Question

QUESTION 1

A firm uses a single input labour to produce output q according to the production

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The commodity sells for R150 per unit and the wage rtae is R75 per hour.

a) Find the profit maximizing quantity of L (3)

b) Find the profit maximizing quantity of q and the level of maximum profit. (5)

c) Suppose now the firm is taxed at R30 per unit of output and that the wage rate is subsidized

at a rate of R15 per hour. Assume that the firm is a price taker, so the price of the producer

remains at R150. Find the new profit maximizing levels of L, q and profit. (6)

d) Nowsupposethefirmisrequiredtopaya20percenttaxonitsprofits,Findthenewprofit maximizing levels of L, q and profit.

Answer 1

Solution:

a.). The profit maximizing quantity of L: This is where the firm’s marginal revenue product of labor (MRP) equals the wage (W).

$MRP_{L} = W$

$MRP_{L} = MPL\times P$

$P = Price \;of\;the\;commodity = 150$

$W = 75$

$MPL = \frac{\partial Q} {\partial L} = \frac{4}{L^{1/2} }$

$Therefore:$

$MRP_{L} = W$

$(\frac{4}{L^{1/2} })\times 150 = 75$

$\frac{600}{L^{1/2} } = 75$

$75L^{1/2} = 600$

$L^{1/2} = \frac{600}{75} =8$

$Square\; both \;sides:$

$L = 64$

$The \;quantity\; of \;labor \;that \;will \;maximize\; profit = 64$

b.). Profit maximizing quantity of q and the level of maximum profit:

$Q = 8\sqrt{L}$

$Substitute\; L \;in\; the \;equation:$

$Q = 8\sqrt{64}$

$Q = 8\times8 = 64$

$Profit \;maximizing\; quantity\; of\; q = 64$

$The \;level\; of\; maximum\; profit = Price\times Quantity$

$Profit = 150\times64 = 9,600$

$The\; level\; of\; maximum\; profit = R9,600$

c.). $MRP_{L} = W$

$MRP_{L} = MPL\times P$

$P = Price \;of\;the\;commodity = 150$

$W = 75 - 15 = 60$

$MPL = \frac{\partial Q} {\partial L} = \frac{4}{L^{1/2} }$

$Therefore:$

$MRP_{L} = W$

$(\frac{4}{L^{1/2} })\times 150 = 60$

$\frac{600}{L^{1/2} } = 60$

$60L^{1/2} = 600$

$L^{1/2} = \frac{600}{60} =10$

$Square\; both \;sides:$

$L = 100$

$The \; new\;quantity\; of \;labor \;that \;will \;maximize\; profit = 100$

To get the new profit maximizing quantity of q and the level of maximum profit:

$Q = 8\sqrt{L}$

$Substitute\; L \;in\; the \;equation:$

$Q = 8\sqrt{100}$

$Q = 8\times10 = 80$

$Profit \;maximizing\; quantity\; of\; q = 80$

$The \;level\; of\; maximum\; profit = Price\times Quantity$

$Profit = 150\times80 = 12,000$

$The\; level\; of\; maximum\; profit = R12,000$

d.). $New\;price = 150 + (20 \% \;tax\;increase) = 180$

$MRP_{L} = W$

$MRP_{L} = MPL\times P$

$P = Price \;of\;the\;commodity = 180$

$W = 75 - 15 = 60$

$MPL = \frac{\partial Q} {\partial L} = \frac{4}{L^{1/2} }$

$Therefore:$

$MRP_{L} = W$

$(\frac{4}{L^{1/2} })\times 180 = 60$

$\frac{720}{L^{1/2} } = 60$

$60L^{1/2} = 720$

$L^{1/2} = \frac{720}{60} =12$

$Square\; both \;sides:$

$L = 144$

$The \; new\;quantity\; of \;labor \;that \;will \;maximize\; profit = 144$

To get the new profit maximizing quantity of q and the level of maximum profit:

$Q = 8\sqrt{L}$

$Substitute\; L \;in\; the \;equation:$

$Q = 8\sqrt{144}$

$Q = 8\times12 = 96$

$Profit \;maximizing\; quantity\; of\; q = 96$

$The \;level\; of\; maximum\; profit = Price\times Quantity$

$Profit = 180\times96 = 17,280$

$The\; level\; of\; maximum\; profit = R17,280$