Economics question ...Hard !!!?

Question

For each of the following production functions, first, express
it in per capita terms and then obtain an expression for the average product of capital (APK). Then show under which conditions the APK decreases, increases or remains unchanged as k = K L varies. Include graphs.

• Y = AK ^α L^1−α .
• Y = AK ^β
.

Answer 1

The first equation assumes, the production function has constant returns to scale (Since, sum of exponents =1).

The second function seems to be simplifying and taking out the labor component from it.

Both functions will yield 3D graphs. I suggest using Mathematica, Matlab, or even TI-89 should do. Or you could try plotting by hand (I suggest using engineering drawing paper for beginners). Good luck.

Answer 2

I am confused by "per capita" do you mean per unit of labor. If so just divide by L that is Y/L
APK is Y/K or maybe Y/(KL) depending on how you read the question.
let L=k/K and substitute it in the expression for APK

The first expression in a Cobb-Douglas function, so if you do a web search you can probably find something helpful.