EconS 50 - Micro Theory I Recitation #7 - Competitive Markets Exercise. Exercise.5, NS: Suppose that the demand for stilts is given by Q = ; 500 50P and that the long-run total operating costs of each stilt-making rm in a competitive industry are given by C(q) = 0:5 0q. Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given byq s = 0:5w where w is the annual wage paid. Suppose also that each stilt-making rm requires one (and only one) entrepreneur (hence, the quantity of entrepreneurs hired is equal to the number of rms). Long-run total costs for each rm are then given by: C(q; w) = 0:5 0q + w (a) What is the long-run equilibrium quantity of stilts produced? How many stilts are produced by each rm? What is the long-run equilibrium price of stilts? How many rms will there be? How many entrepreneurs will be hired, and what is their wage? (b) Suppose that the demand for stilts shifts outward to Q = ; 48 50P How would you know answer the questions posed in part a. (c) Because stilt-making entrepreneurs are the cause of the upward-sloping long-run supply curve in this problem, they will receive all rents generated as industry output expands. Calculate the increase in rents between parts (a) and (b). Show that this value is identical to the change in long-run producer surplus as measured along the stilt supply curve. Solution: This problem introduces the concept of increasing input costs into long-run analysis by assuming that entrepreneurial wages are bid up as the industry expands. Solving part (b) can be a bit tricky; perhaps an educated guess is the best way to proceed. (a) The equilibrium in the entrepreneur market requires Q s = 0:5w = Q D = n or, solving for w, we obtain w = 4n. Hence, given C(q; w) = 0:5 0q + w, the marginal cost is MC = q 0 and the average cost is AC = 0:5q 0 + w q = 0:5q 0 + 4n q In long run equilibrium the MC = AC, thus: Felix Munoz-Garcia, School of Economic Sciences, Washington State University, Pullman, WA, 9964-60. Email: fmunoz@wsu.edu.
q 0 = 0:5q 0 + 4n q q 0:5q = 4n q 0:5q = 4n q = 8n then q = p 8n Total output is given in terms of the number of rms by Q = nq = n p 8n Now in terms of pro t-maximization in perfectly competitive markets requires P = M C, or q = P + 0 Hence aggregate supply becomes Q S = nq = n(p + 0) Since total output requires Q = n p 8n and aggregate supply requires Q = n(p + 0), we have n p 8n = n(p + 0) P = p 8n 0 Therefore, we can use the demand function Q D evaluated at the equilibrium price P = p 8n 0, Q D = 500 50P = 500 50( p 8n 0) = 000 50 p 8n Then since in equilibrium we require that demand is equal to supply, Q D = Q S, we obtain 000 50 p 8n = n p 8n thus n = 50 entrepreneurs Finally, we can also calculate: Total output: Q = n p 8n = 000 Individual output by every rm: q = Q n = 0 Equilibrium price: P = q 0 = 0
w = 4n = 00. (b) Using the results of the previous part and if the demand function is Q = ; 48 then, 50P (n p 8n) = ; 48 + 50( p 8n 0) (n p 8n) + 50 p 8n = ; 98 (n + 50) p 8n = ; 98, therefore n = 7 and, we can then recalculate: Q = n p 8n = 78 q = Q n = 4 P = q 0 = 4 w = 4n = 88. So, as the demand shifts outward, the number of rms in the industry increases, the total production and rm production increases, the price increases and the wages increase. (c) The long-run supply curve is upward sloping because as new rms enter the industry the cost curves shift up: AC = 0:5q 0 + 4n q as n increases the average cost also increases. Using linear approximations, the increase in the producers surplus (PS) from the supply curve is given by 4 000 + 0:5 78 4 = 5456. If we use instead the supply curve for entrepreneurs the area is 88 50 + 0:5 88 = 5368. These two numbers agree roughly. To get exact agreement would require recognizing that the long-run supply curve here is not linear it is slightly concave. 3
Exercise. Exercise.9, NS: Given an ad valorem tax rate (ad valorem tax is a tax on the value of transaction or a proportional tax on price) of t (t = 0:05 for a 5% tax), the gap between the price demanders pay and what suppliers receive is given by P D = (+t)p S. Solution: (a) Show that, for an ad valorem tax, d ln P D = e S dt e S e D and d ln P S = e D dt e S e D (b) Show that the excess burden of a small tax is DW = 0:5 e De S e S e D P 0 Q 0 (c) Compare these results to those for the case of a unit tax. This problem shows that the comparative statics results for ad valorem taxes are very similar to the results for per-unit taxes (a) Given that the gap between the price demanders pay and what suppliers receive is P D = ( + t)p S Then, the introduction of a tax implies a small price change, i.e., dp D = (+t) +dtp S = ( + t) + dt P S, where we can evaluate this expression at t = 0 since the tax is impossed before any tax was present. Hence, the previous expression collapses to dp D = +dtp S =. We know also that e D = @P P Q D and e S = @Q S @P P Q S In equilibrium with a tax rate of t, we will have Q D (P D ) = Q S (P S ) rearranging dp D = @Q S but since dp D = + dtp S then, ( + dtp S ) = @Q S + dtp S = @Q S dtp S = @Q S dtp S = @Q S @Q S = dt P S (*) 4
