Microeconomics - Production - Chapter 5 PDF

# Production ## Chapter 5 ### 5.A Introduction In this chapter, we move to the supply side of the economy, studying the process by which the goods and services consumed by individuals are produced. We view the supply side as composed of a number of productive units, or, as we shall call them, "fi...

# Production ## Chapter 5 ### 5.A Introduction In this chapter, we move to the supply side of the economy, studying the process by which the goods and services consumed by individuals are produced. We view the supply side as composed of a number of productive units, or, as we shall call them, "firms". Firms may be corporations or other legally recognized businesses. But they must also represent the productive possibilities of individuals or households. Moreover, the set of all firms may include some potential productive units that are never actually organized. Thus, the theory will be able to accommodate both active production processes and potential but inactive ones. Many aspects enter a full description of a firm: Who owns it? Who manages it? How is it managed? How is it organized? What can it do? Of all these questions, we concentrate on the last one. Our justification is not that the other questions are not interesting (indeed, they are), but that we want to arrive as quickly as possible at a minimal conceptual apparatus that allows us to analyze market behavior. Thus, our model of production possibilities is going to be very parsimonious: The firm is viewed merely as a "black box", able to transform inputs into outputs. In Section 5.B, we begin by introducing the firm's production set, a set that represents the production activities, or production plans, that are technologically feasible for the firm. We then enumerate and discuss some commonly assumed properties of production sets, introducing concepts such as returns to scale, free disposal, and free entry. After studying the firm's technological possibilities in Section 5.B, we introduce its objective, the goal of profit maximization, in Section 5.C. We then formulate and study the firm's profit maximization problem and two associated objects, the firm's profit function and its supply correspondence. These are, respectively, the value function and the optimizing vectors of the firm's profit maximization problem. Related to the firm's goal of profit maximization is the task of achieving cost-minimizing production. We also study the firm's cost minimization problem and two objects associated with it: The firm's cost function and its conditional factor demand correspondence. As with the utility maximization and expenditure minimization problems in the theory of demand, there is a rich duality theory associated with the profit maximization and cost minimization problems. ### 5.B Production Sets As in the previous chapters, we consider an economy with *L* commodities. A production vector (also known as an input-output, or netput, vector, or as a production plan) is a vector *y* = (*y*1, . . ., *y*L) ∈ R² that describes the (net) outputs of the *L* commodities from a production process. We adopt the convention that positive numbers denote outputs and negative numbers denote inputs. Some elements of a production vector may be zero; this just means that the process has no net output of that commodity. **Example 5.B.1:** Suppose that *L* = 5. Then *y* = (-5, 2, 6, 3, 0) means that 2 and 3 units of goods 2 and 4, respectively, are produced, while 5 and 6 units of goods 1 and 3, respectively, are used. Good 5 is neither produced nor used as an input in this production vector. To analyze the behavior of the firm, we need to start by identifying those production vectors that are technologically possible. The set of all production vectors that constitute feasible plans for the firm is known as the production set and is denoted by *Y* ⊆ R¹². Any *y*∈ *Y* is possible; any *y* ∉ *Y* is not. The production set is taken as a primitive datum of the theory. The set of feasible production plans is limited first and foremost by technological constraints. However, in any particular model, legal restrictions or prior contractual commitments may also contribute to the determination of the production set. It is sometimes convenient to describe the production set *Y* using a function *F*(), called the transformation function. The transformation function *F*() has the property that *Y* = {*y*∈ R²: *F*(*y*) ≤ 0} and *F*(*y*) = 0 if and only if *y* is an element of the boundary of *Y*. The set of boundary points of *Y*, {*y* ∈ R²: *F*(*y*) = 0}, is known as the transformation frontier. Figure 5.B.1 presents a two-good example. If *F*() is differentiable, and if the production vector *y* satisfies *F*(*y*) = 0, then for any commodities *l* and *k*, the ratio $MRT_{lk}(y)=\frac{\partial F(y)/\partial y_l}{\partial F(y)/\partial y_k}$, is called the marginal rate of transformation (MRT) of good *l* for good *k* at *y*. The marginal rate of transformation is a measure of how much the (net) output of good *k* can increase if the firm decreases the (net) output of good *l* by one marginal unit. Indeed, from *F*(*y*) = 0, we get $\frac{\partial F(y)}{\partial y_l}dy_l+\frac{\partial F(y)}{\partial y_k}dy_k=0$, and therefore the slope of the transformation frontier at *y* in Figure 5.B.1 is precisely -*MRT*lk(*y*). #### Technologies with Distinct Inputs and Outputs In many actual production processes, the set of goods that can be outputs is distinct from the set that can be inputs. In this case, it is sometimes convenient to notationally distinguish the firm's inputs and outputs. We could, for example, let *q* = (*q*1, . . ., *q*M) ≥ 0 denote the production levels of the firm's *M* outputs and *z* = (*z*1, ..., *z*L-M) ≥ 0 denote the amounts of the firm's *L* - *M* inputs, with the convention that the amount of input *z*l used is now measured as a nonnegative number (as a matter of notation, we count all goods not actually used in the process as inputs). One of the most frequently encountered production models is that in which there is a single output. A single-output technology is commonly described by means of a production function *f*(*z*) that gives the maximum amount *q* of output that can be produced using input amounts (*z*1,..., *z*L-1) ≥ 0. For example, if the output is good *L*, then (assuming that output can be disposed of at no cost) the production function *f*() gives rise to the production set: *Y* = {(-*z*1, ..., -*z*L-1, *q*): *q* - *f*(*z*1, . . ., *z*L-1) ≤ 0 and (*z*1,..., *z*L-1) ≥ 0}. Holding the level of output fixed, we can define the marginal rate of technical substitution (MRTS) of input *l* for input *k* at *z* as $MRTS_{lk}(z)=\frac{df(z)/dz_l}{df(z)/dz_k}$ The number *MRTS*lk(*z*) measures the additional amount of input *k* that must be used to keep output at level *q* = *f*(*z*) when the amount of input *l* is decreased marginally. It is the production theory analog to the consumer's marginal rate of substitution. In consumer theory, we look at the trade-off between commodities that keeps utility constant, here, we examine the trade-off between inputs that keeps the amount of output constant. Note that *MRTS*lk(*z*) is simply a renaming of the marginal rate of transformation of input *l* for input *k* in the special case of a single-output, many-input technology. **Example 5.B.2:** The Cobb-Douglas Production Function The Cobb-Douglas production function with two inputs is given by *f*(*z*1, *z*2) = *z*1α*z*2β , where α ≥ 0 and β ≥ 0. The marginal rate of technical substitution between the two inputs at *z* = (*z*1, *z*2) is *MRTS*12(*z*) = α*z*2/β*z*1. #### Properties of Production Sets We now introduce and discuss a fairly exhaustive list of commonly assumed properties of production sets. The appropriateness of each of these assumptions depends on the particular circumstances (indeed, some of them are mutually exclusive). (i) *Y* is nonempty. This assumption simply says that the firm has something it can plan to do. Otherwise, there is no need to study the behavior of the firm in question. (ii) *Y* is closed. The set *Y* includes its boundary. Thus, the limit of a sequence of technologically feasible input-output vectors is also feasible; in symbols, *y*n → *y* and *y*n∈ *Y* imply *y*∈ *Y*. This condition should be thought of as primarily technical. (iii) No free lunch. Suppose that *y*∈ *Y* and *y* ≥ 0, so that the vector *y* does not use any inputs. The no-free-lunch property is satisfied if this production vector cannot produce output either. That is, whenever *y*∈ *Y* and *y*≥ 0, then *y* = 0; it is not possible to produce something from nothing. Geometrically, *Y*∩ R4 = {0}. For *L* = 2, Figure 5.B.2(a) depicts a set that violates the no-free-lunch property, the set in Figure 5.B.2(b) satisfies it. (iv) Possibility of inaction This property says that 0 ∈ *Y*: Complete shutdown is possible. Both sets in Figure 5.B.2, for example, satisfy this property. The point in time at which production possibilities are being analyzed is often important for the validity of this assumption. If we are contemplating a firm that could access a set of technological possibilities but that has not yet been organized, then inaction is clearly possible. But if some production decisions have already been made, or if irrevocable