Microeconomic Theory I Assignment #7 - Answer key

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1 Miroeonomi Theory I Assignment #7 - Answer key. [Menu priing in monopoly] Consider the example on seond-degree prie disrimination (see slides 9-93). To failitate your alulations, assume H = 5, L =, and =. (a) Uniform priing. Assuming uniform priing, nd the monopolist s optimal prie, output, and pro ts from serving oth types of ustomers. () Still under a uniform prie setting, if the monopolist were to fous on high-demand uyers alone, whih is the optimal prie, output and pro ts. For whih values of proaility does the rm prefer to serve oth types of uyers? [Hint: Find the uto for,, suh that when satis es < expeted pro ts are higher serving oth types of uyers.] () Two-part tari. Consider now that the monopolist sets a two-part tari (F H ; H ) and (F L ; L ). Find the optimal two-part tari. (d) Consider a proaility = 3, whih should satisfy the ondition you found in part 4 (), <. Hene, under a uniform prie, the rm prefers to serve oth types of uyers. Con rm that expeted pro ts are higher when the rm praties a twopart tari, followed y serving oth types of ustomers under uniform priing, and followed y serving high-type uyers alone under uniform priing. Please see answer key for this exerise in the handout on Two-part tari s in the EonS 503 wesite (numerial example in pages 0-3).. [Monopolist with intertemporal network e ets.] Consider a diret demand funtion x(p; w) = p + where represents the units of the good purhased in previous periods and > 0 denotes the network e ets that exist in this industry. For instane, a larger pool of ustomers in previous periods makes the good more valuale for urrent ustomers, thus produing a rightward shift in the demand funtion. Network e ets arise in industries suh as operating systems, game onsoles, et. wherey the larger the population that uses a spei type of devie the more useful it eomes for new users who will e ale to exhange more les and design more programs. Assume that ; > 0. Hene, solving for p, we otain the indiret utility funtion p() = x + For ompatness, let us denote a ; ; and, whih redues the aove inverse demand funtion to the more familiar expression p() = a x +, where now measures the network e ets. Also assume that marginal osts are onstant and < a.

2 (a) Seond period. Determine a monopolist s optimal prodution level, and the resulting pries, if units were sold in the market during the previous period. Find the monopoly pro ts as a funtion of in the seond period. [Hint: For simpliity, assume only two time periods.] We start y setting up a pro t imization prolem. The monopolist s prolem is = p() x x = ax x + x x x With a rst = a x + = 0 Solving for x, we otain seond-period monopoly output x () = (a ) + whih inreases in oth and. (Note that when = 0, this output level oinides with that of a standard monopolist without network e ets, a.) Plugging x () into the inverse demand funtion, we nd the seond period prie (a ) + p() = a x + = a + = a + + And solving for the monopolist s pro ts, a + + (a () = p()x() x() = ) + (a ) + (a + ) = 4 () First period. If the monopolist antiipates that no rm will enter into the industry in future periods, how muh does the monopolist produe in the rst period, assuming a rst-period inverse demand urve p() = a x. Setting up a pro t imization prolem over two separate periods, = p () + p () x () a + + a + = (a ) + whih simpli es to 4 ( + x ()) a (a ) + (a ) + 4

3 Taking rst order onditions with respet to rst-period = + (a ) + = 0 4 Solving for, we otain the monopolist s rst-period output = a Plugging into the seond-period output funtion x (), we otain a (a ) + x () = = a thus implying that the monopolist seeks to smooth its prodution deisions in oth periods, y produing the same output level, = x( ) = a, whih is inreasing in network e ets,. Similarly as in part (a) of the exerise, note that if network e ets are asent, = 0, the output level redues to = a, whih also oinides with the standard monopoly output, i.e., the monopolist would produe the same amount of output in oth time periods = x = a () Soial optimum. Assume that a soial planner owned this monopoly. Considering that the soial planner imizes the sum of onsumer and produer surplus in oth periods, how muh would it produe in eah period? [Hint: Determine x SO () rst, and then nd SO.] Seond period. Setting up the soial planner s seond-period imization prolem, [(a + ) (a + x)] x + (x) x. = x + (a x + )x x where the rst term represents the onsumer surplus (whih is independent on network e ets) and the last two terms measure the monopolist s seondperiod pro ts. Importantly, this implies that a marginal inrease in rstperiod output only raises the monopolist s revenues ut does not a et onsumer surplus. Simplifying, = x + (a + )x Taking rst order onditions with respet = x + a + = 0 Solving for x, we otain the seond-period soially optimal output as a funtion of, x SO () = a + 3

