Suppose the production function Q = AK^a*L^b , where K is the amount of capital, L is the amount of labor it uses as inputs, A is a constant and represents the technology. The cost per unit of capital is a rental fee r and the cost per unit of labor is a wage w. The price of the output is P.

1. Calculate the technical rate of substitution(TRS, assume K is on the horizontal axis);

2. In the short run, the capital remains unchanged, calculate the conditional labor demand and total cost function in terms of w, r , K and Q;

3. In the long run, both K and L are the variable cost, A=1, a=b=0.5, calculate the conditional labor demand, conditional capital demand and total cost function in terms of w, r and Q.

4. In the long run, both K and L are the variable cost, A=1, a=b=0.5, calculate the average total cost and show that how will the average total cost be impacted if the total output is doubled.

5. In the short run, the capital remains unchanged, calculate the labor demand function and the short-run output level in terms of w, r , K and P;

6. In the long run, both K and L are the variable cost, A=1, a=b=0.25, calculate the labor demand and capital demand function in terms of w, r and P.

7. In the long run, both K and L are the variable cost, A=1, a=b=0.5, try to calculate the labor demand and capital demand function in terms of w, r and P, explain why you cannot get these two demand functions(hint: returns-to-scale).