x 2 λp 2 = 0 x 1 γ 1 λp 2 = 0 (p 1 x 1 +p 2 x 2 w) = 0 x 2 x 1 γ 1 = p 1 p 2 x 2 = p 1 (x 1 γ 1 ) x 1 = w +p 1γ 1 2p 1 w +p1 γ 1 w p1 γ 1 2p 1 2p 2

Transcription

1 Problem Set 7: Welfare and Producer They. F utility function u(x,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 p0 to p and p0. We need both the expenditure function and the indirect utility function. The Lagrangian f the UMP is L (x γ )x λ(p x +p x w) with FONCs The first two equations imply so that Substituting into the budget constraint yields x λp 0 x γ λp 0 (p x +p x w) 0 x x γ p p x p p (x γ ) p x +p (x γ ) w x w +p γ p x w p γ p So the indirect utility function is ( )( ) w +p γ w p γ v(p,w) γ p p and the expenditure function is then v(p,w) (w p γ ) 4p p e(p,u) p γ + 4up p Fixing p, the money metric utility function is then φ( p,p,w) p γ + 4 p p (w p γ ) 4p p See how this is e( p,v(p,w)), so it gives the amount of wealth necessary at prices p to achieve the same utility as at prices p with wealth w. To get to CV and EV, we fix p and compare two different utility levels: CV e(p,v(p,w)) e(p,v(p 0,w)) p γ + 4p p (w 0 p γ ) p γ 4p p 4p p (w γ ) 4 p0

2 EV e(p 0,v(p,w)) e(p 0,v(p 0,w)) γ + 4 p0 (w p γ ) γ 4p p The computation is trivial, but note that they are not the same. 4 p0 (w γ ) 4 p0. Consider a consumer who buys goods x and x has indirect utility function ( ) α α ( ) β β v(p,p,w) w where α+β. i. Derive the expenditure function and Hicksian demand. ii. Suppose the price of good one increases from to p. A policymaker asks you the following question: How can I adjust the price of the second good from to p so that the consumer is as well off as befe the price of good one changed? Characterize the necessary change in p to offset the price change (you do not need to provide a closed-fm solution, but do your best to at least sign the change). How would your answers generalize f a generic expenditure function e(p,u) and Hicksian demands h (p,p,u) and h (p,p,u)? ii. Suppose the policymaker decides that instead of your recommendation, he is going to evaluate the welfare cost of the price change in p using the Area Variation measure (based on Consumers Surplus), since he has no data on e(p,u) h(p,u). Is his choice of p going to be too high, too low, the same as your recommendation? Explain. iii. F a generic e(p,u) and h(p, u), use the implicit function theem to characterize the derivative of the offsetting change in p as a function of where p p0 + ; is it ambiguous unambiguous in sign? F small changes, as 0, compute p lim ( ) 0 Explain your result. i. The expenditure function and Hicksian demands are p p e(p,u) upα pβ α α β β h u α α β βαpα p β h u α α β βpα βpβ ii. We basically want to change p in der to set CV equal to zero. In this case, CV is equal to CV p 0 p e(p,u 0 )dc where C is a smooth contour from p 0 to p. Then CV equals Note that CV e(p 0,u 0 ) e(p,u 0 ) e(p,p,u0 ) e(y,p,u0 ) p e( dy +,y,u0 ) dy +e( p p 0 p,p0,u0 )

3 and substituting this in yields CV Since p e(p,u) h(p,u), we have e(y,p,u0 ) p dy p e(,y,u0 ) dy p To set this equal to zero implies that p CV h (y,p,u0 )dy h (,y,u0 )dy h (y,p,u 0 )dy h (,y,u 0 )dy So we get to control p, which actually appears on both sides. In particular, the left-hand side is positive, but the right-hand side s sign is determined by whether p is greater less than. The integrand is positive, so f the right-hand side to be positive we need p < p0 : To offset the higher price of good one, we must lower the price of good two. BUT p also appears on the left-hand side, don t fget. If we plug in our specific Hicksian demands, we get which equals u 0 α α β βαyα (p )β dy u 0 α α β β(p0 )α βy β dy u 0 α α β β(p )β (p α α ) u0 α α β β(p0 )α (p β β ) (p ) β (p α α ) () α ( β p β ) (p )β p α ( )α β ( p p 0 α β p α ii. To compare CV with a measure based on consumers surplus and Walrasian demand, we use the Slutsky equation. The sht answer is that since ) /β h l p k x l p k + x l x k the Hicksian demand curve is steeper than the Walrasian demand curve f nmal goods since the wealth effect is positive, and flatter than the Walrasian demand curve f inferi goods since the wealth effect is negative. Since CV h (y,p,u0 )dy h (,y,u0 )dy 0, the since of the two integrals will be understated by the AV approximation, but it is unclear which approximation err dominates, so we can t be sure if the welfare loss is under- over-stated by using AV. If we want CV to be equal to zero, this implies CV h (y,p,u0 )dy h (,y,u0 )dy 0 3

