微观经济学讲义(黄有光)6.doc

Advanced Microeconomics Topic 5: Production & CostPrimary Readings: DL Chapter 2 & JR Chapter 5 In this lecture, we will present a general framework of production technology. We will focus on what choices could be made; and the issue of what choices would be made will be deferred to the next lecture when we look into the firms behaviour. The first part will describe production possibilities in physical terms; while the second part will recast this description into a cost function framework. The treatment in this lecture is a bit abstract and quite general. You are required to understand the relevance of this abstract framework in terms of particular technological processes. 5.1Production Possibility SetsThere are many ways to describe the technology of a firm, such as, production functions, graphs, or systems of inequalities. But in mathematical term, these representations can all be expressed as a set. The firm uses and produces a total of m commodities. A particular production plan is y in Rm: yi 0 implies that a net amount yi of i-th commodity is produced; yj 1 and all z. 2. Increasing returns to scale if f(tz) t f(z) for all t 1 and all z. 3. Decreasing returns to scale if f(tz) 1 and all z. The most natural case of decreasing returns to “scale” is the case where we are unable to replicate some inputs. In fact, it can always be assumed that decreasing returns to scale is due to the presence of some fixed input. To see this, let f(z) be a production function with decreasing returns to scale. Suppose that we introduce another new input and measured by z0. Now define a new production function: F(z0, z) = z0 f(z/z0).It is easy to see that F exhibits constant returns to scale. In this sense, the original decreasing returns technology f(z) can be thought as a restriction of the constant returns technology F(z0, z) that results from setting z0 = 1. Elasticity of Scale The elasticity of scale is a local measure of returns to scale. It, defined at a point, specifies the instantaneous percentage change in output as a result of 1 percent increase in all inputs: We say that returns to scale are locally constant, increasing, or decreasing when m(x) is equal to, greater than, or less than one. Constant Returns to Scale and the Marginal Productivity Theory of DistributionFrom the definition of a homogenous production function, differentiation with respect to k, evaluated at k=1, we have sy = xifi where fi f/xi.A production function homogenous of degree s, the marginal product of each factor is homogenous of degree s-1. To show this, differentiate with respect to xi. 5.3The Cost FunctionBasic Settings: output vector: q R+n; input vector: z R+m; input factor price vector: w R+m; Recall that for the given output vector q, the input requirement set is defined as V(q) = z: (-z, q) Y Cost Function. The cost function of a firm is the function c(w, q) = min wz s.t. z V(q)defined for all w 0, q 0. If there is a single output and the production technology is fully represented by the production function q = f(z), then c(w, q) = min wz s.t. f(z) q If z(w, q) solves this minimization problem, then c(w, q) = wz(w, q) The solution z(w, q) is referred to as the firms conditional input demand functions (also known as conditional factor demand functions), since it is conditional on the level of output q, which at this point is arbitrary and so may or may not be profit-maximizing. The inequality constraint can usually be replaced by the equality. Calculus Analysis of Cost MinimizationConsider the following cost-minimizing problem: c(w, q) = min wz s.t. f(z) = q Then the corresponding Lagrange function is L(z, l) = wz - l (f(z) - q) Factor 2 (z2)C = w1 z1 + w2 z2 (Isocost)f(z1, z2) = q (Isoquant)Factor 1 (z1)which leads to the geographical illustration of the cost minimization (tangency condition) indicated as below. The above figure indicates that there is also a second-order condition that must be checked, namely, the isoquant must lie above the isocost line. This, for the case of two inputs, leads to that the bordered Hessian matrix of the Lagrangian, has a negative determinant. Examples: Cost function for the Cobb-Douglas technology: q = K1/2 L1/2, where K is the capital (with a unit price of w1 - rental) and L is the labor (with a unit price of w2 - wage). Then the corresponding cost function is For the general Cobb-Douglas production function: The corresponding cost function is given by: Cost function for CES Technology: q = (az1r + bz2r)1/r, by using the first-order Lagrangian conditions, we can derive the cost function given by: Cost function for Leontief Technology: Its cost function is given by: c(w, q) = q wa. General Properties of Cost Functions If the production function f is continuous and strictly increasing, then c(w, q) is 1. Zero when q = 0. 