Question

3. Suppose the production function for widgets is given by:Q = f(K,L) = KL − 0.4K² − 0.4L²

1. (a) Suppose K = 10 (is fixed), derive an expression for and graph the total product of labor curve (the production function for a fixed level of capital) and the average productivity of labor curve.

2. (b) At what level of labor input does the average productivity reach a maxi- mum? How many widgets are produced at this point?

3. (c) Again, assuming K = 10, derive an expression for and graph the MPLcurve. At what level of labor input does MPL =0?

4. (d) Does this production function exhibit constant, increasing or decreasing returns to scale?

Answer 1

a) Q = f(K,L) = KL − 0.4K² − 0.4L²

K =10, Q = 10L − 0.4*100 − 0.4L²
Q = 10L − 40 − 0.4L²

Total product of labor is the total production given different amount of labor at fixed capital

Q is at Y axis and labour at X axis

Average productivity is the total product per unit labor,
APL = 10 − 40/L − 0.4L

Q is at Y axis and labour at X axis

b) APL is maximum where dAPL/dL = 0,
40/L^2 = 0.4
L = 10
Q = 10 − 40/10 − 0.4*10 = 2
As per the graph as well, L =10 and Q = 2

c) MPL = change in output/ change in labor = dQ/dL
MPL: 10 − 0.8L

MPL = 0
10 − 0.8L = 0
L = 12.5

d) In the part of the curve where
Marginal product of labor >1, increasing
MPL = 1, constant
MPL < 1, decreasing

Constant returns to scale if Q = f(aK,aL) = af(K,L)
Increasing returns to scale if Q = f(aK,aL) >  af(K,L)
Decreasing returns to scale if Q = f(aK,aL) < af(K,L)
f(K,L) = KL − 0.4K² − 0.4L²
f(aK,aL) = a^2KL - 0.4a^2K^2 - 0.4a^2L^2
af(K,L) = aKL − 0.4aK² − 0.4aL²