25. A firm produces the output y using two inputs x₁ and x2 in non-negative quantities through the production relationship: y = g(x1,x2)=xx₁₂ The firm obtains a price of py > 0 per unit of y that it sells. It has available an inventory of K₁ units of the input x₁ and K2 units of the input x2. More units of x₁ and x2 may be purchased from the market at the unit prices of p1 > 0 and p2 > 0, respectively. Alternatively, the firm can also sell any unused amount of its inputs to the market at these prices. 6 (a) Describe the firm's profit-maximization problem, and derive the equations that define the criti- cal points of the Lagrangean L. (b) Assuming py = P₁ = P2 = 1, K₁ = 4, and K₂ = 0, solve for the firm's optimal level of output y. (c) Assume again that py = p₁ = p2 = 1, but suppose now that the values of K₁ and K₂ are interchanged, i.e., we have K₁ = 0 and K₂ = 4. Is the firm's new optimal output level different from the old level? Why or why not?
25. A firm produces the output y using two inputs x₁ and x2 in non-negative quantities through the production relationship: y = g(x1,x2)=xx₁₂ The firm obtains a price of py > 0 per unit of y that it sells. It has available an inventory of K₁ units of the input x₁ and K2 units of the input x2. More units of x₁ and x2 may be purchased from the market at the unit prices of p1 > 0 and p2 > 0, respectively. Alternatively, the firm can also sell any unused amount of its inputs to the market at these prices. 6 (a) Describe the firm's profit-maximization problem, and derive the equations that define the criti- cal points of the Lagrangean L. (b) Assuming py = P₁ = P2 = 1, K₁ = 4, and K₂ = 0, solve for the firm's optimal level of output y. (c) Assume again that py = p₁ = p2 = 1, but suppose now that the values of K₁ and K₂ are interchanged, i.e., we have K₁ = 0 and K₂ = 4. Is the firm's new optimal output level different from the old level? Why or why not?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:25. A firm produces the output y using two inputs x₁ and x2 in non-negative quantities through the
production relationship:
y = g(x1,x2)=xx₁₂
The firm obtains a price of py > 0 per unit of y that it sells. It has available an inventory of K₁ units of
the input x₁ and K2 units of the input x2. More units of x₁ and x2 may be purchased from the market
at the unit prices of p1 > 0 and p2 > 0, respectively. Alternatively, the firm can also sell any unused
amount of its inputs to the market at these prices.
6
(a) Describe the firm's profit-maximization problem, and derive the equations that define the criti-
cal points of the Lagrangean L.
(b) Assuming py = P₁ = P2 = 1, K₁ = 4, and K₂ = 0, solve for the firm's optimal level of output y.
(c) Assume again that py = p₁ = p2 = 1, but suppose now that the values of K₁ and K₂ are
interchanged, i.e., we have K₁ = 0 and K₂ = 4. Is the firm's new optimal output level different
from the old level? Why or why not?
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