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Intermediate Micro HW 16 Producer Theory Let y and x be outputs and intputs, and let p and c be the price of otuput and the cost of input, respectively. Let the production function be f(x) = √x. The firm’s problem is max x,y py – cx s.t. y = √x.. 1. Substitute the constraint into the objective function and solve this problem for y*(p,c) and x*(p,c). y*(p,c) is supply or the supply curve, and x* is input demand. 2. Take the derivative of supply with respect to price, ∂y*/∂p. Does supply increase or decrease with an increase in price? Hint: It increases if the derivative is positive, and decreases if it is negative. 3. Take the derivative of supply with respect to the input cost, ∂y*/∂c. Does supply increase or decrease with an increase in the input cost? 4. Take the derivative of input demand with respect to the input cost, ∂x*/∂c. Does input demand increase or decrease with an increase in the input cost? 5. Take the derivative of input demand with respect to price, ∂x*/∂p. Does input demand increase or decrease with an increase in the price? 6. Compare your answers to 3 and 5. What do you notice? 7. Substitute the supply and input demand into the objective function to get the profit function, π(p,c) = py* + cx*. 8. Take the derivative of the profit function with respect to price, ∂π*/∂p. Do profits increase or decrease with an increase in price? 9. Take the derivative of the profit function with respect to the input cost, ∂π*/∂p. Do profits increase or decrease with an increase in the input cost?