Question 4 Consider the production function given by $f(x_1, x_2) = x_1^{1/3}x_2^{1/3}$, where
$MP_1 = \frac{1}{3}x_1^{-2/3}x_2^{1/3}$ and $MP_2 = \frac{1}{3}x_1^{1/3}x_2^{-2/3}$. Suppose that $w_1$ is the price of factor 1 and $w_2$ is the price of factor 2.
a) What can you say about the returns to scale of this production function?
b) What is the equation of the isoquant?
c) Find the technical rate of substitution. Is it diminishing?
d) In the long run, what are the optimal levels of $x_1$ and $x_2$ needed to produce y units in the cheapest way possible?
e) What is the cost of producing y units at factor prices $w_1$ and $w_2$? The average cost?
f) From part e, we can calculate $MC(w_1, w_2, y) = 3w_1^{1/2}w_2^{1/2}y^{1/2}$. Suppose that $p$ is the price of output and find supply as a function of $p$, $w_1$, and $w_2$. If the price of factor 1 is $4 and the price of factor 2 is $9, what is the supply function?
