Production Possibility Frontier

Transcription

1 Division of the Humanities and Social Sciences Production Possibility Frontier KC Border v ::1410 This is a very simple model of the production possibilities of an economy, which was formulated by Abba P Lerner [1] There are m outputs y 1,, y m and n productive factors x 1,, x n Each output is produced according to the production function y i = f i (x i ) There are no intermediate goods, no joint production, and only one (industry) production function for each output The supplies of each productive factor in the economy are fixed at levels x 1,, x n Assume that for each i, the production function satisfies f i : R n + R is continuous, C 2 on R n ++, f i 0 on R n ++, and that the Hessian matrix [ ] fkj i is negative definite on the subspace orthogonal to f i k=1,,n,,n (Throughout this note, subscripts on the production functions denote partial derivatives) These assumptions guarantee that all the second order conditions hold as strict inequalities 1 Production possibility frontier The production possibility set (PPS) is { y R m : 0 y i f i (x i ), i = 1,, m, and } x i x Note that the PPS is compact since the f i s are continuous and the PPS is the continuous image of the compact set { } (x 1,, x m ) R nm : x i 0, i = 1,, n, and x i x (This implicitly assumes free disposal of factors and outputs, but this is not crucial) The production possibility frontier (PPF) is the outer boundary of the PPS The production possibility frontier can be described by the following maximization problem (Here each x i R n + and x i j denotes the quantity of factor j used to produce good i) maximize f 1 (x 1 ) subject to x 1,,x m f i (x i ) = η i i = 2,, m x i j = x j j = 1,, n x i 0 i = 1,, m 1

2 KC Border Production Possibility Frontier 2 The Lagrangean for this maximization is: L(x, λ, µ; η, x) = f 1 (x 1 1,, x 1 ( n) + λ i f i (x i 1,, x i ) n ) n) η i + µ j ( x j x i j Q: Are the gradients of the constraints (wrt x) linearly independent? The answer is yes λ 2 λ m µ 1 µ n x x 1 n x 2 1 f x 2 n fn x m f1 m 1 0 x m n 0 0 fn m 0 1 Figure 1 The columns are the gradients of the constraints To see this it might help to consult Figure 1 Suppose λ 2,, λ m, µ 1,, µ n yield a linear combination of the gradients that adds up to the zero vector Then clearly µ 1 = = µ n = 0 Thus since each fj i > 0, we get λ i = 0, for all i Thus by the Lagrange Multiplier Theorem the first order conditions are (assuming each x i j > 0): λ i fj i i = 1,, m µ j = 0 j = 1,, n where for symmetry we define λ 1 = 1 This implies λ i = f 1 j for any factor j = 1,, n Let y 1 (w, x) be the optimal value function Then by the Envelope Theorem, the slope of v ::1410

3 KC Border Production Possibility Frontier 3 the PPF satisfies y 1 = L η i η i = λ i = f 1 j for any j = 1,, n In other words, λ i is the marginal opportunity cost of a unit of y i in terms of y 1 Also note that f i k = µ j µ k, which is independent of i That is, in every industry the slopes of the isoquants are the same 11 Second order conditions While we re at it, let s check the second order conditions The Hessian of the Lagrangean is the mn mn block-diagonal matrix λ 1 f11 1 λ 1 f 1 1n λ 1 fn1 1 λ 1 f 1 nn λ 2 f11 2 λ 2 f 2 1n λ 2 fn1 2 λ 2 f 2 nn H = λ m f11 m λ m f1n m λ m fn1 m λ m fnn m Let v = (v 1,, v m ) belong to R nm The second order condition is that the quadratic form v Hv is negative semidefinite on the subspace orthogonal to the gradients of the constraints Again referring to Figure 1, it is straightforward to see that this means λ i fjkv i jv i k i 0 for all nonzero v satisfying 1=1 k=1 f i v i = fjv i j i = 0, i = 2,, m, v ::1410

4 KC Border Production Possibility Frontier 4 and vj i = 0 j = 1,, n What about the case i = 1? If we can show that f 1 v 1 = 0, then by our assumption on the gradients of the f i s, each λ i > 0, so by the assumption on the Hessian of the f i s, each bracketed term is nonpositive, and at least one is strictly negative (since at least one v i 0) To see that f 1 v 1 = 0, observe that for each j, v 1 j = m i=2 v i j Thus f 1 v 1 = fj 1 vj 1 = fj 1 vj i i=2 = λ i fjv i i j i=2 = 0 The penultimate equality follows from the first order condition that λ i = µ j = f 1 j for all i 2 Relation to cost minimization Assume that each producer faces the same wages w = (w 1,, w n ) for the factors and minimizes costs To ease notation in this section, I shall suppress the superscripts denoting the particular output The cost minimization problem is to minimize w x subject to y f(x) Form the Lagrangean L(x, γ; w, y) = w x + γ ( y f(x) ) The value function is the cost function c(w, y) By the Envelope Theorem, the marginal cost is MC = c y = L y = γ We also have the first order conditions (check the gradient of the constraint): w j γf j = 0, j = 1,, n assuming each x j > 0 (Note that these implies γ > 0) In other words, f j = w j MC v ::1410

5 KC Border Production Possibility Frontier 5 Now back to the PPF If all firms face the same wages and minimize costs, then y 1 η i = λ i = f 1 j = fj i w j MC 1 w j MC i = MC i MC 1 That is, the marginal opportunity cost of one unit of y i expressed in terms of y 1 is exactly the ratio of the marginal cost of a unit of y i (calculated in terms of wages) relative to the marginal cost of a unit of y 1 What this tells us is that marginal costs (derived from wages) indicate real opportunity costs! 21 Extensions What if there are several production functions for each y i? Call them f i,1,, f i,p 1 Then λ i,k f i,k j µ j = 0, and we proceed as before What if there are joint products? Describe feasibility as where each T y i < 0 and each T x i j T (y 1,, y m, x 1 1,, x 1 n,, x m 1,, x m n ) 0 > 0, and consider the Lagrangean ) y 1 + λt (y 1, η 2, η m, x 1 1,, x 1 n,, x m 1,, x m n ) + µ j ( x j x i j Also how do we deal seriously with the nonnegativity constraints? References [1] Lerner, A P 1934 The concept of monopoly and the measurement of monopoly power Review of Economic Studies 1(3): v ::1410