Question

Suppose labor (L) and capital (K) are fixed proportion inputs, and that each unit of output must be produced with exactly 5 units of labor and 2 unit of capital. The price of a unit of labor is w, and the price of a unit of capital is r.

a. Write down the production function.

b. Derive the equation for the expansion path.

c. Derive the equation for the long-run total cost function.

d. Graphically depict the labor demand curve.

e. Assume that a unit of the firm’s output can be produced using either 5 units of labor and no capital, or 2 units of capital and no labor. That is, labor and capital are perfect substitutes. Moreover, assume w = r.

i. Write down the firm’s production function.

ii. Derive the equation for the firm’s long-run total cost function.

Answer 1

as - the fixed proportion prod" func in: :.. Q = min{ 4, } ais per unit labor requirement bu " u capital a .::.Q : min { } . by At the equilibrium: & :.L = 2:5ķ This is the equation for expansion path. Then in the long run , Q = L = 2.5k :: LEQ, K=0.4.2 Total cost of the firm: . c = WL trk :.C=wQ+r(0:42) ...LRTC = (w +0:4r) a d) The labor demand in: MRPL = w or, P.MPL fox, P. and sw WA La - w/p=1 Labor demand in infinitely elastic at real wage (W/P) . exis The perfect swestitute production function in Q = . (11) A+ equilibrium; as, m 04< /

or, mert a mer Then as this implies each dollar invested in labor give lower retuen than that of K. Then the firm choses only k. : Q = K TC = WL Ark - wxo + TK = r. 2Q :.LRTC 2ra