4 The second integral is no problem, since h (,y,u 0 ) and using the Slutsky equation, h (,y,u 0 ) which equals yielding h (,z,u0 ) p dz +h (,,u 0 ) x (,z,e(p0,z,u0 )) + x (,z,e(p0,z,u0 )) x (p 0 p,z,e(,z,u 0 ))dz h (,y,u0 ) x (,y,e(p0,y,u0 ))+ h (,y,u 0 )dy x (,y,e(,y,u 0 ))+ +x (,p0,e(p0,u 0 )) x (,z,e(p0,z,u0 )) x (,z,e(p0,z,u0 ))dz x (,z,e(p0,z,u0 )) x (p 0,z,e(,z,u 0 ))dzdy The first integral, however, requires me caution because the integrand is h (y,p,u0 ), involving u 0 and p. To switch to Walrasian demand x(p,w), we need to substituted in the crect expenditure function f w. The crect one is e(y,p,u0 ), so that h (y,p,u0 ) x (y,p,e(y,p,u0 )) and we can then play the same trick with the Slutsky equation, yielding Then h (y,p,u 0 )dy CV x (,y,e(y,p,u 0 ))+ x (,y,e(,y,u 0 ))+ x (,y,e(y,p,u 0 ))+ x (z,p,e(z,p,u0 )) x (z,p,e(z,p,u 0 ))dzdy x (,z,e(p0,z,u0 )) x (p 0,z,e(,z,u 0 ))dzdy Setting this equal to zero and re-arranging yields p x (,y,e(,y,u 0 ))+ x (,y,e(y,p,u 0 ))+ x (z,p,e(z,p,u0 )) x (z,p,e(z,p,u 0 ))dzdy x (,z,e(p0,z,u0 )) x (p 0,z,e(,z,u 0 ))dzdy x (z,p,e(z,p,u0 )) x (z,p,e(z,p,u 0 ))dzdy In the AV measure, we throw away the wealth effect terms on both sides, leaving the integrals of two Walrasian demand functions. However, it is unclear what effect this will have. To see this, let s suppose that good one is nmal. Then p x (,y,e(,y,u 0 ))+ x (,z,e(p0,z,u0 )) x (p 0,z,e(,z,u 0 ))dzdy > x (,y,e(y,p,u0 ))dy 4

5 To throw away the wealth effect on the left-hand side but maintain the inequality, we must assume that good two is inferi, so that the wealth effect is negative. Then p x (,y,e(,y,u 0 ))dy x (,y,e(y,p,u 0 ))dy > 0 So that we must be under-compensating the agent f the change using a welfare measure based on Walrasian demand (since CV 0, but the AV above evaluated at the prices that achieve CV 0 is positive). In the case where both goods are nmal, we cannot wk out a string of inequalities like the one above, so we cannot say whether we are over- under-adjusting the price of good two. Or, if CV 0, we have y AV + p x (,z,e(p0,z,u0 )) x (,z,e(p0,z,u0 ))dzdy y x (z,p,e(z,p,u0 )) x (z,p,e(z,p,u 0 ))dzdy 0 and since the two integral terms can go in opposite directions, we cannot decide whether the agent is being under- over-compensated. iii. In the generic situation, we must have CV + Differentiating with respect to yields h ( + +,p ( ),u0 ) h (y,p ( ),u0 )dy ( ) h (,y,u0 )dy 0 h (y,p ( ),u0 ) p ( ) p dy h (,p ( ),u0 ) p ( ) 0 p ( ) h ( +,p ( ),u0 ) + h (y,p ( ),u0 )/ p dy +h (,p ( ),u0 ) Taking the limit as 0 yields p (0) h (,p (0),u0 ) h (,p (0),u0 ) which equals the ratio of Walrasian demands at the initial prices, p (0) x (p 0,w) x (p 0,w) So f small price changes, the adjustment in prices is equal to the ratio of quantities they are consuming. 3. Consider a consumer with expenditure function e(p, u) and Hicksian demands h(p, 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) 5