Hence, we can express this ratio in terms of elasticities, as follows, dt P S = d ln P S dt = @Q S = @Q S P 0 Q P 0 Q P 0 = e D e S e D Q (b) A linear approximation of the DWL accompanying a small tax dt is given by: as depicted in the next gure. DW L = 0:5(P 0 dt)(dq) Figure. Introducing a tax in a competitive equilibrium market. Since e D = @P P Q D = dlnq dlnp Q then dq = e 0 D dp and substituting into DWL P DW L = 0:5(P 0 dt) e D Q 0 P dp DW L = 0:5 P 0 (dt) e D Q 0 dp dt but from (*) we know dp D dt P D = e S e D e S then DW L = 0:5 P 0 (dt) e D Q 0 e D e DW L = 0:5 D e S e D e S (dt) Q 0 P 0 P e S e S 5
We can now generalize this result for any small t: e DW L = 0:5 D e S e D e S Q 0 P 0 (c) The unit tax described in this chapter is equivalent to the value of the ad-valorem tax. In other words, the unit tax is equal to the ad-valorem tax multiplied by Ps. Therefore, the results obtained using the ad-valorem tax are equivalent to the ones obtained using the unit tax. Exercise 3 3. (Ramsey rule) Consider a three-good economy (k = ; ; 3) in which every consumer has preferences represented by the utility function U = x + g(x ) + h(x 3 ), where the functions g() and h() are increasing and strictly concave. Suppose that each good is produced with constant returns to scale from good, using one unit of good per unit of good k 6=. Let good be the numeraire and normalize the price of good to equal. Let t k denote the (speci c) commodity tax on good k so the consumer price is q k = ( + t k ). (a) Consider two commodity tax schemes t = (t ; ; t 3 ) and t = (t 0 ; t 0 ; t 0 3). Show that if ( + t 0 k ) = ( + t k) for k = ; ; 3 for some scalar > 0, then the two tax schemes raise the same amount of tax revenue. (b) Argue from part (a) that the government can without cost restrict tax schemes to leave one good untaxed. (c) Set t = 0, and suppose that the government must raise revenue of R. What are the tax rates on goods and 3 that minimize the welfare loss from taxation? (d) Show that the optimal tax rates are inversely proportional to the elasticity of the demand for each good. Discuss this tax rule. (e) When should both goods be taxed equally? Which good should be taxed more? Solution: (a) The budget constraint for the consumer with tax scheme t = (t ; ; t 3 ) is: ( + )x + ( + t 3 )x 3 = ( + t )x where the consumer sells units of the numeraire in order to purchase units of goods and 3. (Note that all prices take into account taxes.) The above budget constraint can be alternatively writen as ( + t )x + ( + )x + ( + t 3 )x 3 = 0 6
Hence tax revenue R is: R t x + x + t 3 x 3 = (x + x + x 3 ) Similar reasoning shows that with tax scheme t 0 = (t 0 ; t 0 ; t 0 3) the tax revenue R 0 is: R 0 t 0 x 0 + t 0 x 0 + t 0 3x 0 3 = (x 0 + x 0 + x 0 3) But the demand for each commodity is homogeneous of degree zero (Recall that, by proposition 3.D. in MWG, Walrasian demand is homogeneous of degree zero in prices). Hence, we have that x i = x i ( + t ; + ; + t 3 ) x i = x i (( + t ); ( + ); ( + t 3 )) x i = x i ( + t 0 ; + t 0 ; + t 0 3) = x 0 i Therefore (x + x + x 3 ) = (x 0 + x 0 + x 0 3) and R = R 0. (b) The value for can be chosen arbitrarily. In particular, a tax system with a tax t k on good k can be shown to be equivalent to one with no tax on good k by choosing = +t k (c) The optimization decision for the consumer is max U = x + g(x ) + h(x 3 ) fx ;x ;x 3 g s.t. x + ( + )x + ( + t 3 )x 3 = 0 Substituting the constraint into the objective function for x reduce the F.O.C. to g 0 (x ) ( + ) = 0 and h 0 (x 3 ) ( + t 3 ) = 0 7
These necessary conditions result in demand functions x = x ( + ) and x 3 = x 3 ( + t 3 ), where x is a function of the e ective price of good, +, and similarly for the demand of good 3. Thus, the above budget constraint can be written as a function of these demands x = ( + )x ( + t 3 )x 3. The optimization of the government can now be written as max f ;t 3 g U = ( + )x ( + t 3 )x 3 {z } x + g(x ) + h(x 3 ) s.t. R = x + t 3 x 3 where x and x 3 are, in turn, functions of ( + ) and ( + t 3 ) respectively. The solution to this problem provides the tax rates that minimize welfare loss. The necessary conditions are: g 0 x 0 x ( + )x 0 (x + x 0 ) = 0 h 0 x 0 3 x 3 ( + t 3 )x 0 3 (x 3 + t 3 x 0 3) = 0. From the consumer s choice problem g 0 = + and h 0 = + t 3. These allow the implicit solutions = x + and t x 0 3 = x 3 + x 0 3 (d) The elasticity of demand for good k is de ned as by this de nition, and " d k = (+t k)x 0 k x k + = + " d t 3 +t 3 = +. " d 3 The tax rate on good k is therefore inversely proportional to the elasticity of demand for that good. Setting the relative taxes in this way minimizes the excess burden resulting from the need to raise the revenue R. This relationship between tax rates and elasticity is often referred to as the "inverse elasticity rule" (e) This tax rule implies that the good with the lower elasticity of demand should have higher tax rate. The two goods should be taxed at the same rate only if they have the same elasticity of demand. 8
