1University of California at Berkeley Spring 1999Econ 100A Professor PerloffSuggested solution for problem set 31. A) Isoquant drawingBy rearranging the production function, we obtain 2121−=LqK .For example, q = 1: 21121−=LK , q = 2: 21221−=LK . Isoquants for q = 1, q = 2 and q = 3 belowhave the shape of downward sloping and convex to the origin. K q = 1 q = 2 q = 3 1 2 3 LB) With the capital level fixed at K, the output level is determined by only L, LKq )21( += . Hence,the average product of labor is )21( KLqAPL+=≡ . As the capital level increases, so does the averageproduct of labor.C) i) The use of calculus yields the marginal product of labor, )21( KLqMPL+=∂∂≡ .ii) Alternatively, )21()21()1)(21(| KLKLKLqMPgivenKL+=+−++=∆∆≡ .D) To determine the degree of returns to scale, let’s increase L and K by αtimes where 1>α. Then newoutput level qLKLLKLKLL αααααααα =+>+=+ 22))((22. In other words, increase of L andK by α times raises output level by more than α times, which implies this production function hasincreasing returns to scale.2. A) omittedB) First, how much units of labor and leather are required to produce output level ‘q’? Note that half unitof labor and one unit of leather makes one unit of belt. Hence, the shape of the production function is},2min{ TLq= where L and T denote labor and leather respectively. At cost minimizing combination,L = q/2 and T = q. For example, the graph below shows isoquants for q = 1 and q = 2.2 T q = 1 q = 2 cost minimizing combination of L and T 2 1 1/2½ 1 L Therefore, qqqqTPqWLqVCT10)(5)(10)()()(21=+=+= . By adding the fixed cost, we obtainthe total cost function below :10010)()()()( +=++=+≡ qFqTPqWLFqVCqCT. C(q) 120 110 100 1 2 qC) 10)( =≡dqdCqMC . Alternatively, 10}10010{}100)1(10{)( =+−++=∆∆≡ qqqCqMC . AndqAFCqAVCqCqAC10010)()( +=+=≡ .D)Q MC(q) AC(q) AVC(q) AFC1 10 110 10 1002 10 60 10 503 10 43 1/3 10 33 1/34 10 35 10 255 10 30 10 206 10 26 2/3 10 16 2/37 10 24 2/7 10 14 2/78 10 22.5 10 12.59 10 21 1/9 10 11 1/910 10 20 10 103E) Curves P 110 100 10 20 ACMC=AVC=10 AFC 10 q3. The imposition of tax ‘g’ on energy increases the price of energy such that gPPEE+→ . Without lossof generality, let’s assume energy is a fixed input in the short run. Since the firm can not adjust the level ofenergy use in the short run, tax ’g’ raises total cost of production by increasing the fixed cost. However,firm chooses more labor and less energy for a given level of output in the long run where it can adjustenergy input to the optimal level. With the assumption of decreasing marginal rate of technical substitution,a flatter input price ratio entails new combination of more labor and less energy. The graph below showsthe initial LR path(before the tax on energy), intermediate SR path(after the tax) and new LR path(after thetax). E Initial LR expansion path Intermediate SR expansion path New LR expansion path Isoquant for q L Initial input price ratio New input price ratio after the tax44. First, consider the output q1, the intersection of price and the marginal cost in the regime of decreasing.Additional sale of output brings marginal revenue, P, which is greater than the additional cost to produceone more unit, MC. The firm has incentive to increase the output. Second, suppose the firm produces theoutput level q2, the intersection of P and the MC in the regime of increasing. If the firm increases its outputlevel, marginal revenue from additional sale, P, is less than the additional cost for that unit. Definitely thefirm would not increase its output level. Therefore, q2 is the profit maximizing output level. In fact, outputq1 is the profit minimizing level. MC(q) MC(q) P q1 q2 q5. In the long run where there is no fixed input, the profit function is characterized asqqACPqCqP⋅−=−⋅=))(()(π. If a firm shutdowns its production, ‘q = 0’, 0=π. As long as)(qACP≥, the positive level of output, ‘q > 0’, brings non-negative profit. Otherwise, when)(qACP<, the production, ‘q > 0’, causes the economic ‘loss’. In other words, the shutdown price inthe long run is equal to the minimum of the long run average cost curve. By contrast, the short run costfunction is defined as FqVCqPqCqPsss−−⋅=−⋅= )()(π . If the fixed costs are sunk, theshutdown price is below the minimum of the short run average cost curve, the breakeven price. BecauseFqqAVCPs−⋅−= ))((π implies that ‘q > 0’ when )(qAVCP≥ yields at least ‘- F ’, which isequal to the profit level from the shutdown. Now suppose the fixed costs are avoidable in the short-run.Then the firm can recover the fixed cost by shutdown even in the short run. In other words, the shutdownprice is now equal to the breakeven price.6. Since each firm is identical in this competitive market, we can focus on a representative firm’sproduction decision. Total supply, the horizontal summation of individual firm’s supply isPnPnqPqPqPqQSSnSSS2121)()()()( ==+⋅⋅⋅++= . The last equality comes from the profitmaximizing condition PPqqqMCPS21)(2)( =⇔== . By equating the market demand to themarket supply, SDQnPPQ ==−=2124 , we obtain the market equilibrium price and the individualfirm’s supply in terms of the number of the
View Full Document
Unlocking...