(q2) = Þ2(q2) q2 represent the revenues in markets 1 and 2, and Þ1(q1) and Þ2(q2) the prices in these markets, respectively. The total production of the firm is q = q1 + q2, and the cost of production is PC(q). What are the requirements for profit maximization for the discriminating monopolist? In which market will the price be lowest?
27. What are the results of Problem 26 for the case where Þ1(q1) = 100 − 6.33q1, Þ2(q2) = 100−20q2 and PC(q1+q2) = 25(q1+q2). What is the profit of the firm? How do the results of this problem compare with those in Problem 23? Show that the aggregation of the two markets in this example results in Problem 23.
28. For an oligopolist production with two firms (generally called a duopoly) the total quantity of production is a function of price p = g(q) = g(q1 + q2) where q1 and q2 are the production levels of the two firms with production costs PC1(q1) and PC2(q2).
a. What is the expression for the profit of each firm?
b. What is the requirement for each profit to be maximized with respect to the output of the firm?
c. What is the total profit of the two firms?
d. What is the requirement that this total profit be maximized?
29. What do the results of Problem 28 become if g(q1+q2) = 100−5(q1+q2) and PC1(q1) = 20q1, PC2(q2) = 3q2?
30. An oligopsonist purchases factor inputs to production according to W(x) = W(x1+x2) = g(x1+x2), where x1 and x2 are the factor inputs for the two productive units whose production functions are q1 = f1(x1) and q2 = f2(x2).
a. What is the equation for the profit of each firm?
b. What is the requirement that the profit of each firm be maximized by choice of the factor input for each firm?
c. What is the total profit of both firms?
d. What is the requirement that this profit be maximized with respect to the factor inputs x1 and x2?
31. What are the results of Problem 30 for the case where
32. It is sometimes stated that a firm, rather than maximizing profits, will seek to maximize revenue subject to the constraint that profits be above some minimum level. Formulate this problem and determine optimality conditions for
a. a perfectly competitive firm,
b. a firm that practices monopoly and monopsony, and
c. the monopolistic firm of Problem 23.
33. A potential manufacturer is contemplating the purchase of a factory. Two options are available: (a) q = αKL/(0.6L + 0.4K) and (b) q = βK0.25L0.75, where K is capital and L is labor. Determine the appropriate decision rules to enable the selection of a production process in terms of the wages for capital and labor, and α and β.
34. Suppose that the demand curve for a product is q = 10−p. The cost of making the product is c(q) = 5q and the firm is a monopoly.
a. What will be the equilibrium price that maximizes profit, the profit, and the quantity produced?
b. Suppose that price ceilings p = 4.5 and 6 are set by the government. What will be the production level and profit under these conditions?
c. Sketch these results graphically.
d. What would the response of a firm in perfect competition be to the price ceilings of b? Compare the resulting profit with that for the monopolist.
35. Advertising can generally act to increase the price at which a given quantity of a product will sell. Only firms with monopoly capability will find it desirable to advertise. Suppose that the demand function for a monopolist is q = 25 + A0.5−p where A is the cost of advertising. The production and advertising cost of the product is C = 5q+q2+A.
a. What are the optimum values of p, q, and A to maximize profit?
b. What is the effect of the advertising?
c. What would be the results for parts a and b for the firm if it operates under perfect competition conditions?
36. Consider a monopolist who produces a product using labor only, with q = L+ 0.1L2 − 0.0001L3. The wage rate for labor is given by w = 20 + 0.5L. The price for the product q is p = 3. Determine the relations for revenue and profit. Determine the conditions that yield maximum profit.
37. Suppose that there are a large number of possible oil firms in the world and each is equally efficient, such that each has the production cost function PC(q) = 1 + q + q2, where the cost is measured in millions of dollars and q is oil production in millions of barrels. Suppose that the world demand for oil is p = 10−q/10.
a. What is the perfectly competitive equilibrium for the oil industry? What is the resulting number of firms in competitive equilibrium?
b. This number of firms now agrees to form a cartel. What price do they now set? At what production level? What is the resulting profit per firm? The form of control for the cartel is the quota. Members are given a maximum production level.
c. What happens if the form of control used to restrict output is the price that is achieved by pooling revenue? What is the best output and pooled revenue per firm?
d. Suppose that a new firm decides not to join the cartel; rather, it decides to sell as much as desired at the price at which the cartel is selling. What are the conditions under which the new firm operates?
e. The government makes the activities in d illegal. It gives the cartel the right to force all oil firms into the cartel. What is the new equilibrium number of firms in the cartel under these conditions?
f. The cartel falls out of favor with the government, which then forbids the restrictive setting of output quotas. The same number of firms is in the cartel as in part f. What are the new equilibrium conditions?
g. Generalize the results of this example to the extent that you find meaningful. Write a few sentences concerning the general lessons learned from this example.
BIBLIOGRAPHY AND REFERENCES
This chapter is a very basic one and there are many economic texts that discuss the theory of the firm. Some of the more classic recent discussions appear in
Baumol WJ, Blinder AS. Microeconomics: principles and policy. 9th ed. Mason Ohio: South-Western; 2009.
Besanko D, Braeutigam RR. Microeconomics: an integrated approach. Hoboken New Jersey: Wiley; 2001.
Jehle GA, Reny PJ. Advanced microeconomic theory. 2nd ed. Reading Massachusetts: Addison Wesley; 2000.
Pindyck RS, Rubinfeld DL. Microeconomics. 5th ed. Englewood Cliffs, New Jersey: Prentice Hall; 2000.
Samuelson PA. Microeconomics. New York: McGraw Hill; 1995.
Varian HR. Microeconomic analysis. New York: W.W. Norton & Company; 1992.
Varian HR. Intermediate microeconomics: a modern approach. 8th ed. New York: W.W. Norton & Company; 2009.
1We say that vector x1 is greater than or equal to vector x2 when each component of x1 is greater than or equal to each component of x2, i.e., 
2If H(x) is negative definite, f(x) is strictly concave. If H(x) is negative semidefinite, then f(x) is (quasi)concave. Normally, concavity is all that is required in the relevant economic region. We recall that a convex function f(x) is one in which for 0 ≤ a ≤ 1, we have af(x1) + (1 − a)f(x2) ≥ f[a(x1) + (1 − a)x2]. A function f(x) is concave if for 0 ≤ a ≤ 1, we have f[ax + (1 − a)y] ≥ a f(x) + (1 − a)f(y). The concept of a convex set is also of interest. If S is convex and if x ∈ S, y ∈ S, then we have ax + (1 − a)y ∈ S, ∀a ∈ (0, 1). The graph of a concave function will always lie below all tangent lines for the function. A function is quasi-concave if A − {x:f(x) ≥ t} is a convex set.
3We will encounter several types of elasticity in this book. In all cases, elasticity will be the ratio of a marginal quantity to an average quantity. We will generally use subscripts and superscripts to denote the type of elasticity when this distinction is needed.
4Generally, we can assume that the fixed costs are equal to zero, since we obtain optimality by differentiating total cost. The constant fixed costs will not influence the conditions for optimality. This simply says that no input represents a true economic cost unless it is possible to change it by varying the input. Fixed costs influence profit, of course, and thus influence whether or not a firm will produce at all.
5See Problem 18 at the end of this chapter. These two problems are mathematical equivalents of one another.
6The conditions for optimality do not depend on FC, so we can let FC = 0 without loss of generality, as we have noted previously.
7These are obtained by minimizing the cost of production, given by C = wTx, for a prescribed output production level q = f(x) and then finding the minimum cost C(q) in terms of this production level.