6 that comes directly out of the agent s wealth, w T. The deadweight loss of commodity taxation is equal to w T e(p 0,u ) T EV(p 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. Deadweight loss is DWL(t) +t ii. To compare it to Walrasian demand, note that +t DWL(t) h (z,p,u ) h ( +t,p,u )dz +t z +t z h (y,p,u ) +t p +t x p + x x dydz x (z,p,u ) x ( +t,p,u )+ DWL(t)+ +t z z dz +t + x x dydz +t + x x dydz where DWL(t) is the deadweight loss measure based on Walrasian demand. Since z < +t, the second term is negative f a nmal good (Walrasian-based DWL understates the true deadweight loss) and positive f an inferi good (Walrasian-based DWL overstates the true deadweight loss of an inferi good). iii. Remember Hicksian demand is steeper f nmal goods and flatter f inferi goods. iv. DWL (t) h ( +t,p,u ) h ( +t,p,u +t ) h ( +t,p,u ) dz p t h ( +t,p,u ) p Which is negative as long as h / p > 0. Since e(p,u) is concave in p and e(p,u) h(p,u), then e(p,u) h(p,u), so all the own-price effects of Hicksian demand are negative, since they are the diagonal terms of a negative semi-definite matrix. Therefe, DW L is always negative. 6

7 4. Suppose a firm s technology is characterized by y y y < 0 F(y,y ) 0 y 0 y y y > 0 i. Sketch the production set Y. Verify that it is convex, satisfies irreversibility, non-increasing returns to scale, and is closed. ii. Suppose p < p. Solve the profit maximization problem. Verify that π(p) is homogeneous of degree one in p. Verify that y(p) is homogeneous of degree zero and is the derivative of the profit function. Show that the law of supply holds and that y k (p)/ p k 0. iii. Show that if y and y, there exists a price vect f which this production plan is optimal. Show that if y 4 and y, there exists a price vect f which this production plan is optimal. Show that whenever y < 0, p < p, and y (p) is an efficient plan f that y, there is a price vect f which (y (p),y ) is optimal. i. Note that when y < 0 and F(y,y ) 0 implies that y y (likewise, y > 0 and F(y,y ) 0 implies y y ). This uses y as an input to produce y. Note that if α >, αy < α y < α y, so that we have non-increasing returns to scale (actually, decreasing returns to scale). The production set Y is closed, since F(y,y ) 0 is a continuous function, so that the set F(y,y ) 0 contains all of its limit points (, the complement of Y is open because any point y that is not in Y has a ball around it which contains no points of Y. Since the complement of Y is open, then Y must be closed). If (y,y ) is in Y and y < 0, then the inverse of this process would be ( y, y ). Let s take y 4 which implies y ; then y and y 4 should be part of the production set. But using y as an input yields y ( ) < 4. So the process is irreversible due to the decreasing returns to scale. I thought showing convexity was the hardest part, personally. Let s start with two feasible production plans y and y which both have y < 0. Then we have the two inequalities Multiplying by λ and λ and adding yields y y 0 y y 0 λy +( λ)y λ y ( λ) y 0 Since λ (0,), we get λy +( λ)y λy ( λ)y λy +( λ)y λ y ( λ) y 0 and since a b a+b, we get λy +( λ)y λy ( λ)y λy +( λ)y λy ( λ)y 0 which implies that the production plan λy +( λ)y is feasible when y < 0. This argument also wks f y + y and y < 0, and since if 0 y,y, then 0 λy +( λ)y, we can break the situation into three cases: y 0,y 0, y 0,y 0, and y,y < 0. But note that if we take a vect y in one case and a vect y in another case, the convex combination 7

