if production is Q=(a^c) min(K,L) where a>0, c>1, price of capital(r)=£10 and price of labour (w)= £10

what is the optimal combination of capital and labour?
does the production have increasing/constant/decreasing returns to scale?

To find the optimal combination of capital and labor, we need to maximize the production function Q with respect to both K (capital) and L (labor).

First, let's calculate the partial derivatives of Q with respect to K and L:

∂Q/∂K = a^c * min(1, L) (since the partial derivative of K with respect to K is 1)
∂Q/∂L = a^c * min(1, K) (since the partial derivative of L with respect to L is 1)

To find the optimal combination, we set these partial derivatives equal to 0 and solve for K and L:

∂Q/∂K = a^c * min(1, L) = 0
∂Q/∂L = a^c * min(1, K) = 0

Since a^c is always positive (given that a>0 and c>1), we can ignore it for now and focus on the min(1, L) and min(1, K) terms.

For ∂Q/∂K = a^c * min(1, L) = 0:
Either min(1, L) = 0 or min(1, L) = 1

- If min(1, L) = 0: This means L = 0, which implies that no labor is used.
- If min(1, L) = 1: This means L > 0, so labor is used.

Similarly, for ∂Q/∂L = a^c * min(1, K) = 0:
Either min(1, K) = 0 or min(1, K) = 1

- If min(1, K) = 0: This means K = 0, which implies that no capital is used.
- If min(1, K) = 1: This means K > 0, so capital is used.

Therefore, the optimal combination of capital and labor depends on the values of a, c, and the prices of capital and labor.

Regarding the returns to scale, we need to examine how the production function changes when all inputs (K and L) are multiplied by a constant factor.

The production function Q = (a^c) * min(K, L) exhibits constant returns to scale if doubling all inputs (K and L) results in a doubling of output Q.

To determine whether the production function has increasing, constant, or decreasing returns to scale, you would need to evaluate the production function for different values of K and L. Specifically, you would compare Q for various combinations of K and L when the inputs are proportionally increased (e.g., doubling K and L) to see how the output changes.