Problem 5:
Consider the following variation of the aggregate production function. Now
firms must use oil $M$ to produce output (in addition to labor and capital). The
price of a unit of oil is $p$
$\max \Pi_f = AK^\alpha L^\beta M^\gamma - wL - rK - pM$
(a) Find a first-order condition for the firm's demand for oil.
(b) What must be true about the parameters $\alpha$, $\beta$, and $\gamma$ if this production
function exhibits constant returns to scale?
(c) If the price of oil $p$ rises, what would you expect to happen to carbon
intensity (the ratio of oil per unit output: $M/Y$) in this economy? What
happens to the revenue share of oil (the ratio of total oil payments to
output: $pM/Y$)?