伦敦经济学院高微讲义-消费者理论.ppt

,The consumer aims to maximise utility.,subject to the budget constraint,Constraint set,This is known as the primal problem.,Solution to the primal problem,x*,Contours of objective function,Alternatively.,the consumer aims to minimise cost.,. subject to a utility constraint,Constraint set,u,This alternative approach is known as the dual problem,The solution to the dual problem,But.,x*,cost minimisation by the firm,Contours of objective function,p1 x1 +p2 x2 = constant,Comparing cost-minimisation for the firm,Lets run through the formalities of the solution.,and for the consumer.,. the difference is only in notation,so their solution functions and response functions must be the same,Use the FOC,At x* we find.,xi* = Hi(p,u),C(p,u),Problem.,.solution,response:,Cost-minimisation: the consumer,Cost-min problems are identical,l U1(x*) = p1 l U2(x*) = p2 . . . l Un(x*) = pn,u = U(x*), ,goods MRS = price ratio,If both i and j are bought.,C(p, u) :=,vector of goods prices,Specified utility level,min pi xi,U(x) u,The solution is the consumers cost function or expenditure function,Example:Cobb-Douglas utility two goods,Solve for conditional demands,C(p, u) =,C(p, u)=,Non-decreasing in every price. Increasing in at least one price Increasing in utility u. Concave in p Homogeneous of degree 1 in all prices p. Shephards lemma.,Properties of The Cost Function,(Check that all this is just a “translation” from the lecture 2 on the firm),so we can link up their solution functions and response functions,and the dual problem,are equivalent,Lets run through the primal problem,Again use the FOC,At x* we find.,The primal problem,How to solve the problem,Now that we know the constraints on the consumer and the objective function,The obvious thing to do might be to set up a standard Lagrangean and solve it:,Objective function,Constraint,parameter,parameters,decision variable,Lagrange multiplier,U1(x*) =n p1 U2(x*) =n p2 . . . Un(x*) =n pn, ,A set of n +1 conditions,goods MRS = price ratio,If both i and j are bought.,V(p, M) :=,Si=1 pi xi M ,n,money income,vector of goods prices,max U(x),The solution is the indirect utility function,The Utility function is U(x1,x2),Use First order conditions to get demand functions,X1=D1(P1,P2,M), and X2=D1(P1,P2,M),Substitute these back into Utility function,Gives indirect utility v(P1,P2,M),Example:Cobb-Douglas utility two goods,Solve for conditional demands,indirect utility v(P1,P2,M) =,indirect utility v(P1,P2,M) =,The indirect function works like an “inverse“ to the cost function,The two solution functions have to be consistent with each other. Two sides of the same coin,. so the indirect utility function maps prices and budget into maximal utility,Sometimes funny identities like these can be useful,Just as the cost function maps prices and utility into minimal budget.,Non-increasing in every price. Decreasing in at least one price Increasing in income M. Quasi-convex in prices p Homogeneous of degree zero in (p , M) Roys Identity,The Indirect Utility Function has some familiar properties.,u = V(p, M),(.use the definition of the optimum),= V(p, C(p,u),0 = Vi(p, C(p,u) + VM(p, C(p,u) Ci(p,u),(.Differentiating),0 = Vi(p, M) + VM(p, M) xi*,(.using Shephards Lemma),and so.,The ordinary demand function for commodity i,Roys Identity.?,Cobb-Douglas utility two goods,Utility maximisation: summary,Cost-minimisation: the consumer,A long(ish) summary,A lot of the basic results of the consumer theory can be found without too much hard work. We need two “tricks” 1. A simple relabelling exercise: cost minimisation is reinterpreted from output targets to utility targets. 2. The insight that utility maximisation subject to budget is equivalent to cost minimisation subject to utility.,THE FIRM min cost of inputs subject to output target Solution is of the form C(w,Q),THE CONSUMER min budget subject to utility target Solution is of the form C(p,u),1. Cost minimisation: two applications,PRIMAL max utility subject to budget constraint Solution is a function of (p,M),DUAL min budget subject to utility constraint Solution is a function of (p,u),2. The consumer: two equivalent approaches,Hi(p,u),Di(p,M),V(p,M),compens