# Finding economy's factor price for perfect complement production function

Q.Consider an economy producing a single good by a production function $$Y = min [K, L]$$ where Y is the output of the final good. K and L are input use of capital and labour respectively. Suppose this economy is endowed with 100 units of capital and labour supply is Ls is given by the function

Ls = 50w

where w is the wage rate. Assuming that all markets are competitive find the equilibrium wage and rental rate.

Actually I am having doubt in assessing if the economy would use its full capital endowment. Also how we are going to calculate marginal product of labour and capital for finding wage rate and rental rate. I think if we use isoquants and isocost lines, then optimal solution should come where $$K = L = Y$$. But I think I am missing something, please help.

• Solution to this problem is posted on this youtube video: youtube.com/…
– Amit
Apr 25, 2017 at 13:26

The planner is trying to maximize: $$\Pi = Y - r\cdot K - w\cdot L= min [K, L] - r\cdot K - w\cdot L$$

In perfect competition factors receive their marginal product as price. If $0<w <2$ then $0 < L_s < 100$. This means that the marginal product of capital (MPK) is zero and therefore the price of capital ($r$) should be zero. In this region the marginal product of labor (MPL) is 1, and so there is only one wage that represents a competitive equilibrium, $w*=1 \rightarrow L^* = 50, Y^* = 50, r^* = 0$.

What about of $w\geq 2$? In this region the MPL is 0 so this can't be a competitive equilibrium because $2< w \neq MPL$.

Complementing @BKay's answer, let's consider the labor market, which per assumptions is perfectly competitive. The equilibrium wage adjusts so that labor supply and labor demand equate and we have

$$L^s = L^d \implies 50w^* = L^d \implies w^* = L^d/50$$

Now the question is, how much will labor demand be? Given the specification of the production function and the fact that $K =100$, and assuming that we will either employ all $K$ or none at all, then we see that Labor Demand cannot exceed the value of $100$, because, if its was, say $101$, then we would have $Y = \min \{K,L\}=K$ and said Labor would not be used in production at all, so it is nonsensical to "demand" it. So we conclude that the maximum observable wage is $\max w = 2$. For the wage range $(0,2]$ labor supply will always be smaller than $100$, and so also labor demand and labor employed, and production will be carried using only labor and no capital, and the rest are @BKay's.

Note that this can be considered suboptimal, because if we employed all capital and no labor, we would have output equal to $100$, while we now have output of only $50$. Even if we assume that working does not generate disulity, higher output would be preferable since it expands the boundaries of the consumption set. So the central planner solution would be different than the market outcome.

Could an entity like a government use tax and subsidies to increase output? In other words assume that the model is

$$\max _{K,L^d}\pi = min [K, L^d] - r\cdot K - w\cdot L^d + s\cdot K$$

$$L^s = 50\cdot (1-\tau)w$$

$$L^s = L^d =L^*$$

and a balanced budget for the government

$$sK = \tau wL^*$$

Would we obtain something different?