2. Continuous on its domain3. For any all w 0, strictly increasing and unbounded above in y. 4. Increasing in w. 5. Homogenous of degree one in w. 6. Concave in w. 5.4Conditional Input Demand Functions For the given cost minimization problem, the solution z(w, q) is the firms conditional input demand function. Applying the usual argument, z(w, q) must satisfy the first-order conditions: Differentiating these identities with respect to w we will have the following: which, in term of matrices, become: From this, we can solve for the substitution matrix z(w, q) by taking the inverse of the bordered Hessain matrix: This result is in fact associated with comparative statics of the conditional input demand functions with respect to the input prices. Shephards Lemma: (The derivative property) Let z(w, q) be the firms conditional input demand function. Assume that c(w, q) is differentiable at w with w 0, and This result is a direct application of Envelope Theorem. Shephards lemma implies that the cost function is a non-decreasing function of input prices. Note: It is easy to see that the conditional input demand functions are homogeneous of degree 0. 5.5Short-Run & Long-Run CostsRecall that the cost function can be expressed as the value of conditional input demands: c(w, q) = wz(w, q). In short-run, some of the inputs are fixed. Let zf be the vector of fixed inputs, zv the vector of variable inputs. We also break the vector of input prices w = (wv, wf). The short-run conditional input demand functions: zv(w, q, zf). Short-run cost function is then given by: sc(w, q, zf) = wv zv(w, q, zf) + wf zf SVC + FC = STC Other derived cost concepts are Short-run average cost (SAC) = sc(w, q, zf)/q Short-run average variable cost (SAVC) = SVC/q Short-run average fixed cost (SAFC) = FC/q Short-run marginal cost (SMC) = sc(w, q, zf)/q In the long-run, all production factors are variable. In this case, the firm must optimize in the choice of zf. We can express the long-run cost function in terms of the short-run cost function in the following way. Let zf(w, q) be the optimal choice of the fixed inputs, and let zv(w, q) = zv(w, q, zf(w, q). Then the long-run cost function is given by: c(w, q) = wv zv(w, q) + wf zf (w, q) = sc(w, q, zf(w, q). Similarly, we will have two derived long-run cost concepts: Long-run average cost (LAC) = c(w, q)/q Long-run marginal cost (LMC) = c(w, q)/qProposition (Cost Envelope) - Relationship Between Short-run & Long-Run CostsFor ease of presentation, we drop the argument of w (the fixed input prices) assume a single fix input z. Then, c(q) = sc(q, z(q). For a given output q*, let z* = z(q*) is the associated (optimal) long-run demand for the fixed input. Then it is clear that The short-run cost, sc(q, z*), must be least as large as the long-run cost, c(q, z(q), for all levels of output. The short-run cost will equal to the long-run cost at the output q = q*, i.e., sc(q*, z*) = sc(q*, z(q*) = c(q*).This implies that the long-run and the short-run cost curves must be tangent at q*. The above result follows from the following argument, which comes from the envelope theorem: since z* is the optimal choice of the fixed input at the output level q*, which implies that Finally, note that if the long-run and short-run cost curves are tangent, the long-run and short-run average cost curves must also be tangent. In other word, the long-run average cost curve is the lower envelope of the short-run cost curves. The geometric illustration of the above result is as follows: AC SAC LACAC(q*,z*)q* qTechnical Appendix of Topic 5Derivation of Cobb-Douglas Production Function from CES FormIt suffices to consider the case for n = 2. Note that We need to work out the limit of f as r 0. Since (a (z1)r + (1- a) (z2)r ) 1 as r 0, the limiting problem of f as r 0 becomes an indeterminate form of 0. To find the limit of this nature, we have to find the limit of the logarithm of f: (Here we have used two results from Calculus: (1) the formula for the derivative of ax: d(ax)/dx = (ln a)(ax); and (2) LHopitals Rule.) Therefore, we have For the general case, the proof is similar. Additional References: Arrow, K. J., H. Chenery, B. Minhas, and R. M. Solow (1961) Capital-Labor Substitution and Economic Efficiency, The Review of Economics and Statistics, vol. 43, 225-250. Blackorby, C. and R. R. Russell (1989) Will the Real Elasticity of Substitut