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.
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 $0&lt;w &lt;2$ then $0 &lt; L_s &lt; 100$. 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 $2&lt; w \neq MPL$.