Problem Set 5: Expenditure Minimization, Duality, and Welfare. Suppose you were given the following expenditure function: ) ep,ū) = ūp p where 0 <, < and =. Derive Hicksian demand, Walrasian demand, and the indirect utility function. What is the utility function that this expenditure function was derived from? Be sure to briefly explain each calculation along the way. Since p ep,u) = hp,u), Hicksian demand is ) h p,u) = up p h p,u) = up p ) Since ep,vp,w)) = w, the indirect utility function is ) w = vp,w)p p vp,w) = w p p ) p p and by oy s identity, x p,w) = vp,w)/ p w ) vp,w)/ w = p p ) yielding x p,w) = w p x p,w) = w p If you look f the Walrasian demands in the indirect utility function, you get ) w ) w ) vp,w) = p p which equals vp,w) = ) ) x x ) ux,x ) = x ) x ) K so this was generated by a Cobb-Douglas utility function.
. Consider a consumer with constant elasticity of substitution preferences CES), ux,x ) = x x )/ where <. i) Solve the expenditure minimization problem f Hicksian demand and derive the expenditure function, ep,u). Verify that p ep,u) = hp,u). ii) Use the expenditure function to derive the indirect utility function. iii) Use the indirect utility function to derive Walrasian demand. iv) Verify that the Walrasian demands you derived in part iii match the solution to the UMP by solving the UMP from scratch. i) The Lagrangian f the EMP is with FONCs The first two equations imply that Solving f x yields substituting into the constraint yields L = p x p x λu x x ) p λx = 0 p λx = 0 u x x ) = 0 x = p = x p p u = x p x ) / ) x p p ) / ) x yielding so that ep,u) = p u ep,u) = u h p,u) = u h p,u) = u p / ) p / ) p / ) p / ) p / ) p / ) p / ) p / ) p / ) p / ) p / ) p / ) ) / p u ) / u p ) / ) / ep,u) = u p/ ) p / ) p / ) p / ) ) / p / ) p / ) p / ) / ) p / ) p / ) ) / ) /
and letting / ) = σ, ) ep,u) = u p / ) p / ) / ep,u) = up σ pσ )/σ Taking derivatives with respect to p and p yields the Hicksian demands. ii. To get the indirect utility function, vp,w) = wp σ pσ ) /σ iii) Use oy s identity on the indirect utility function to get the Walrasian demands. iv) The Lagrangian f the UMP is with FONC s The first two equations imply substituting into the constraint yields so that and the indirect utility function is vp,w) = w L = x x λp x p x w) x λp = 0 x λp = 0 p x p x w) = 0 ) / ) p x = x p ) / ) p w = p x p x p wp / ) = p x p / ) p / ) p x x = x = p / ) p / ) p / ) vp,w) = w wp / ) p / ) p / ) wp / ) p / ) p / ) ) p/ ) p / ) p / ) p / ) p / ) p / ) p / ) ) / ) vp,w) = w p / ) p / ) )/ ) ) / 3
3. A linear functional fx) is a mapping f : x N satisfying fx) = x = x =< x, > depending on your notational preferences. Let K be a convex set in N. The suppt functional of K is S) = sup x x K Note that S) is a functional mapping linear functionals, fx) = x, into real numbers, S : N. The suppt functional S) asks you to find the x that maximizes x, taking as given, so it is really a value function. i) Graph the unit circle, K = {x,x ) : x x }, and f =,0), =,) and 3 =,), graph x = u f some different values of u and solve the constrained maximization problem sup x K x. ii) Now, think of S) as the value function associated with the constrained maximization problem sup x K x where is allowed to vary, in the case of the unit circle. Show that by varying, we can recover K and x ). Explain how this is also true f all convex sets K. iii) Explain why the recovery idea from part ii) will not wk f a non-convex set K, but will wk f the convex hull of K, where we define cok) = {x : x = λy λ)y,{y,y } K,λ 0,)} S) = sup x x cok) Explain the relationship between randomization and the closed convex hull, and then explain how we can convexify choice sets that are non-convex so that our duality ideas still wk. iv) Explain how these ideas are a generalization of the relationship between the UMP and EMP, and oy s identity. i. Solution to the maximum problem: gives first-der necessary conditions L = x λx x ) λx = 0 λx = 0 x x ) = 0 Then the first two equations imply that /x = /x, x = x /. Substituting into the constraint gives x x = x = sign ),x = sign ) ii. Now, if we start with the value function, S) = sgn ) sgn ) 4
