Question

The aggregate production function for a country is $Y = 10N - 0.005N^2$, where $N$ is the quantity of labor input. The aggregate labor supply function is $N = 55 + 5w$, where $w$ is the real wage rate. Assuming perfectly competitive labor and product markets, the equilibrium real wage is ___________. (in integer) 
 

Hint:

In labor market equilibrium, set marginal product of labor equal to the real wage rate to determine both $w$ and $N$.

Answer 1

Correct Answer: 9

Step 1: Labor demand from marginal productivity. \[ Y = 10N - 0.005N^2 \Rightarrow \frac{dY}{dN} = 10 - 0.01N. \] In competitive equilibrium, \[ w = 10 - 0.01N. \quad \text{(Labor demand)} \]
Step 2: Labor supply. \[ N = 55 + 5w. \]
Step 3: Substitute demand into supply. \[ N = 55 + 5(10 - 0.01N) \Rightarrow N = 55 + 50 - 0.05N. \] \[ 1.05N = 105 \Rightarrow N = 100. \]
Step 4: Compute equilibrium wage. \[ w = 10 - 0.01(100) = 9. \] \[ \boxed{w = 9.} \]