Finding economy's factor price for perfect complement production function
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 $2&lt; w \neq MPL$.