The production function given is Q = LK², where Q is the output, L is labor, and K is capital.
If the firm wants to produce 20 units of output, we can set Q = 20 and solve for the optimal combination of L and K that will minimize the cost of production.
We can use the cost function C = wL + rK to find the cost of production, where w is the cost of labor and r is the cost of capital. In this case, w = 12 birr/unit and r = 6 birr/unit.
To minimize the cost, we can take the partial derivatives of C with respect to L and K, set them equal to zero and solve for L and K.
∂C/∂L = w - 0 = 12
∂C/∂K = r - 0 = 6
We can substitute the values of Q, L, K and w, r in the equation to solve for L and K
Q = LK²
20 = L(6)²
20 = 36L
L = 20/36
L = 5/9
K = √(Q/L)
K = √(20/5/9)
K = √(40/5)
K = 2√(8)
So the optimal combination of inputs is L = 5/9 and K = 2√(8)
The type of returns to scale can be determined by comparing the percentage change in output to the percentage change in inputs.
If the percentage change in output is greater than the percentage change in inputs, then the firm is experiencing increasing returns to scale.
If the percentage change in output is less than the percentage change in inputs, then the firm is experiencing decreasing returns to scale.
If the percentage change in output is equal to the percentage change in inputs, then the firm is experiencing constant returns to scale.
In this case, since the output is directly proportional to the square of the inputs, the firm is experiencing increasing returns to scale.