S) = = Differentiating S) with respect to yields x ) = sign ),x ) = sign ) and these equations jointly trace out a circle of radius as you vary. iii. Non-convexities involve points on the interi of the convex hull, which will never maximize a linear functional, since we can always move to a higher indifference curve and improve the value of the objective. Upon taking the convex hull, this problem goes away, since the extreme points of the simplex can always be made to be maximizers of some linear functional by invoking the separating hyperplane theem. iv. This exactly how ep,ū) = hp,ū) allows us to vary prices to recover the indifference curve ux) = ū. So when the feasible set is convex, a value function based on a linear objective contains all the infmation necessary to recovery infmation about the feasible set: Just use the envelope theem to get the optimal controls, then vary the parameters to trace out the set. This problem is like the EMP, but we might wonder also how it relates to the UMP. The dual of the above problem is min x,x x x subject to x x r. By varying here, we trace out a set of solutions, but it won t recover the budget set. There is something like a wealth vs. substitution effect that arises when changing the constraint. This is me like what occurs in the UMP, where we end up using oy s identity to recover the maximizers from the indirect utility function by way of its connection with the EMP. 4. F utility function ux,x ) = x γ )x and budget constraint w = p x p x, derive the agent s money-metric utility function. Provide a general expression f EV and CV, and compute these f changes from p 0 = p0 = to p = and p0 =. The expenditure function is ep,u) = p p up γ and the indirect utility function is ) w p γ vp,w) = p p So the money-metric utility function is ep a,vp b,w)) = p a pa p b pb w p b γ )p a γ Then p 0 EV = p0 p w p γ )p 0 p 0 γ p0 p p 0 w p 0 γ ) p 0 p 0 γ = p0 p0 p w p γ ) w p 0 γ ) p 5
and p CV = p p w p p γ )p γ p p p 0 w p 0 p0 γ ) p γ = w p γ p p p 0 w p 0 p0 γ ) and plugging in the numbers gives: So they are similar but different. EV = /w γ ) w γ ) CV = w γ w γ ) 5. Consider a consumer with expenditure function ep, u) and Hicksian demands hp, u). Suppose the government puts a tax on the first good, so that p = p0 t, without adjusting any of the other prices. The tax revenue is then T = tx p,w). This method of taxation is disttionary because it changes the relative prices of goods. Instead, consider a lump-sum tax T equal to tx p,w) that comes directly out of the agent s wealth, w T. The deadweight loss of commodity taxation is equal to w T ep 0,u ) = T EVp 0,p,w) i. Derive an expression f the deadweight loss in terms of integrals of the Hicksian demand function. ii. Does the deadweight loss measure based on Walrasian, rather than Hicksian demand, understate overstate the deadweight loss? iii. Sketch a graph of the Hicksian demand and the deadweight loss. Sketch the Walrasian demand on the same graph, and explain your results from part ii graphically. iv. Compute the derivative of the deadweight loss with respect to the tax, t, and sign it. i. The deadweight loss is DWL = T EVp 0,p,w) = ep,u ) ep 0,u ) th p,u ) = Then p = p0 t equals = t p 0 ez,p,u ) p dz p 0 t p0 )h p 0 t,p,,u ) = p 0 t So this is just the areas behind the Hicksian demand curve. ii. If we rewrite it as t t h y,p,u )) dydz z p we can then use the Slutsky equation t t{ x y,p,w) ) p t p 0 p 0 z { x z,p,w) x p 0 t,p,w) } t p 0 p p 0 ez,p,u ) p dz th p,u ) h z,p,u ) h p 0 t,p,u )dz x } y,p,w) x y,p,w) dydz w z x y,p,w) x y,p,w)dydz w 6
The first term in braces integrates to deadweight loss based on Consumers Surplus/Area Variation. So if the good is nmal, then the second term is negative, and throwing it away will reduce our estimate of the loss. So DWL based on CS/AV underestimates the loss in the case of a nmal good, and overstates it in the case of an inferi good. iii. Sketched in class. iv. The derivative is DWL t) = t h p 0 t,p,u ) p > 0 And you can see again that DWL will tend to understimate in the case of a nmal good, since { DWL x y,p,w) t) = t x } y,p,w) x y,p,w) t x y,p,w) p w p so that the increase in the loss is greater f a nmal good when measured with Hicksian demand/ev versus Walrasian demand/av/cs. 7