Answer the following questions using the production function

Answer the following questions using the production function q=AK^a L^b, where A,a,b>0.

1. Does labor exhibit diminishing marginal product if b=0.2? Show me how you know.

2. Find the rate of technical substitution (RTS) for an isoquant of this production function. Show your work—b does not equal 0.2 for this question. Find the RTS in terms of the parameters and variables (i.e., it won’t yield a numerical answer at this stage).

3. Pick your own a and b, each between 0 and 1, and then explain to me whether your assumed values give us constant, increasing, or decreasing returns to scale. (You can use words or mathematics, but if you use words, be very precise!)

4. Using your values of a and b from above, pick a value for A that is greater than zero, then solve for the function of the q=100 isoquant. Show your work.

Solution

1.

It can’t be said.

The powers “a” and “b” are the output elasticities of capital and labor respectively. Production function depends on both of them.

If a + b = 1, the production function is constant.

If a + b < 1, the production function is decreasing.

If a + b > 1, the production function is increasing.

Getting the value of one elasticity without knowing the other one can’t satisfy any of the above equation/ inequations. Therefore, marginal product status can’t be said.

2.

MRTS = (a / b) × (L / K) = aL / bK

3.

If a + b = 1, the production function has constant return to scale

If a + b < 1, Return to scale is decreasing.

If a + b > 1, Return to scale is increasing.