2350pqset_3an

2350pqset_3an - York University AS/ECON2350 V Bardis...

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Unformatted text preview: York University - AS/ECON2350 - V. Bardis Answers to Practice Set 3 1. (a) and (b) See notes or text on how to do these. Here pm = 30 and Qm = 40. (b) The increase in an input price will raise the ﬁrm’s cost. If we write the cost function as C (Q) = cQ then the parameter c would rise (from 10 to, say, 11). On a graph, this amounts to an upward shift of the M C curve. If the industry were perfectly competitive then the consumer price would rise by 100% since we have constant (average) cost and so a ‘constant cost’ industry. In the case of monopoly, the proﬁt maximizing price is pm = (A + c)/2 (where A = 50 here). From this we get ∂pm = 1/2, that is, the ∂c monopoly price would rise by only 50% of the cost increase. On a graph, we can show that the amount by which M C shifts up is less than the rise in the price in the monopoly case. It would seem that this is a general conclusion when comparing monopoly and P.C. but it is not. For a counterexample see the textbook’s discussion of a monopolist facing constant elasticity demand (instead of the linear demand we assumed here). 2. (a) p = AC = M C ; M C = AC ⇒ 2q = q + 1/q ⇒ q = 1. Therefore p = 2 and Q∗ = 980, n∗ = 980. (b) T S ∗ = CS ∗ + P S ∗ = (1/2(10))9802 + 0 = 48, 020 (c) If the ﬁrm uses n plants will allocate output such that M C is the same across plants. This means the same output will be produced in each plant that is q = Q/n where Q is the total. It follows that the ﬁrm’s cost is given by C (Q, n) = n((Q/n)2 + 1). Thus the cost minimizing number of plants is nm = Q which in turn implies C (Q) = 2Q and M C (Q) = 2. R = pq = (100 − Q/10)Q ⇒ M R = 100 − Q/5. Setting M R = M C gives 100 − Q/5 = 2 ⇒ Qm = 490 and pm = 51. (d) DW L = T S ∗ − T Sm , where T Sm = πm + CSm = 36, 015 so DW L = 12, 005 3. See notes or text. 4. 1 . eno neerG kraD eht suht dna eniL neerG thgiL eht enimreted su spleh sihT . )der niht( RM lautca eht sa sixa latnoziroh eht no tniop emas eht hguorht ssap osla tsum tI . tQ ta )eulB krad( CM tcesretni tsum )neerG thgiL( xat selas eht HTI W dnamed eht fo RM eht , xat tinu rep eht htiw sa Q emas eht teg ot re dro ni ereh , suhT . A tniop dnuora stovip )der kciht( dnamed eht )remusnoc eht ot degrahc( xat selas a htiw , lareneg nI RM stI = eniL neerG thgiL xat selas htiw tsiloponom gnicaf dnameD = eniL neerG kraD xa t tinu rep eht si t erehw t + CM = eniL eulB thgiL tsoC lanigraM = eniL eulB kcihT RM stI = eniL deR nihT dnameD tekraM = enil deR kcihT )T( xat selas rednu remusnoc ot degrahc ecirp=sP . )T( xat selas eht DNA )t( xat tinu rep rednu remusnoc yb diap ecirp = tP A . sP)T+1 (=tP , eroferehT tQ x t > tQ x sP x T , eroferehT xat selas htiw EUNEVER XAT = tQxsPxT=tQx )sPUtP(=AERA YERG xa t tinu rep htiw EUNEVER XAT = tQ x t = AERA WOLLEY tQ t+CM CM sP tP 5. (a) If transportation costs are very large the two markets are in eﬀect separated and the monopolist can discriminate (3rd degree P.D.). The optimal price and quantity are p∗ = 8, Q∗ = 7 in market 1 and 1 1 p∗ = 10.5, Q∗ = 9.5 in market 2. 1 2 (b) The higher price is charged in the more inelastic market |ǫ2 | = p/(20 − p) < p/(10 − p) = |ǫ1 |. (c) Then consumers would travel to the place where the good is cheaper and the monopolist would not be able to discriminate. The price diﬀerence is not expected to survive. Instead the monopolist would charge a uniform price. Using the method discussed in class the optimal uniform price is pu = 9.25. (d) Under price discrimination Tˆ = 208.875; under uniform pricing T S u = 210.4375. Therefore the eﬃciency S loss due to P.D. is T S u − Tˆ = 1.5625. S 6. We wish to induce A to buy the 2-unit package and B to buy the 1-unit package because A is the high wtp buyer. We charge p1 = 5 for 1-unit package (the maximum B is willing to pay) and p2 = 10 − (6 − 5) = 9 for the 2-unit package. The latter price is chosen such that A is indiﬀerent between buying the 1-unit package and the 2-unit package, that is, 6 − p1 = 10 − p2 ⇒ p2 = 10 − (6 − 5) = 9 So A gets the 2 units at 1 dollar less than he would be willing to pay, a discount equal to his surplus if he bought the 1-unit package. (It is clear that A receives a quantity discount since for each unit he pay (in eﬀect) 9/2 = 4.5 < 5.) Here the maximum proﬁt is the one from discrimination: n(5 + 9) = 14n, where n is the number of consumers of each type. There are several other alternatives. Inspection shows that the second best alternative (from the ﬁrm’s point of view) would be to sell the 2-unit package at a price of 6 so that both types buy and the proﬁt is 2n(6) = 12n < 14n. The scheme is not eﬃcient since both types have M W T P > M C = 0 for the second unit (4 for A and 1 for B ) but only A ends up getting it. 7. See notes or text. 2 ...
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