Department of Economics Microeconomic TheoryUniversity of California, Berkeley Economics 101AJanuary 31, 2007 Spring 2007Economics 101A: Microeconomic TheorySection Notes for Week 31 The Envelope TheoremThink back to the derivations of Roy’s Identity and Shepard’s Lemma (from week 2 section)and the interpretation of the LaGrange multiplier (from week 1 section).LaGrange Multiplier as a shadow price of loosening the budget constraint.∂V∂I= λ∗(p1, p2, I)Roy’s Identity∂V∂p2= −λ∗(p1, p2, I)x∗1(p1, p2, I)Shepard’s Lemma∂e∂p2= xc1(p1, p2, u0)The derivations of these identities are applications of the envelope theorem. To illustratethe envelope theorem, we return to the general constrained maximization setup from week1 section, however instead of focussing s olely on the endogenous variables (x1, x2), we alsoconsider an exogenous variable ρ which can potentially affect the objective function f or theconstraint function g. (In the context of the UMP or EMP the exogenous variable could bep1, p2, or I.)maxx1,x2f(x1, x2, ρ) s.t. g(x1, x2, ρ) = kThe LaGrangian is thenL(x1, x2, λ, ρ) = f(x1, x2, ρ) − λ[g(x1, x2, ρ) − k]and the first order necessary condidtions (FOC’s)L1(x∗1, x∗2, λ, ρ) = f1(x∗1, x∗2, ρ) − λ∗g1(x∗1, x∗2, ρ) = 0L2(x∗1, x∗2, λ, ρ) = f2(x∗1, x∗2, ρ) − λ∗g2(x∗1, x∗2, ρ) = 0L3(x∗1, x∗2, λ, ρ) = g1(x∗1, x∗2, ρ) − k = 0Note, there is no FOC from the partial derivative of the LaGrangian with respect to ρsince it is an exogenous variable, and thus can not b e manipulated by the person doing the1optimization. Solving the maximization leads to optimal values of the choice (or endogenous)variables as functions of the exogenous variable,x∗1(ρ)x∗2(ρ)if we were solving the UMP these would be the ordinary demand functions. Substitutingthese back into the objective function gives the value function (or indirect utility functionfor the UMP)F (ρ) = f(x∗1(ρ), x∗2(ρ), ρ)which is the maximum (or minimum if the objective is to minimze) possible value of theobjective function given a certain value of the exogenous variable ρ. Now we can take thetotal derivative of the value function with respect to ρ to getdF (ρ)dρ=∂f∂x∗1·dx∗1dρ+∂f∂x∗2·dx∗2dρ+∂f∂ρNext, we can use the FOC’s to substitute in for∂f∂x∗1and∂f∂x∗2.dF (ρ)dρ= λ∗g1(x∗1, x∗2, ρ) ·dx∗1dρ+ λ∗g2(x∗1, x∗2, ρ) ·dx∗2dρ+∂f∂ρ= λ∗g1(x∗1, x∗2, ρ) ·dx∗1dρ+ g2(x∗1, x∗2, ρ) ·dx∗2dρ+∂f∂ρwe can also take the total derivative of the constraint g1(x∗1, x∗2, ρ) = k which gives∂g∂x∗1·dx∗1dρ+∂g∂x∗2·dx∗2dρ+∂g∂ρ= 0∂g∂x∗1·dx∗1dρ+∂g∂x∗2·dx∗2dρ= −∂g∂ρwhich can be substituted into the expression above to givedF (ρ)dρ= −λ∗(ρ)∂g(x∗1, x∗2, ρ)∂ρ+∂f(x∗1, x∗2, ρ)∂ρremember this relationship holds at the optimum and that the optimal LaGrangian multiplierwill generally be a function of the exogenous variable ρ. Note that this expression is equalto the partial derivative of the LaGrangian with respec t to the exogenous variable evaluatedat the optimum.