8 will cross over cases. That is fine: pick any convex combination, λy + ( λ)y. Then the convex combination cresponds to a vect in exactly one case. But we can express this vect as a convex combination of vects that are within that case, f which the above wk applies (thus showing that any convex combination of any two vects is feasible). ii. If p < p, then the slope of the hyperplane p y +p y is less than one, and the firm will use y as an input to produce y (tangency is not possible otherwise). This implies an objective maxp y +p y subject to y y. Substituting the constraint into the objective yields and maximizing yields y maxp y +p y p ( y ) / +p 0 ( p ) ( ) p p p y p p These are HOD-0 in p, since the prices only appear as the ratio (αp )/(αp ) p /p. Since the controls are HOD-0, the profit function satisfies π(αp) αp y(αp) αp y(p) απ(p) so π() is HOD-. It is straightfward to show that the law of supply, Hotelling s lemma, and y k (p)/ p k 0. iii. If we have y and y, the plan is efficient, since ( ), so it lies on the production frontier. This implies using our expressions f the profit-maximizing production plans y p p so that any prices satisfying p p make the production plan (, ) optimal. Likewise, if y 4 and y, the production plan is efficient, and y p p so that any prices satisfying 4p p make the production plan ( 4,) optimal. Now, if we take any efficient plan (y y,y ), we can make it profit maximizing by selecting prices so that ( ) p y p y p p so we only need p y p. (This is a version of the second fundamental theem of welfare economics: any efficient plan can be profit maximizing). 5. Consider a firm with production function q f(z) facing a price p f its output and prices 8

9 w f its inputs z (z,z,...,z L ). Assume that f(z) is twice differentiable, negative definite at all z, and weakly increasing in all its arguments. Say an input z k is inferi at (w,p) if the fact demand satisfies z k (w,p) p and say that z k is nmal at (w,p) otherwise. Suppose that L i. Find necessary conditions f fact z to be an inferi input at (w,p). What do the conditions imply about the relationship between z and z? ii. If z is an inferi input at (w,p), can z be an inferi input at (w,p)? Explain why why not. iii. Let (z,z ) be the profit-maximizing fact demands at (w,p). If q f(z,z ), show that < 0 q z p If z is inferi, what does this imply about how the firm s output responds to changes in prices and input costs? Suppose that L > iv. Show that if p increases, the profit-maximizing level of output always increases, and the demand f at least one input must increase. v. Show that, if q f(z ), where z (z,z,...,z L ) is the profit-maximizing fact demand at (w, p), q z k k p Interpret this relationship economically in terms of how the optimal supply q responds to a change in the price of the k-th input, w k, both when k is inferi and when k is nmal. i. Consider the maximization problem The first-der conditions are and π(p,w) pf(z,z ) w z w z pf (z ) w 0 pf (z ) w 0 The second-der conditions are automatically satisfied since the Hessian is the matrix of second-derivatives of f(), which is negative definite by assumption. Meover, there is a unique solution z, so we can freely use the implicit function theem. Then totally differentiating with respect to p yields z f +pf p +pf z p 0 in matrix terms [ pf pf pf pf And z f +pf p +pf z p 0 ][ z / p z / p ] [ f f z p f pf +pf f det(h(f)) ] 9

10 Since f is negative definite, the denominat is positive because the determinants of negative definite matrices alternate in sign. F the numerat to be negative, then, we need f < f f f Or f < 0, since f < 0 by the negative definiteness of f. Therefe, z and z must be substitutes it is imptant you say something like this, since that is the economic relationship between good and good.. ii. This is not possible. The reason is that output q is increasing in p; this can be quickly shown using a revealed preference argument: pf(z) w z pf(z ) w z p f(z ) w z p f(z) w z (p p )(f(z) f(z )) 0 So that if p p, then f(z) f(z ), q q. But if both goods are inferi and input prices go up, the firm must then hire less of both inputs, and its output would fall. This would be a contradiction, so not all inputs can be inferi at once. iii. Using the chain rule, q f (z ) z +f (z ) z To get the partial derivatives z / and z /, consider the system [ pf pf pf pf ][ z / z / Using the implicit function theem twice yields z pf det(h(f)) ] [ 0 ] and z pf det(h(f)) Insert these into q f (z ) z +f (z ) z, and you get which is what we got above f z / p. The expression q q pf f det(h(f)) f pf det(h(f)) z p means that if z is inferi at (w,p), the input price of z can increase, and the firm s output will also increase. 0

