Please, provide all the steps... I need to be able to do this on my own after yo
ID: 1188979 • Letter: P
Question
Please, provide all the steps... I need to be able to do this on my own after you show me. Thank you!
A firm produces output according to the production function:
Q = F(K,L) = 4K + 8L.
a. How much output is produced when K = 2 and L = 3?
b. If the wage rate is $60 per hour and the rental rate on capital is $20 per hour, what is the cost-minimizing input mix for producing 32 units of output?
Capital: ?
Labor: ?
c. If the wage rate decreases to $20 per hour but the rental rate on capital remains at $20 per hour, what is the cost-minimizing input mix for producing 32 units of output?
Capital: ?
Labor: ?
Explanation / Answer
Production Function: Q = F(K,L) = 4K + 8L.
a) when K = 2 and L = 3; Q = 4(2) + 8(3) = 8 + 24 = 32 units
b) w = $60 and r = $20.
At cost minimizing output or at optimal output the following condition needs to fulfilled:
MP of capital / MP of Labor = Rental rate / wage rate
MP refers to Marginal Product.
and r/w = 20/60 = 1/3
thus MPk / MPl = 1/3
3MPk = MPl
3dQ/dK = 3*4 = MPl = 12
Thus L = 12/8 = 1.5
and K = 5 units.
c) if wage reduced to $20.
r/w = 20/20 = 1
thus MPk/MPl = 1
MPk = MPL
Thus K = 4 and L = 2.