A firm uses a single input, labor, to produce output $q$ according to the production function $q=8 \sqrt{L}$. The commodity sells for $\$ 150$ per unit and the wage rate is
$\$ 75$ per hour.
a. Find the profit-maximizing quantity of $L$
b. Find the profit-maximizing quantity of $q$
c. What is the maximum profit?
d. Suppose now that the firm is taxed $\$ 30$ per unit of output and that the wage rate is subsidized at a rate of $\$ 15$ per hour. Assume that the firm is a price taker, so the price of the product remains at $\$ 150$ Find the new profit-maximizing levels of $L, q,$ and profit.
e. Now suppose that the firm is required to pay a 20-percent tax on its profits. Find the new profitmaximizing levels of $L, q,$ and profit.