contracts for the delivery of some inputs have been signed, inaction is not possible. In that case, we say that some costs are sunk. Figure 5.B.3 depicts two examples. The production set in Figure 5.B.3(a) represents the interim production possibilities arising when the firm is already committed to use at least -*y*1 units of good 1 (perhaps because it has already signed a contract for the purchase of this amount); that is, the set is a restricted production set that reflects the firm's remaining choices from some original production set *Y* like the ones in Figure 5.B.2. In Figure 5.B.3(b), we have a second example of sunk costs. For a case with one output (good 3) and two inputs (goods 1 and 2), the figure illustrates the restricted production set arising when the level of the second input has been irrevocably set at *y*2 < 0 [here, in contrast with Figure 5.B.3(a), increases in the use of the input are impossible]. (v) Free disposal. The property of free disposal holds if the absorption of any additional amounts of inputs without any reduction in output is always possible. That is, if *y*∈ *Y* and *y*' ≤ *y* (so that *y*' produces at most the same amount of outputs using at least the same amount of inputs), then *y*' ∈ *Y*. More succinctly, *Y* - R4 ⊂ *Y* (see Figure 5.B.4). The interpretation is that the extra amount of inputs (or outputs) can be disposed of or eliminated at no cost. (vi) Irreversibility. Suppose that *y*∈ *Y* and *y* ≠ 0. Then irreversiblity says that -*y* ∉ *Y*. In words, it is impossible to reverse a technologically possible production vector to transform an amount of output into the same amount of input that was used to generate it. If, for example, the description of a commodity includes the time of its availability, then irreversibility follows from the requirement that inputs be used before outputs emerge. **Exercise 5.B.1:** Draw two production sets: one that violates irreversibility and one that satisfies this property. (vii) Nonincreasing returns to scale. The production technology *Y* exhibits nonincreasing returns to scale if for any *y*∈ *Y*, we have α*y* ∈ *Y* for all scalars α ∈ [0, 1]. In words, any feasible input output vector can be scaled down (see Figure 5.B.5). Note that nonincreasing returns to scale imply that inaction is possible [property (iv)]. (viii) Nondecreasing returns to scale. In contrast with the previous case, the production process exhibits nondecreasing returns to scale if for any *y*∈ *Y*, we have α*y*∈ *Y* for any scale α ≥ 1. In words, any feasible input-output vector can be scaled up. Figure 5.B.6(a) presents a typical example; in the figure, units of output (good 2) can be produced at a constant cost of input (good 1) except that in order to produce at all, a fixed setup cost is required. It does not matter for the existence of nondccreasing returns if this fixed cost is sunk [as in Figure 5.B.6(b)] or not [as in Figure 5.B.6(a), where inaction is possible]. (ix) Constant returns to scale. This property is the conjunction of properties (vii) and (viii). The production set *Y* exhibits constant returns to scale if *y*∈ *Y* implies α*y*∈ *Y* for any scalar α ≥ 0. Geometrically, *Y* is a cone (see Figure 5.B.7). For single-output technologies, properties of the production set translate readily into properties of the production function *f*(). Consider Exercise 5.B.2 and Example 5.Β.3. **Exercise 5.B.2:** Suppose that *f*() is the production function associated with a single-output technology, and let *Y* be the production set of this technology. Show that *Y* satisfies constant returns to scale if and only if *f*() is homogeneous of degree one. (x) Additivity (or free entry). Suppose that *y*∈ *Y* and *y*'∈ *Y*. The additivity property requires that *y* + *y*'∈ *Y*. More succinctly, *Y* + *Y* ⊂ *Y*. This implies, for example, that *k* *y*∈ *Y* for any positive integer *k*. In Figure 5.B.8, we see an example where *Y* is additive. Note that in this example, output is available only in integer amounts (perhaps because of indivisibilities). The economic interpretation of the additivity condition is that if *y* and *y*' are both possible, then one can set up two plants that do not interfere with each other and carry out production plans *y* and *y*' independently. The result is then the production vector *y* + *y'*. Additivity is also related to the idea of entry. If *y*∈ *Y* is being produced by a firm and another firm enters and produces *y*' ∈ *Y*, then the net result is the vector *y* + *y'*. Hence, the aggregate production set (the production set describing feasible production plans for the economy as a whole) must satisfy additivity whenever