The production function is Q = AL a K b , where a > 0 and b > 0. a. The marginal
ID: 1255153 • Letter: T
Question
The production function is Q = ALaKb, where a > 0 and b > 0.
a. The marginal product of labor is MPL =
b. The marginal product of capital is MPK =
c The marginal rate of technical substitution is MRTS =
d. Show that the isoquants for this production function are convex. [hint: Show that MRTS diminshes as L increase. Why?]
For the production function above let the price of labor be w and the price of capital be r.
a. the efficient usage function for labor is L* =
b. The efficient usage function for capital is K* =
c. Find the long run cost functions. LTC, LAC, and LMC
d. Show that both LAC and LMC increase at any Q when either w or r increase.
Explanation / Answer
The production function is Q = AL^aK^b, where a > 0 and b > 0. a. The marginal product of labor is MPL = aAL^a-1K^b b. The marginal product of capital is MPK = bAL^aK^b-1 c The marginal rate of technical substitution is MRTS = MPK/MPL=(bAL^aK^b-1)/(aAL^a-1K^b) d. Show that the isoquants for this production function are convex. [hint: Show that MRTS diminshes as L increase. Why?] take 2nd derivative of either MPK or MPL MPK''=(b-1)bAL^aK^b-2 =((b^2-b)AL^a)/(AK^b+2)Related Questions
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