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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?