4 whih, similarly to the seond-period output funtion of the unregulated monopolist, it inreases in rst-period network e ets (as measured y ). Sustituting into the inverse demand funtion, p() = a a + + = whih indiates that under a soial optimum, the prie harged for eah unit of output would oinide with its marginal ost. First period. Setting up the soial planner s rst-period imization prolem, [a (a )]+()+ (a + ) (a + x SO ()) x SO ()+(x SO ()) = +(a ) + a + a + + a + whih also indiates that rst-period onsumer surplus is independent on. Simplifying, we otain + (a ) + a + Taking rst order onditions with respet = + a + + = 0 Solving for, we otain the rst-period soial planner output SO = a and an assoiated rst-period euilirium prie of a a p() = a = Plugging SO into the seond-period soial planner s output funtion x SO (), yields an output of x SO ( SO ) = a + a = a whih is inreasing in network e ets,. Hene, similarly as the unregulated monopolist in part () of the exerise, the regulator produes the same soially optimal output in oth periods, i.e., SO = x SO ( SO ), and this output is inreasing in network e ets. In addition, soially optimal output levels are stritly larger than those seleted y the unregulated monopolist, i.e., a > a, for all parameter values. 4

5 (d) Numerial example. Consider parameter values a = = and = 0. Fidn output and x( ) as a funtion of. Find the soial optimum SO and x SO ( SO ). Compare. Evaluating our aove results in these parameter values, we otain the monopolist produes = x( ) =. In ontrast, the soial planner would selet SO = x SO ( SO ) =. Hene, the soially optimal output is higher than the output that the monopolist would hoose for all network e ets. Figure depits the rst- and seond-period output that the unregulated monopolist selets, along with the output that soial planner would selet. Figure. Output deisions under network e ets. 3. [Advertising in monopoly] Consider a monopolist faing inverse demand urve p() = a, where a; > 0. The vertial interepta satis es a = p A where A 0 represents the monopolist s advertising expenditure, and [0; ] denotes the sensitivity of demand to one additional dollar of advertising. In addition, assume that the monopolist s prodution ost is T C() =, where > 0; and its advertising ost is T C(A) = A p A, where A > 0. (a) Assuming that the monopolist hooses and A simultaneously, write down the monopolist s pro t-imization prolem. Find the optimal output and advertising level. Sine a = p A, the inverse demand funtion an e expressed as p() = a = p A. In this setting, the monopolist solves h p i p A A A ;A Taking FOC with respet to and A, we otain p A = 0, and A p = 0 A Rearranging, and solving for and A, we nd A = 4 A and A = 5 4 A

6 () Assume now that the monopolist hooses and A seuentially ( rst A, and then ). Write down the monopolist s pro t-imization prolems (one for A, and one of ). Using akward indution, nd the optimal output and advertising level. Seond stage. We analyze the last stage, when the monopolist hooses its output for any given value of advertising A seleted in the rst stage. The monopolist nds the optimal, y solving h p A Taking FOC with respet to, we otain i A A p A = 0 Rearranging, and solving for, we nd the funtion (A) = p A First stage. The monopolist antiipates the optimal value of (A) that it will hoose in the suseuent stage. Hene, the rm an insert (A) into the pro t funtion of the rst stage, as follows A h p A. i p (A) (A) (A) A A thus otaining a pro t funtion that depends on advertising A alone. Taking FOC with respet to A, and solving for A, we nd A = 4 A Hene, the optimal output evaluated at A = 4 A is (A ) = () Compare your results in parts (a) and (). 4 A 8 A Comparing advertising, note that advertising is larger in the simultaneous game than in its seuential version > 4 A 4 A sine > 0. Last, eause output is inrease in A, output is also larger in the simultaneous game than in its seuential version. 4. [Regulating a natural monopoly] A water supply ompany provides water to Pullman. The demand for water in Pullman is p() = 0, and this ompany s osts are () = +. 6

