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Unformatted text preview: Economics 310 Microeconomic Theory: A Mathematical Approach Fall 2008 Solutions to Problem Set 6 Question 1 (a) The production possibility frontier shows the efficient combination of goods x and y with this countrys endowment of labor and capital. By drawing the Edgeworth box, we can easily see that the efficient allocation of resources should satisfy (1) k x = l x for l x [0 , 16]; (2) for l x = 16, l y = 0 and k y = z , z [0 , 8]. By the production function of y, we can easily get the PPF y = p (24- x )(16- x ) where x [0 , 16]. (b) RPT =- dy dx = 20- x y (c)First, given ( v,w,x ), we solve firm 1s input demand. The problem is min k x ,l x vk x + wl x ,s.t.k x x so, c ( v,w,x ) = ( v + w ) x and k x = l x = x . Second, given p 1 = 6 and ( v,w ), firm 1 maximizes his profit: max x [6- ( v + w )] x So, firm 1s output supply and input demands will be infinity if 6 > v + w ; any quantity if 6 = v + w ; 0 if 6 < v + w . Note: some students take the output supply and input demands to be 16 if 6 > v + w and any quantity in [0 , 16] if 6 = v + w . This is not correct, however. In the factor market, the supply of labor is 16. What we solved is the demand for labor and capital. Given the input demand, we have the output supply. So, the output supply shouldnt satisfy any resource constraint, either. (d)To solve firm 2s input demand, we use the Lagragian method: l = vk y + wl y + ( y 2- k y l y ) By the first order condition, we get that l y = v w k y . Putting this into y 2 = k y l y , we get k y = y p w v and l y = y p v w . 1 Then, the firms problem is max y 2 5 y- 2 vwy So, the output supply and input demands ( y,k y ,l y ) = ( , , ) if 5 > vw ; ( z,z p w v ,z p v w ) for any nonnegative z if 5 = vw ; (0,0,0) if 5 < vw ....
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This note was uploaded on 01/15/2011 for the course ECO 310 at Princeton.