∂L(x∗1, x∗2, λ∗, ρ)∂ρ=∂f(x∗1, x∗2, ρ)∂ρ− λ∗(ρ)∂g(x∗1, x∗2, ρ)∂ρ=dF (ρ)dρ1.1 Example: Utility Maximization Problem (UMP)In the UMP the objective is to maximize utility sof(x1, x2, ρ) = u(x1, x2)2and the constriant is to do so within the budget sog(x1, x2, ρ) = p1x1+ p2x2− INext thing to note is that for the UMP we have a special name for the the value function.We call it the indirect utility functionF (ρ) = f(x∗1(ρ), x∗2(ρ), ρ) = V (p1, p2, I) = u(x∗1(p1, p2, I), x∗1(p1, p2, I))Lets consider the impact of an exogenous change in p1. We can evaluate the partials of fand g with respect to the exogenous variable p1at the optimum.∂f(x∗1, x∗2, ρ)∂ρ=∂u(x∗1, x∗2)∂p1= 0∂g(x∗1, x∗2, ρ)∂ρ=∂(p1x∗1+ p2x∗2)∂p1= x∗1then using the envelope condition we getdVdp1= 0 − λ∗x∗1which is Roy’s Identity and which we dervied in week 2 of section.1.2 Example: E xpense Minimization Problem (UMP)In the EMP the objective is to minimize expenditures sof(x1, x2, p1) = p1x1+ p2x2and the constriant is to do so with in the budget sog(x1, x2, p1) = u(x1, x2) = u0For the EMP we also have a special name for the value function. We call it the expenditurefunction.F (ρ) = f(x∗1(ρ), x∗2(ρ), ρ) = e(p1, p2, u0) = p1xc1(p1, p2, u0) + p2xc2(p1, p2, u0)Lets consider the impact of an exogenous change in p1. We can evaluate the partials of fand g with respect to the exogenous variable p1at the optimum.∂f(x∗1, x∗2, ρ)∂ρ=∂(p1xc1+ p2xc2)∂p1= xc1∂g(x∗1, x∗2, ρ)∂ρ=∂u(xc1, xc2)∂p1= 0then using the envelope condition we getdedp1= xc1− λ∗· 0 = xc1which is Shepard’s Lemma.32 Other Identitiese(p1, p2, V (p1, p2, I)) = IMeaning that the minimal expenditure needed to achieve the maximal amount of utility thatcan be achieved at prices p1, p2and income I is I.V (p1, p2, e(p1, p2, u0)) = u0Meaning that that maximal amount of utility that can be achieved at prices p1, p2and incomelevel equal to the minimal level of expenditure required to achieve utility u0is (of course)u0.xc1(p1, p2, u0) = x1(p1, p2, e(p1, p2, u0))xc2(p1, p2, u0) = x2(p1, p2, e(p1, p2, u0))Meaning that ordinary and compens ated demand will be equal if the amount of income thatthe consumer has happens to be equal to the minimal amount necessary to achieve utilitylevel u0. (Note I am denoting the ordinary demand function with no superscript.)x1(p1, p2, I) = xc1(p1, p2, V (p1, p2, I))x2(p1, p2, I) = xc2(p1, p2, V (p1, p2, I))Meaning that ordinary and compensated demand will be equal if the utility level that theconsumer must achieve is equal to the maximal utility level possible given that they haveincome I.2.1 ExampleLet’s examine the UMP for the specific utility function u(x1, x2) = x1+ log x2maxx1,x2x1+ log x2s.t. p1x1+ p2x2The FOCs are1 − λ∗p1= 01x∗2− λ∗p2= 0p1x∗1+ p2x∗2= IThe tangency condition isp1p2= x∗2The ordinary demand functions arex1(p1, p2, I) =I − p1p1x2(p1, p2, I) =p1p24Pluggin the ordinary demand functions into the utility function we get the indirect utilityfunctionV (p1, p2, I) =I − p1p1+ logp1p2Now as long as it is easy to solve for I we can use our identities from above to find theexpenditure function quicklyV (p1, p2, e(p1, p2, u0))