11 iv. The slickest way to do this is the revealed preference argument given above, that (p p )(q q ) 0. This establishes that output is increasing in p. Then if we totally differentiate optimal output, dq dp k f(z) z k z k p 0 Since the inequality must hold and f(z)/ z k 0 f all k, some good must be nmal at (w,p), implying z k / p 0 f some k. Suppose you didn t do that. What s a me MWG-style argument? Consider the cost minimization problem: min z w z subject to f(z) q. Solving this yields a cost function c(w,q). If we maximize over q in the first-der condition is maxpq c(w,q) q p c q (w,q) 0 and q is increasing in p if (by the implicit function theem) c qq (w,q) dq dq 0 dp dp c qq (w,q) 0 So we need c qq (w,q) 0 f output to be increasing in price; i.e., we need the cost function to be convex in q. Let s go back to the cost minimization problem and prove c(w,q) is convex in q. To prove c(w,q), we need the fact that f(z) is concave. Let q > q. Then we want to show that if z is the cheapest way of attaining q and z is the cheapest way of attaining q, then Start by noting that c(w,λq +( λ)q ) λc(w,q)+( λ)c(w,q ) λc(w,q)+( λ)c(w,q ) λw z +( λ)w z w (λz +( λ)z ) So we can consider the candidate λz + ( λ)z as a potential solution to at λq + ( λ)q. Since f() is concave, f(λz +( λ)z ) λf(z)+( λ)f(z ) λq +( λ)q so that mix of inputs achieves the right level of output. But by definition of c(w,q), So finally, w (λz +( λ)z ) c(w,λq +( λ)q ) λc(w,q)+( λ)c(w,q ) c(w,λq +( λ)q ) And the convex chd lies above the function; i.e., c(w,q) is convex in q. v. There are a number of ways to get this done. The easiest follows the duality arguments we usually use. Let π(p,w) max z pf(z) w z. Then D w π(p,w) z

12 by the envelope theem. Similarly, D p π(p,w) f(z) q also by the envelope theem. Now differentiate the first with respect to p to get π(p,w) p k z k p Now differentiate the second with respect to w k to get π(p,w) k p q k The profit function is twice-differentiable, so it is continuous, and the cross-partials are equal by Young s theem (that s what people usually call that result), so z k p π(p,w) p k π(p,w) k p q k And we re done. Again, the interpretation is the surprising fact that if the price of an inferi input goes up, the firm s output will increase. F a nmal input, output will always fall after the price of the input goes up. 6. There is a firm who uses a vect of inputs z (z,z,...,z K ) with prices w (w,w,...,w k ) to produce output q f(z), where f(z) is strictly concave, increasing, and f(0) 0. The price of the final product is p. Consider the cost minimization problem (CMP) c(w, q) min z w z subject to f(z) q, and the profit maximization problem (PMP), π(w,p) maxpq w z z subject to f(z) q. c(w, q) i. Show that the solution to the CMP and PMP are the same if and only if p at the q optimum f both problems. ii. Show that the conditional fact demands z k (w, q) are decreasing in w k and that output q(w,p) is increasing in p. iii. Suppose now that the fact prices w k vary across time randomly, and the firm observes the prices befe choosing its inputs. If the government proposed a program to fix input prices at the expected value of w k, w k, would the firm be better wse off? Explain. i. The CMP has Lagrangian L w z λ(q f(z)) and FONC s w+λ f(z) 0

13 (q f(z)) 0 while the PMP has Lagrangian and FONC s L pq w z λ(q f(z)) p λ 0 w+λ f(z) 0 (q f(z)) 0 Note that both systems require that w λ f(z) and q f(z), but the PMP has the added equation p λ. If we evaluate the PMP equations at the solution to the CMP, however, by the envelope theem we know that c(w, q) q L q λ Consequently, p λ c(w,q) q so the price equals marginal cost rule holds even f firms with many inputs. ii. F w k, use a revealed preference argument, show concavity/convexity of the objective and use a version of Hotelling s Shephard s lemma. F p, use the implicit function theem. iii. Wse off. The cost minimization problem is concave in w: c(λw +( λ)w,q) λw z +( λ)w z λc(w,q)+( λ)c(w,q) Hence, by Jensen s inequality, E[c(w,q)] c(e[w],q) so the firm s expected costs are lower than the cost of the expectation. 3