i) If there are 100 identical firms, this would mean the resources are divided from the total number of men or acres over the total number of firms. Thus 9 men per firm and 6 acres per firm: 

\(\frac{900}{100} = 9 , \frac{600}{100} = 6\)

ii) The marginal product of men is equal to the derivative of their total product:

\(\frac{dQ}{dM} = 3.33(9)^{-2/3}(6)^{2/3} \approx ~2.5437\) units of Q

The marginal product of acres is equal to:

\(\frac{dQ}{dA} = 10(9)^{1/3}(\frac{2}{3})(6)^{-1/3} \approx 7.6314\) units of Q

iii) In equilibrium, factors are paid their marginal product. Since Q is numeraire, the wage and rent will be equal the marginal product of men and acres, about 2.5 and 7.6 units of Q respectively. 

iv) Marginal cost is found by taking the derivative of the total cost function. Since we know the simple economy is optimizing, we can assume the marginal costs of the two factors are identical and take it just with respect to M:

\(TC = 2.5M(Q) + 7.6A(Q)\)

 \(MC = \frac{\partial TC}{\partial Q} = \frac{\partial TC}{\partial M} \cdot \frac{\partial M}{\partial Q} = 2.5Q \cdot \frac{2}{5Q} = 1\) unit of Q is the marginal cost of producing 1 unit of Q. 

This is confirmed because MC should also be equal to the price, 1 unit of Q.  

v) The absolute shares of each factor can be found by:

\(TP = 10(900^{1/3}) (600^{2/3}) = 6868.29\)

The share paid to men: 

\(2.5437(900) = 2289.33\)

The share paid to the acreage:

\(7.6314(600) = 4578.84\)

Please note, the sum of the shares (6868.17) is a little off since the marginal product of each input was rounded to the nearest decimal place.  

Part B

i) Men would be hired up to the point where \(MP_M = w\), that is, at 5 units of Q. Individual firms would still rent 6 acres of land since this is the number that maximizes profit, thus the full number of acres would be employed. The marginal product of M and A would still be such that:

\(\frac{dQ}{dM} = 3.33(M)^{-2/3}(600)^{2/3} = 5 \)

\(M ^{-2/3} = 0.0211 \Longrightarrow M \approx 326\) 

With the enforced minimum wage, only 326 men would be employed, thus about 3.26 men per firm. The amount of acres would stay the same (6).   

ii) Now to find the marginal product of men and of acres, we can put the new figures into our previous equations:

\(\frac{dQ}{dM} = 3.33(3.26)^{-2/3}(6)^{2/3} \approx ~5\) units of Q

\(\frac{dQ}{dA} = 10(3.26)^{1/3}(\frac{2}{3})(6)^{-1/3} \approx 5.44\) units of Q

iii) Again, we know that men and acres are paid their marginal product, thus the wage is 5 (the minimum wage) and the rent is 5.44 units of Q, lower than before because of the artificially high minimum wage.  

iv) Marginal cost should be the same as before, but we can also calculate it differently by dividing the wage/rent by the respective marginal product of the input:

\(\frac{w}{MP_M} = {5}{5} = 1\) and likewise \(\frac{r}{MP_A} = \frac{5.44}{5.44} = 1\)

As previously seen, the marginal cost is 1 unit of Q, the amount it takes to produce an additional unit of Q. 

v) The total production is now 

 \(TP = 5(326) + 5.44(600) \approx 4894\)

Using this, we can see that the share amount paid to men is roughly 1630 while the share amount paid to acres is 3264. The fractional shares are equal to their exponents, 1/3 for men and 2/3 for acres. Thus, due to the minimum wage, the total product has been reduced as well as the shares paid to the factors of production.

Part C

i) The quantity of men employed is still equal to the point where the marginal product equals the maximum wage: 

\(\frac{dQ}{dM} = 3.33(M)^{-2/3}(600)^{2/3} = 2\)

\(M^{-2/3} = 0.0084 \Longrightarrow M \approx 1289\)

This would mean that the number of men desired per firm is roughly 12. Therefore, since the market can only employ 900 men, the amount of acres each firm employs will change:

\((1.67)^{3/2}(A) = 900 \approx 417.03\) 

So each firm will employ 9 men and 4.17 acres. 

ii) Finding the marginal product of men and of acres:

\(\frac{dQ}{dM} = 3.33(9)^{-2/3}(4.17)^{2/3} \approx ~2\) units of Q

\(\frac{dQ}{dA} = 10(9)^{1/3}(\frac{2}{3})(4.17)^{-1/3} \approx 8.6\) units of Q

iii) The wage is equal to the marginal product of men, being 2 units of Q. The rent will be equal to the marginal product of each acres, 8.6 units of Q. 

iv) Finding the marginal cost of men and acreage will the the same as previously:

\(\frac{w}{MP_M} = \frac{2}{2} = 1\) and likewise \(\frac{r}{MP_A} = \frac{8.6}{8.6} = 1\) unit of Q.

v) To find the absolute shares, we first calculate the total product:

\(TP = 2(900) + 8.6(417) \approx 5386\)

Therefore, we know the share of men is 1800, which is a third of the total product and acres are paid 3586, which is two thirds of the total product. Due to a maximum wage, there is a lower total product and product paid to each share than in the absence of market controls.