Advanced Microeconomics

Transcription

1 Example November 20, 2012

2 Cost minimization Derive the cost function and conditional factor demands for the Cobb-Douglas utility function of the form: The cost minimization problem is: q = f (z 1, z 2 ) = z α 1 z β 2 Min w 1 z 1 + w 2 z 2 subject to q = f (z 1, z 2 ), z 1,z 2 where q is an arbitrary output level. The Lagrange function is: L = w 1 z 1 + w 2 z 2 λ(z α 1 z β 2 q)

3 The rst order conditions (FOCs) Dividing (1) by (2) we get: L z 1 = w 1 λαz α 1 1 z β 2 = 0 (1) L z 2 = w 2 λβz α 1 z β 1 2 = 0 (2) L λ = zα 1 z β 2 q = 0 (3) w 1 = α z 2 w 2 β z 1

4 Get z 1 and z 2 Solve for z 2 : w 1 β w 2 α z 1 = z 2 Substitute to (3): z α 1 ( ) β w1 β w 2 α z 1 = q z 1 ( ) β w1 β = q w 2 α

5 Conditional factor demands Take into account the cost function: C(w 1, w 2, q) = z 1 (q)w 1 + z 2 (q)w 2 C(w 1, w 2, q) = z 1 (q)w 1 + w 1 w 2 β α z 1w 2 = = z 1 (w 1 + w 1 β α ) = z 1 w 1 ( α + β α ) ( ) α C(w1, w 2, q) z 1 (w 1, w 2, q) = α + β w 1 ( ) β C(w1, w 2, q) z 2 (w 1, w 2, q) = α + β w 2

6 But what is the cost function? Take the production function: q = f (z 1, z 2 ) = z α 1 zβ 2 Substitute conditional factor demands: ) C(w1, w 2, q) q = (( α α + β w 1 ) α (( β α + β (( ) α 1 q = C(w 1, w 2, q) α + β w 1 ) C(w1, w 2, q) w 2 ) β ) α (( β ) ) β 1 α + β w 2 (( ) ) α/() (( ) ) β/() α + β α + β C(w 1, w 2, q) = w 1 w 2 q 1 α β

7 Cost function (( ) ) α/() (( ) ) β/() α + β α + β C(w 1, w 2, q) = w 1 w 2 q 1 α β or: ( α + β C(w 1, w 2, q) = α ) α ( α + β β ) α w α 1 w α 2 q 1 where θ = ( ) α α C(w 1, w 2, q) = θw α ( β ) α 1 w α 2 q 1 is just a parameter.

8 The cost function or even: C(w 1, w 2, q) = θw α 1 w α 2 q 1 C(w 1, w 2, q) = q 1 θφ(w1, w 2 ), where φ(w 1, w 2 ) = w α/() 1 w β/() 2.

9 Notes C(w 1, w 2, q) = q 1 θφ(w1, w 2 ), In the cost function, only the q 1 scalar if prices are xed. is a function of q. The rest is a α + β is the measure of returns to scale, if > 1 IRS, if = 1 CRS, if 0 < α + β < 1 DRS If α + β = 1, then the function becomes (should remind you the expenditure function for a Cobb-Douglas utility function): C(w 1, w 2, q) = qθw α 1 w 1 α 2, with α + β = 1, MC = AC = θw α 1 w 1 α 2. In this case, θ = α α β β = α α (1 α) (1 α).

10 Prot maximization with cost function We have can now state the prot maximization problem in the following way: FOC: Max q π(p, w 1, w 2, q) = pq C(w 1, w 2, q) = pq q 1 θφ(w1, w 2 ) π q = p ( 1 α + β )q1/() 1 θφ(w 1, w 2 ) = 0 Solving for q from the above will give us the prot maximizing output (the second term after the minus is the marginal cost). However, we have to check SOC

11 Prot maximization with cost function 2 π q 2 = ( 1 α + β 1)( 1 α + β )q1/() 2 θφ(w 1, w 2 ) 0 whenever 0 < α + β 1. so we need to have either decreasing or constant returns to scale. However, with CRS the q disappears from the FOC and we cannot determine the prot maximizing q

12 Prot maximization with cost function any q such that P = MC is in fact prot maximizing. We can get the prot function π(p, w) by substituting out q in π(p, w 1, w 2, q) We can get factor demands by taking derivatives of π(p, w) with respect to w 1 and w 2.

13 Calibration of a Cobb-Douglas producer (1) In the applied economic models, we want to nd parameters that reect the reality. Usually in input output tables and national accounts we only observe values - not prices and quantities. The usual assumption: values=quantities and prices=1. In order for this to be plausible, we need homothethic functions with constant returns to scale. We assume that the observed data is the initial optimal choice by the producer. We choose parameters so that it is indeed the case - calibration.

14 Calibration of a Cobb-Douglas producer (2) Producer has the production function: q = Az α1 1 zα2 2 with α 1 + α 2 = 1. The corresponding cost function is: TC = q 1 A ( w1 α 1 ) α1 ( w2 α 2 ) α2, MC = AC = 1 A ( w1 α 1 ) α1 ( w2 α 2 ) α2, Use Shephard Lemma to get the conditional factor demands: TC q MC z l = α l w 1 = α l w 1 With CRS and price taking, we have p = MC, and therefore z l = α l q p w 1 and p = 1 A ( w1 α 1 ) α1 ( w2 α 2 ) α2 So let say in the data output equals 100 and 70 is spent on factor 1 and 30 on factor 2.

15 Calibration of a Cobb-Douglas producer (2) So let say in the data output equals 100 and 70 is spent on factor 1 and 30 on factor 2. Remember that we assumed that: p = 1, w 1 = 1, and w 2 = 1. Therefore we need MC = 1 and q = 100 and z 1 = 70, z 2 = 30. Using demands that gives us: α 1 = 0.7, and α 2 = 0.3. We need to calibrate A. But we know that p = 1 = MC = 1 A ( )0.7 ( )0.3. So A = ( w1 α 1 ) α1 ( w2 α 2 ) α2 = ( )0.7 ( )0.3 and everything is calibrated. At the same time 100 = Q = Az α1 1 zα2 2 = which gives the same thing. Now we can x the parameters and play with the prices to see the reaction of the producer.