7 (a) Depit the following in a gure: the demand urve p(), the assoiated marginal revenue MR(), the marginal ost of prodution MC() and the average ost of prodution AC(). Disuss why this situation illustrates a natural monopoly. Figure depits the information provided in the exerise. The average ost urve is dereasing in output, implying that multiple produers are more ostly than a single monopolist, i.e., the sum of their average osts will e larger than the monopolist s average osts. Then, a monopolist naturally eomes more ost-e ient than having several produers with a similar ost struture as the monopolist. Figure. Natural monopoly. () Unregulated monopolist. Find the amount of water that this rm will produe if left unregulated as a monopolist. Determine the orresponding pries and pro ts for the rm. The monopolist imizes (0 ) ( + ) Taking rst order onditions with respet to, we nd 0 m = 0. Solving for we otain a monopoly output of m = 4 units, whih are sold at a prie of p m = 0 4 = $6, with assoiated monopoly pro ts of m = (0 4)4 ( + ( 4)) = $5: () Marginal ost priing. Determine the amount of water that this rm will produe if a regulatory ageny in Pullman fores the rm to prie aording to marginal ost (i.e., to produe an amount of output that solves p( ) = MC( )). Find the orresponding pries and pro ts for the rm. In that ase, the monopolist sets 0 =, i.e., = 8 units, at a prie p = 0 8 = $, with orresponding losses of = (0 8)8 ( + ( 8)) = : 7

8 This result arises eause the presene of dereasing average osts, i.e., in gure 6.5 the prodution of = 8 units yields per unit losses of AC(8) M C(8) = 8 + = $0:5: (d) Prie disrimination. Consider now that the regulatory ageny allows the monopoly to harge two di erent pries: a high prie p for the rst units, and a low prie p( ) for the remaining ( ) (i.e., the units from up to the output level you found in part (), ). In addition, the regulatory ageny imposes the ondition that the rm annot make any pro ts, = 0, when harging these two pries.. Find the value of and the assoiated value of p( ). First, note that the value of must satisfy the no pro ts ondition, that is = [p( ) AC( )] + [p( ) AC( )] ( ) = 0 and sine we know from part () that = 8 units, and that p() = 0 and AC() = + = +, we an rewrite the aove ondition as = (0 ) + + (0 8) 8 + (8 ) = 0 whih simpli es to = 0 Solving for we otain two solutions: First solution: = 0:8 with a orresponding prie of p = $9:8. This means that the rst 0.8 units are sold at $9.8 eah, while the remaining 7.8 (up to 8 units) are sold at a prie of $. Seond solution: = 0:30 with a orresponding prie of p = $9:7. This means that the rst 0.30 units are sold at $9.7 eah, while the remaining 9.7 (up to 8 units) are sold at a prie of $.. Depit these two pries and uantities in a gure and shade the areas of ene ts and losses for the rm. Figure 3 depits out aove results: for the rst units, the monopolist makes a pro t of p( ) AC( ) per unit. For instane, if = 0:8, then the monopolist makes a pro t of (0 0:8) + = $5:97 per unit. 0:8 In ontrast, for the remaining 8 units, the monopolist inurs a loss measured y the distane etween the average and marginal ost urves, i.e., AC(8) M C(8) = 8 + = $0:5 8

9 per unit. Figure 3. Disriminating natural monopoly. 9