\(\frac{10}{3}*[\frac{6}{1.5^{1.5}}]^{-\frac{2}{3}}6^{\frac{2}{3}}= \frac{10}{3}*[\frac{6}{\frac{6}{1.5^{1.5}}}]^{\frac{2}{3}} = \frac{10}{3}*[1.5^{\frac{3}{2}}]^{\frac{2}{3}} = 5\) \(\Longrightarrow\) Since Q is the numéraire, the marginal productivity of M is equal to 5.
\(\frac{\partial Q}{\partial A} = \frac{20}{3} M^{\frac{1}{3}}A^{-\frac{1}{3}}\)
\(\frac{20}{3}*[\frac{6}{1.5^{1.5}}]^{\frac{1}{3}}6^{-\frac{1}{3}}=\frac{20}{3}*1.5^{-0.5} \approx 5.443\) \(\Longrightarrow\) Since Q is the numéraire, the marginal productivity of A is equal to about 5.443.
The higher price per man reduces the number of men hired such that the price paid for them equals their marginal productivity. The lower number of men decreases the marginal productivity of acres.
iii)
Men will be paid 5 units of Q because of the minimum wage and A will be paid about 5.443 because of its new, lower marginal productivity.
iv)
Total cost is equal to:
\(TC=5M(Q) + [\frac{20}{3}*1.5^{-0.5}]A(Q)\)
Since \(\frac{\partial Q}{\partial M} = 5\) Then \(\frac{\partial M}{\partial Q} = \frac{1}{5}\)
\(\frac{\partial TC}{\partial Q} = 1\)
The marginal cost of producing 1 more Q is still 1.
v)
\(Q=10M^{\frac{1}{3}}A^{\frac{2}{3}}\) \(\Longrightarrow\) \(Q=10*[\frac{600}{1.5^{1.5}}]^{\frac{1}{3}}*600^{\frac{2}{3}} \approx 4898.979\) is the total revenue in the economy.
The amount paid to men is equal to the wage time the number of men:
\(5*M = 5*[\frac{600}{1.5^{1.5}}] \approx 1632.993\)
The share going to labor is equal to the amount of numéraire paid to men divided by total revenue:
\(\frac{5*[\frac{600}{1.5^{1.5}}] }{10*[\frac{600}{1.5^{1.5}}]^{\frac{1}{3}}*600^{\frac{2}{3}}}=\frac{0.5}{1.5} = \frac{1}{3}\)
This means that the minimum wage does not change the share of income going to labor.
Logically, our result means that the share paid to land must be equal to two thirds. The share of income going to A is equal to:
\(\frac{20}{3}*1.5^{-0.5}*A = \frac{20}{3}*1.5^{-0.5} *600 \approx 3265.986\)
\(\frac{\frac{20}{3}*1.5^{-0.5} *600}{10*[\frac{600}{1.5^{1.5}}]^{\frac{1}{3}}*600^{\frac{2}{3}}}=\frac{2}{3}\)