# Suppose that production for good X is characterized by the following production function,...

Suppose that production for good X is characterized by the following production function, {eq}Q=4(K)^{0.5} (L)^{0.5} {/eq}, where K is the fixed input set at K=9 in the short run.If the per-unit rental rate of capital, r, is $25 and the per-unit wage, w, is$15, and the price of the product is $30, how many units of the variable labor should be used? a. 6 b.12 c. 144 d. 9 2. Costco's strategy of offering (non-mandatory) unpaid time off for slow seasons and paying employees a higher hourly rate (than Wal-Mart) may increase Costco's profits due to solving: a. the underinvestment problem b. the hold-up problem and opportunism c. the shutdown problem d. the principal-agent problem ## Variable Cost Variable cost is incurred on the hiring of variable factors in the production of goods and services. It varies as per the extent of the production of output. When the production of output would be zero, then the variable cost would also be zero and it rises as the production of output rises. ## Answer and Explanation: 1. Suppose that production for good X is characterized by the following production function, Q=4(K)0.5(L)0.5Q=4(K)0.5(L)0.5, where K is the fixed input set at K=9 in the short run. If the per-unit rental rate of capital, r, is$25 and the per-unit wage, w, is $15, and the price of the product is$30, how many units of the variable labor should be used c. 144 .

It is because the total revenue is determined as,

{eq}\begin{align*} {\rm{TR = PQ}}\\ {\rm{TR = 30(4}}{{\rm{K}}^{{\rm{0}}{\rm{.5}}}}{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}{\rm{)}}\\ {\rm{TR = 30}}\left[ {{\rm{4(9}}{{\rm{)}}^{{\rm{0}}{\rm{.5}}}}{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}} \right]\\ {\rm{TR = 360}}{{\rm{L}}^{{\rm{0}}{\rm{.5}}}} \end{align*} {/eq}

Now, the cost of the firm is given as,

{eq}\begin{align*} {\rm{TC = wL + rK}}\\ {\rm{TC = 15L + 25(9)}}\\ {\rm{TC = 15L + 225}} \end{align*} {/eq}

Then the profit function of the firm will be determined as,

{eq}\begin{align*} {\rm{Profit = TR - TC}}\\ {\rm{Profit = 360}}{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}{\rm{ - 15L - 225}} \end{align*} {/eq}

Hence, required variable labor be,

{eq}\begin{align*} \frac{{{\rm{d(profit)}}}}{{{\rm{dL}}}}{\rm{ = 360}}\frac{{\rm{1}}}{{{\rm{2}}{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}}} - 15\\ \frac{{{\rm{d(profit)}}}}{{{\rm{dL}}}}{\rm{ = }}\frac{{{\rm{180}}}}{{{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}}} - 15 \end{align*} {/eq}

Put it equal to zero,

{eq}\begin{align*} \frac{{{\rm{d(profit)}}}}{{{\rm{dL}}}}{\rm{ = }}\frac{{{\rm{180}}}}{{{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}}}{\rm{ - 15}}\\ {\rm{0 = }}\frac{{{\rm{180}}}}{{{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}}}{\rm{ - 15}}\\ {\rm{15 = }}\frac{{{\rm{180}}}}{{{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}}}\\ {\rm{144 = L}} \end{align*} {/eq}

Hence, maximum profit by using variable factor will be,

{eq}\begin{align*} \frac{{\rm{d}}}{{{\rm{dL}}}}\left( {\frac{{{\rm{d(profit)}}}}{{{\rm{dL}}}}} \right){\rm{ = }}\frac{{\rm{d}}}{{{\rm{dL}}}}\left( {\frac{{{\rm{180}}}}{{{{\rm{L}}^{{\rm{0}}{\rm{.5}}}}}}{\rm{ - 15}}} \right)\\ \frac{{\rm{d}}}{{{\rm{dL}}}}\left( {\frac{{{\rm{d(profit)}}}}{{{\rm{dL}}}}} \right){\rm{ = 180}}\left( {\frac{{{\rm{ - 1}}}}{{\rm{2}}}} \right){{\rm{L}}^{{\rm{ - 1}}{\rm{.5}}}}\\ \frac{{\rm{d}}}{{{\rm{dL}}}}\left( {\frac{{{\rm{d(profit)}}}}{{{\rm{dL}}}}} \right){\rm{ < 0}} \end{align*} {/eq}

2. Costco's strategy of offering (non-mandatory) unpaid time off for slow seasons and paying employees a higher hourly rate (than Wal-Mart) may increase Costco's profits due to solving b. the hold-up problem and opportunism .

It is because, in this strategy, the labor is not paid anything when there is no chance of production, but take back to the work and pay them the higher wages. Costco sees the opportunity of better production and gains, and then take back employees with their proper incentives. And in slow season, he does not need to pay idle amount to the labor. Hence, Costco?s profit is attained by solving the hold up problem and opportunism as he does not need to hold the labor until the arrival of any kind of opportunity. 