To solve the questions provided regarding the economy's production functions, we need to analyze both parts (a) and (b) systematically.
Part (a): Marginal Products
We are given two production functions:
- \( Q_s = \sqrt{L_S \cdot H_S} \) (for the specific sector)
- \( Q_c = \sqrt{L_C \cdot H_C} \) (for the competitive sector)
where:
- \( L_S \) and \( L_C \) are the amounts of the mobile factor (labor) used in the specific and competitive sectors, respectively,
- \( H_S \) and \( H_C \) are amounts of specific factors in each sector.
We know that the total labor is constrained by \( L_S + L_C = 72 \).
Marginal Product of Mobile Labor (L):
To find the marginal product of mobile labor in both sectors, we differentiate the production functions with respect to \( L_S \) and \( L_C \):
-
For the specific sector: \[ Q_s = \sqrt{L_S \cdot H_S} \] Using the product rule: \[ MP_L^s = \frac{\partial Q_s}{\partial L_S} = \frac{1}{2} (L_S \cdot H_S)^{-1/2} \cdot H_S = \frac{H_S}{2\sqrt{L_S \cdot H_S}} \]
-
For the competitive sector: \[ Q_c = \sqrt{L_C \cdot H_C} \] Similarly: \[ MP_L^c = \frac{\partial Q_c}{\partial L_C} = \frac{1}{2}(L_C \cdot H_C)^{-1/2} \cdot H_C = \frac{H_C}{2\sqrt{L_C \cdot H_C}} \]
Marginal Product of Specific Factors:
Now, we find the marginal products of the specific factors \( H_S \) and \( H_C \):
-
For the specific sector: \[ MP_H^s = \frac{\partial Q_s}{\partial H_S} = \frac{1}{2}(L_S \cdot H_S)^{-1/2} \cdot L_S = \frac{L_S}{2\sqrt{L_S \cdot H_S}} \]
-
For the competitive sector: \[ MP_H^c = \frac{\partial Q_c}{\partial H_C} = \frac{1}{2}(L_C \cdot H_C)^{-1/2} \cdot L_C = \frac{L_C}{2\sqrt{L_C \cdot H_C}} \]
Part (b): Equilibrium Wages and Employment
Given that \( P_s = P_c = 2 \), we can derive the equilibrium wages.
Let \( W \) be the wage for the mobile factor \( L \). In equilibrium, the wage must equal the marginal products of labor in both sectors adjusted for prices:
-
For the specific sector: \[ W = P_s \cdot MP_L^s = 2 \cdot \frac{H_S}{2\sqrt{L_S \cdot H_S}} = \frac{H_S}{\sqrt{L_S \cdot H_S}} \]
-
For the competitive sector: \[ W = P_c \cdot MP_L^c = 2 \cdot \frac{H_C}{2\sqrt{L_C \cdot H_C}} = \frac{H_C}{\sqrt{L_C \cdot H_C}} \]
Since the wage must be the same in both sectors: \[ \frac{H_S}{\sqrt{L_S \cdot H_S}} = \frac{H_C}{\sqrt{L_C \cdot H_C}} = W \]
Next, we need to solve for \( L_S \) and \( L_C \). We know \( L_S + L_C = 72 \).
Using the above equations, we set them equal: \[ H_S \cdot \sqrt{L_C \cdot H_C} = H_C \cdot \sqrt{L_S \cdot H_S} \]
Squaring both sides leads to an explicit relationship that can be solved for \( L_S \) and \( L_C \). Let’s denote them using a variable \( k \) for simplification based on the assumption of equal production, and solve the system by substituting \( L_C = 72 - L_S \) into the equilibrium wage equations, but the exact movements will depend on relative factoricity.
Setting both equations leads to conditions we can solve iteratively or with substitutions directly.
Final Equilibrium Calculation
Speculative calculations assuming certain labor distribution along these conditions lead us to deploy factors and numerical methods ultimately.
Note, to find \( H_S \) and \( H_C \) values as we have fixed values suggested from the calculation leads to specific outputs.
- Solve for \( W \) using input that factors \( H_S, H_C \).
- Deduce from available constant performance approximation equilibriums between wage factors from the earlier equations charters.
This sketch can keep developing as per more specified values for labor allocations once decidable.
So, typically, we'd have to iteratively pinpoint values, so situate and figure each sector's solution iteratively as per emergent relations from the set figures.