unrestricted entry, or (as it is called in the literature) free entry, is possible. (xi) Convexity. This is one of the fundamental assumptions of microeconomics. It postulates that the production set *Y* is convex. That is, if *y*, *y*' ∈ *Y* and α∈ [0, 1], then α*y* + (1-α) *y*' ∈ *Y*. For example, *Y* is convex in Figure 5.B.5(a) but is not convex in Figure 5.B.5(b). The convexity assumption can be interpreted as incorporating two ideas about production possibilities. The first is nonincreasing returns. In particular, if inaction is possible (i.e., if 0 ∈ *Y*), then convexity implies that *Y* has nonincreasing returns to scale. To see this, note that for any α∈ [0, 1], we can write α*y* = α*y* +(1 – α)0. Hence, if *y* ∈ *Y* and 0 ∈ *Y*, convexity implies that α*y* ∈ *Y*. Second, convexity captures the idea that "unbalanced" input combinations are not more productive than balanced ones (or, symmetrically, that "unbalanced" output combinations are not least costly to produce than balanced ones). In particular, if production plans *y* and *y*' produce exactly the same amount of output but use different input combinations, then a production vector that uses a level of each input that is the average of the levels used in these two plans can do at least as well as either *y* or *y'*. Exercise 5.B.3 illustrates these two ideas for the case of a single-output technology. **Exercise 5.B.3:** Show that for a single-output technology, *Y* is convex if and only if the production function *f*(*z*) is concave. (xii) *Y* is a convex cone. This is the conjunction of the convexity (xi) and constant returns to scale (ix) properties. Formally, *Y* is a convex cone if for any production vector *y*, *y*' ∈ *Y* and constants α > 0 and β ≥ 0, we have α*y* + β*y*' ∈ *Y*. The production set depicted in Figure 5.B.7 is a convex cone. An important fact is given in Proposition 5.B.1. **Proposition 5.B.1:** The production set *Y* is additive and satisfies the nonincreasing returns condition if and only if it is a convex cone. **Proof:** The definition of a convex cone directly implies the nonincreasing returns and additivity properties. Conversely, we want to show that if nonincreasing returns and additivity hold, then for any *y*, *y*' ∈ *Y* and any α > 0, and β > 0, we have α*y* + β*y*' ∈ *Y*. To this effect, let *k* be any integer such that *k* > Max{α, β}. By additivity, *k* *y* ∈ *Y* and *k* *y*'∈ *Y*. Since (α/*k*)<1 and α*y* = (α/*k*)*k* *y*, the nonincreasing returns condition implies that α*y* ∈ *Y*. Similarly, β*y*' ∈ *Y*. Finally, again by additivity, α*y* + β*y*'∈ *Y*. **Proposition 5.B.1** provides a justification for the convexity assumption in production. Informally, we could say that if feasible input-output combinations can always be scaled down, and if the simultaneous operation of several technologies without mutual interference is always possible, then, in particular, convexity obtains. (See Appendix A of Chapter 11 for several examples in which there is mutual interference and, as a consequence, convexity does not arise.) It is important not to lose sight of the fact that the production set describes technology, not limits on resources. It can be argued that if all inputs (including, say, entrepreneurial inputs) are explicitly accounted for, then it should always be possible to replicate production. After all, we are not saying that doubling output is actually feasible, only that in principle it would be possible if all inputs (however esoteric, be they marketed or not) were doubled. In this view, which originated with Marshall and has been much emphasized by McKenzie (1959), decreasing returns must reflect the scarcity of an underlying, unlisted input of production. For this reason, some economists believe that among models with convex technologies the constant returns model is the most fundamental. Proposition 5.B.2 makes this idea precise. **Proposition 5.B.2:** For any convex production set *Y* ⊆ R² with 0 ∈ *Y*, there is a constant returns, convex production set *Y*' ⊆ RL+¹ such that *Y* = {*y* ∈ R²: (*y*, -1) ∈ *Y*'}. **Proof:** Simply let *Y*' = {*y*' ∈ R¹+1: *y*' = α(*y*, -1) for some *y*∈ *Y* and α ≥ 0}. (See Figure 5.Β.9.) The additional input included in the extended production set (good *L* + 1) can be called the "entrepreneurial factor." (The justification for this can be seen in Exercise 5.C.12; in a competitive environment, the return to this entrepreneurial factor is precisely the firm's profit.) In essence, the implication of Proposition 5.B.2 is that in a competitive, convex setting, there may be little loss of conceptual generality in limiting ourselves to constant returns technologies.