Question

Equation Description: A firm is attempting to minimize total cost subject to sufficiently employing units of labor and units of capital to produce an output level at least as large as a specified output quota. Total cost equals the cost to employing units of labor plus the cost to employing units of capital. A firm's production function is the product of two terms:  the first term is units of capital raised to the .25 power; and, the second term is units of labor raised to the .75 power. The marginal product of labor equals the product of three terms: the first term is .75; the second term is units of capital raised to the .25 power; the third term is units of labor raised to the -.25 power. The marginal product of capital equals the product of three terms: the first term is .25; the second term is units of labor raised to the .75 power; the third term is units of capital raised to the -.75 power. The wage rate is $60 per unit of labor. The rental rate is $20 per unit of capital. The output quota is 1000 units.

Givens: 60 x L120 * K min L>0, K >0 . s.t. 1000 < K.25 * 1.75 • The marginal product of labor: MPL = .75 * K:25 • The marginal product of capital: MPK = .25* :76

Answer 1

... ..... min Sot 60L + zok 1000 s ko.25 0.75 MPL = 0.75 x 0.25 10.25 MPK = 0.25 0.75 xo.75 At cost menimizing cond" MPL mer 0.75 0.25 20:25 0.25 0.75 I ko.751 3 075 0.75 10.25 0.75 so 3 20 0.25 0.25 3(x) = 3 TK=L K= L in put proon th

1000 = 1000 = k0.25 10-75 10.25 0.75 .. L = 1000 since kal ther, lost minimizing amount of Labor & capital is most ܘ ܘ ܘܙ - ܬ RE 10 со tence min. c = = = 60lt zok Go(10007 & 20 (2000) 1000 (60+20) [06 x 85 = 80,000