In equilibrium, factors are paid their marginal products, thus wage per man is about 2.544 and rent per acre is about 7.631.
iv) The marginal cost.
Total cost is equal to:
\(TC=[\frac{10}{3}*(\frac{6}{9})^{\frac{2}{3}}]M(Q) + [\frac{20}{3}*(\frac{9}{6})^{\frac{1}{3}}]A(Q)\)
Since \(\frac{\partial Q}{\partial M} = \frac{10}{3}M^{-\frac{2}{3}}A^{\frac{2}{3}}=\frac{10}{3}*(\frac{6}{9})^{\frac{2}{3}}\) Then \(\frac{\partial M}{\partial Q} = \frac{3}{10}*(\frac{6}{9})^{-\frac{2}{3}}\)
\(\frac{\partial TC}{\partial Q} = \frac{\partial TC}{\partial M} \frac{\partial M}{\partial Q} = [\frac{10}{3}*(\frac{6}{9})^{\frac{2}{3}}]*[\frac{3}{10}*(\frac{6}{9})^{-\frac{2}{3}}] = 1\)
The marginal cost of producing 1 more Q is of 1.
v) The absolute shares of the two factors.
\(Q=10M^{\frac{1}{3}}A^{\frac{2}{3}}\) \(\Longrightarrow\) \(Q=10*900^{\frac{1}{3}}*600^{\frac{2}{3}} \approx 6868.285\) is the total revenue in the economy.
The amount paid to men is equal to the wage time the number of men:
\([\frac{10}{3}*(\frac{6}{9})^{\frac{2}{3}}]*M = [\frac{10}{3}*(\frac{6}{9})^{\frac{2}{3}}]*900 \approx 2289.428\)
Hence the share going to labor is equal to the amount of numéraire paid to men divided by total revenue:
\(\frac{[\frac{10}{3}*(\frac{6}{9})^{\frac{2}{3}}]*900}{10*900^{\frac{1}{3}}*600^{\frac{2}{3}}} = \frac{\frac{10}{3}}{10}*\frac{(\frac{6}{9})^{\frac{2}{3}}*900}{900^{\frac{1}{3}}*900^{\frac{2}{3}}*(\frac{6}{9})^{\frac{2}{3}}} = \frac{1}{3}\)
Logically, this means that the share paid to land must be equal to two thirds.
Indeed, the total amount of rent paid is equal to:
\([\frac{20}{3}*(\frac{9}{6})^{\frac{1}{3}}]*A = [\frac{20}{3}*(\frac{9}{6})^{\frac{1}{3}}]*600 \approx 4578.857\)
\(\frac{[\frac{20}{3}*(\frac{9}{6})^{\frac{1}{3}}]*600}{10*900^{\frac{1}{3}}*600^{\frac{2}{3}}} = \frac{\frac{20}{3}}{10}*\frac{(\frac{9}{6})^{\frac{1}{3}}*600}{600^{\frac{2}{3}}*600^{\frac{1}{3}}*(\frac{9}{6})^{\frac{1}{3}}} = \frac{2}{3}\)
b) Do the same, on the assumption that a minimum wage equal to 5 units of Q has been legally established and enforced. Explain.
i) Firms will hire men to the point where \(MPM = P_M\) in our case \(P_M = 5\).
\(\frac{\partial Q}{\partial M} = \frac{10}{3}M^{-\frac{2}{3}}A^{\frac{2}{3}}=5\)
\(\frac{A}{M} = 1.5^{1.5}\)
\(M= \frac{A}{1.5^{1.5}}\)
The firms will still hire 6 units of A because that's what maximizes profits given the constraints. Since \(M= \frac{A}{1.5^{1.5}}\) :
\(M=\frac{A}{1.5^{1.5}} = \frac{6}{1.5^{1.5}} \approx 3.266\)
The number of men hired will be lower with the minimum wage.
ii)
\(\frac{\partial Q}{\partial M} = \frac{10}{3}M^{-\frac{2}{3}}A^{\frac{2}{3}}\)