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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 } , where 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 , fi nd Toyota’s input demands of K and L necessary to produce a quantity Q of cars. Also, fi nd Toyota’s cost function. ( ii ) Suppose Toyota takes the price of its cars as given. If w = 10 and v = 10 , fi 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 . At what price would the 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 ff ectively, Toyota is the only supplier of Toyota cars, and therefore will price its cars as a monopolist, taking into account the e ff ect of the price on demand. What quantity will Toyota choose to produce in the long run? ( v ) In the short run, K is fi xed at 40 , and input prices are fi x at w = 10 and v = 10 . As a 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 fi 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 fi rm fi nds it optimal not to produce. Whenever P > 20 , Toyota would want to produce an in fi nite amount. Whenever P = 20 , Toyota is indi ff erent between all quantities (pro fi ts are necessarily equal to zero). The long-run supply curve is fl 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 ) = v + w , i.e. given by the input prices. An increase in the input prices would lead to an increase in the market-clearing price (Toyota’s supply function shifts to a higher level). Toyota’s total pro fi ts in equilibrium are zero. ( iv ) If Toyota acts as a monopolist, it will set its price and quantity taking into account the market demand. The inverse demand function for Toyota’s cars is P = 200 Q 5 , and 1
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Toyota’s pro fi t function becomes Π ( Q ) = P ( Q ) · Q C ( Q ) = Q · 200 Q 5 20 ¸ = Q · 20 Q 5 ¸ Taking FOC’s with respect to Q gives Π 0 ( Q ) = 20 2 Q 5 = 0 = Q = 50 The corresponding market price is P = 200 Q 5 = 30 . Since P = 30 and MC = 20 , Toyota
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