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addprob2sol - Econ 11 - Additional Problems: Solutions 1....

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Econ 11 - Additional Problems: Solutions 1. Toyota’s technology for producing cars is given by the production function F ( K,L )= min { K, L } ,whe re K denotes the amount of capital inputs, and L denotes the amount of labor inputs necessary for producing cars. ( i ) For a given rental rate of capital v and a wage rate w , f nd Toyota’s input demands of K and L necessary to produce a quantity Q of cars. Also, f nd Toyota’s cost function. ( ii ) Suppose Toyota takes the price of its cars as given. If w =10 and v =10 , f nd Toyota’s long-run supply function of cars (as a function of its price P ) in the long-run, when Toyota can adjust both K and L . ( iii ) The demand for Toyota cars is given by Q =200 5 P . A twhatpr icewou ldthe market clear in the long run, if Toyota took the market price as given? In general, how would the market-clearing price depend on Toyota’s input prices? ( iv ) E f ectively, Toyota is the only supplier of Toyota cars, and therefore will price its cars as a monopolist, taking into account the e f ecto fthepr iceondemand .Whatquant ity will Toyota choose to produce in the long run? ( v ) In the short run, K is f xed at 40 , and input prices are f xat w =10 and v =10 .A sa function of P , what is the short-run optimal supply of cars, as a well as the market-clearing price for Toyotas? ( i ) In order to minimize costs for a given quantity of output, Toyota wants to use inputs in the f xed proportions. To produce Q units of output, it needs Q units of capital and Q units of labor. Hence, the inputs necessary for producing Q cars are K ( Q )= Q and L ( Q )= Q , which are independent of the market prices (perfect complements - no substitution between inputs). The Cost function is C ( Q )= vK ( Q )+ wL ( Q )=( v + w ) Q ( ii ) With w = v =10 , the cost function becomes C ( Q )=20 Q . The marginal cost is MC ( Q )=20 (CRS implies constant marginal cost). Whenever P< 20 , the long-run supply is 0 -the f rm f nds it optimal not to produce. Whenever P> 20 , Toyota would want to produce an in f nite amount. Whenever P =20 ,Toyotaisind i f erent between all quantities (pro f ts are necessarily equal to zero). The long-run supply curve is F at at a price level of P =20 . ( iii ) If Toyota takes the market price as given, the market has to clear at P =20 -at that price, Toyota would be willing to produce any quantity Q . The corresponding Q that is demanded in the market is Q =200 5 P =100 . With CRS (and perfect complements), the marginal cost is simply MC ( Q )=
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This note was uploaded on 04/05/2009 for the course ECON 11 taught by Professor Cunningham during the Spring '08 term at UCLA.

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addprob2sol - Econ 11 - Additional Problems: Solutions 1....

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