This preview shows page 1. Sign up to view the full content.
Unformatted text preview: X+ PY Y = Solving these two equations for quantities of goods X and Y demanded, we obtain consumer demands as functions of prices PX , PY, and income, . X = __ 2PX Y = __ 2PY If we substitute these demands into the objective function, we get the optimal value of utility. U = X0.5Y0.5 U = [ __ . __ ]1/2 2PX 2PY U = _________ 2(PX PY)1/2 If we rearrange this result, we get the consumer expenditure function as a function of prices PX , PY , and utility level, U. = 2(PX PY)1/2 U SQUARE ROOT PRODUCTION L = _r_ K w rK + wL = C Solving these two equations for quantities of inputs K and L demanded, we get producer demands as functions of prices r , w , and cost, C. K = _C_ 2r L = _C_ 2w If we substitute these demands into the objective function, we get the optimal value of output. Q = K0.5L0.5 Q = [ _C_ . _C_ ] 1/2 2r 2w Q = ____C____ 2(r w)1/2 If we rearrange this result, we get the producer's cost function as a function of prices r , w, and output level, Q. C = 2(r w)1/2 Q 37 HOMEWORK: 1. Find the MRS for each of the following utility functions: a) b) c) d) e) U(X,Y) = 3X0.3Y0.7 U(X,Y) = 3X0.7Y0.3 U(X,Y) = 0.3X + 0.7Y U(X,Y) = 3X + Y0.7 U(X,Y) = min{3X , 2Y} 2. Show that a CobbDouglas production function defined by Q = f(K,L) = KL with positive constants, > 0 capital share, > 0 labour share, > 0 scale constant, has a marginal rate of technical substitution of: MRTS = L. K 3. Consider the CobbDouglas utility function U = U(X, Y, Z) for the case of three goods X, Y, and Z defined by: U = U(X, Y, Z) = X1/3Y1/3Z1/3 What are the marginal utilities and marginal rates of substitution for this utility function? 4. Show that the MPK of Q = f(K,L) = (K + L)1/ is MPK = (Q / K)1 + . 38 ECON 301 LECTURE #3 DUALITY OF PRODUCTION AND COST What do we mean by the duality of production and cost? Well, essentially there are many distinct similarities in the logical structure of the theories of production and cost. In reality, production and cost are two sides of the same coin. That is, an efficient producer must consider t...
View
Full
Document
This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.
 Spring '10
 sning
 Economics

Click to edit the document details