Exercise 4 4. Consider the utility function U = log(x ) + log(x ) l and budget constraint wl = q x + x. Solution: (a) Show that the price elasticity of demand for both commodities is equal to -. (b) Setting producer prices at p = p =, show that the inverse elasticity rule implies t t = q. (c) Letting w = 00 and + =, calculate the tax rates required to achieve revenue of R = 0. (a) The consumer s demands is solve max log(x ) + log(x ) fx ;x ;lg s.t. wl = q x + x l or equivalently max log(x ) + log(x ) ( q x fx ;x ;lg w + x w ) The F.O.C.s are then x = q and w x = w Then, the demands are x = w q and x = w The elasticity of demand is de ned by " d i = dx i dq i q i x i Calculating this for good obtains " d = w q ( w q ) = Calculating this for good obtains " d = w ( w ) = 9
(b) The inverse elasticity rule (see exercise 3) states that t i +t i = " d i, i = ;. Hence t +t " d = = + " d But " d i = and + t i = q i, so (c) Revenue is de ned by t q = q and rearranging we have t t = q. R = t x + x Using the solutions for the demands x = w q and x = w we have R = t ( w q ) + ( w ) Using the relation t t = q we just found in part b as t = q R = ( q )( w q ) + ( w ) = w[( q )( q ) + ] = w ( + ) Finally, since + t i = q i, + =, R = 0 and w = 00, the optimal tax on good solves which has solution = 9, and hence t = 9. 0 = 00 + 0
Exercise 5 5. Let the consumer have the utility function U = x + x l. (a) Show that the utility maximizing demands are x = h i =[ ] w. h w i =[ ] q and x = (b) Letting p = p =, use hthe inverse i h elasticity i rule to show that the optimal tax rates are related by = + t. (c) Setting w = 00, = 0:75, = 0:5, nd the tax rates required to achieve revenue of R = 0 and R = 300. (d) Calculate the proportional reduction in demand for the two goods comparing the no-tax position with the position after introduction of the optimal taxes for both revenue levels. Comment on the results. Solution: (a) If the consumer maximization problem is max U = x + x l s.t. q x + x = wl Thus we can rewrite the budget constraint l = q x + x and we can replace into the utility w function for an unconstrained optimization problem as: max U = x + x q x w x w Taking rst order condtions with respect to every good, x i, yields @U @x i = i x i i q i w = 0 and rearranging, we obtain i x i i = q i w Solving for x i we get the utility maximizing demands as required. x i = wi i q i (b) The rst step is to calculate the price elasticity using the demand function we just found: " d i = i With p = p = the inverse elasticity rule states that (see previous exercise):
Substituting for the elasticities t +t " d = + " d or + " d = +t t " d + = +t t + = +t t ( + ) = t ( + t ) + = t + t t = t + t t = t + ( ) ( ) nally h i = t + or = + h i t. (c) Using the parameter values gives so = 0:5 + 0:5 t Then, given the revenue constraint = t R = t x + x but we know the optimal values for the demand and using the fact that + t i = q i, then and is also known, then R = t w q =( ) + t w =( ) R = t w +t =( ) + t w + =( )
R = t w +t =( ) + Replacing the values of the parameters R = t 4 75 +t + t t 0 B @ w + 0 B @ 5 + t t which simpli es to 5065t R = 65 ( + t ) + 6t4 4 (x )!! C A C A =( ) The revenue curve has a maximum level of revenue around t = 0:4, this is known as La er property. For R = 0 the solution is t = 0:003 and = 0:006; as depicted in the following gure which illustrates expression () evaluated at R = 0 as a function of t. In the case R = 300 the solution is t = 0:84 and = 0:443. () Figure. Tax revenue R = 0 as a function of t. (d) The proportional reduction in demand for the two goods comparing the no-tax position with the position after introduction of the optimal taxes for both revenue levels is in the next table. R x % x % 0 3; 64 5 0 3; 5 :3 4:69 :4 300 ; 64 48:67 :00 5:00 As we can see, the optimal taxes do reduce demand in approximately the same proportion for both commodities. In this case the interpretation of the Ramsey rule is applicable even when the tax intervention has a signi cant e ect on the level of demand. 3