Assignment 5 solutions Q1-1

# Assignment 5 solutions Q1-1 - Assignment 5 Solutions...

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Unformatted text preview: Assignment 5 Solutions November 3, 2011 Question [1] (A) To ﬁnd an average cost, we ﬁrst need the total cost. The total cost is the combination of ﬁxed and variable costs. We are told ﬁxed costs are 16 and variable costs are (12, so we can say total cost, TC, is: TC'(q) = 16 + (12. To ﬁnd the average cost, we just divide the total by the number of units we want the average for (just like ﬁnding the average of anythingl), so we divide TC by q and we get AC(q) = TC(q)/q = 16/q + q. (B) The minimum point of the average cost curve is found where marginal cost (MC) is equal to average cost (AC). We are given marginal cost as 2q, so we have 2q 2 16/q + q which yields q = 4. (C) The long—run equilibrium price is found at the minimum point of the long—run average cost curve. Since the question tells us this minimum is the same as in the short run, we know the solution will be the average cost that exists at q=4 for the ﬁrm. Plug q = 4 into AC(q) to get AC(4) = 4 + 4 = 8. Thus, the long—run equilibrium price is 8. (D) Lond run equilibrium output will be the output that satisﬁes de— mand at the long—run equilibrium price (which we just found in part C). We are given demand as Q(P)=80—4P, so (2(8) 2 80 — 32 = 48. (E) In this chapter we assume all ﬁrms are identical and that the number of ﬁrms can be represented by n. Therefore, if we need to produce 48 units in the long—run and we know that each ﬁrm produces at its efﬁcient scale of 4 units each then n = Q/q = 48/4 2 12. Thus, there will be 12 ﬁrms in long— run equilibrium. For more on the theory read pages 503—504 in the textbook. (F) We know that ﬁrms choose their quantity such that P = M C in perfectly competitive markets. Thus, P 2 2q <=> q = P/2. We know Q = mg 2 12(P/2) = 6P <=> P = (1/6)Q. (G) The long—run market supply curve is horizontal at P = 8. Firms do not make proﬁt in the long run and they all produce at their minimum efﬁcient scale, so the only no—proﬁt price is where P28. The supply curve is necessarily a horizontal line at P28 because price does not vary with quan— tity in the long run in perfect competition. (H) In the short run, we can ﬁnd consumer and producer surplus in the usual way by using the information provided in the demand and supply curves. See Figure 1 below. Since demand and supply curves are linear, we can use the formula for the area of a triangle to calculate the surpluses. CS 2 .5(48)(12) = 288 and PS 2 .5(48)(8) = 192. AS 2 CS + PS 2 480. (I) Now we have a new equation for the demand curve. Set it equal to the equation for short run supply that we calculated in to get 120—4P 2 GP <=> P = 12. (J) The number of ﬁrms must remain ﬁxed in the short run; there is no time for new ﬁrms to enter. Now that the market demands 72 units, the existing 12 ﬁrms in operation will equally share in this production. q = Q/n = 72/12 = 6. So each ﬁrm will produce 6 units (and will do so at a higher average cost because they are no longer at the efﬁcient scale of 4 units). (K) Here we are asked to calculate the proﬁts of an individual ﬁrm (not the entire industry). Proﬁts 2 Total Revenue — Total Costs 2 Pq — (16+q2) = (12)(6) — 16 — 62 = 20 (L) We found in part (C) that the price in the long run is P28; this is not affected by the increase in demand. So to ﬁnd the new output we do what we did in part Demand is = 120 — 4P so = 120 — 32 = 88. (M) Now that we are in the long run again, we know that each ﬁrm will operate at its efﬁcient scale. Since there has been no change in the cost function of the ﬁrms, the efﬁcient scale continues to be at q = 4. So now that we have determined the market requires 88 units, we will need 22 ﬁrms each producing 4 units to satisfy the demand. (N) Now that the number of ﬁrms has changed7 we will have a new short—run market supply curve. We follow the same approach as in part We know q=P/ 2. As always7 total output can be found by the equation Q = mg 2 (22)(P/2) 2 HP <=> P = (1/11)Q. (O) The proﬁt maximizing level of output for a ﬁrm remains at q = 4. This will not change as long as the cost function does not change. (P) We can use the same approach as we did in part again, the only difference is we now have new demand and supply curves. See Figure 2 below. CS 2 .5(88)(22) = 968 and PS 2 .5(88)(8) = 352. 20 P=(1/5)Q Figure 1: 30 Figure 2: 38 P=(1/11)Q ...
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## This note was uploaded on 02/02/2012 for the course ECON 2410 taught by Professor Prescott during the Fall '11 term at University of Guelph.

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Assignment 5 solutions Q1-1 - Assignment 5 Solutions...

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