(a) Let ƒ : R→ R+ be a a firm's differentiable production function satisfying the usual assumptions. Suppose that x R is a vector of inputs and that w E R are the factor prices. Let p € R+ denote the output price. The profit of the firm is given by: π(p, w) = max [pf (x) - wx] . Derive and explain Shephard's lemma: ƏT -r. (na)
(a) Let ƒ : R→ R+ be a a firm's differentiable production function satisfying the usual assumptions. Suppose that x R is a vector of inputs and that w E R are the factor prices. Let p € R+ denote the output price. The profit of the firm is given by: π(p, w) = max [pf (x) - wx] . Derive and explain Shephard's lemma: ƏT -r. (na)
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![(a) Let f: R→ R+ be a a firm's differentiable production function satisfying the
usual assumptions. Suppose that x = R¹ is a vector of inputs and that w € R¹
are the factor prices. Let p E R+ denote the output price. The profit of the firm
is given by:
π(p, w) = max [pf (x) - wx].
Derive and explain Shephard's lemma:
ƏT
θω;
(b) Now let f: R² → R₁ be defined by:
:-xi (p, w).
==
ƒ (x) = x1 x3.
Suppose this firm is a price taker in the output market and in the factor market.
The output price is p and the factor prices are w = (1,1). Derive the firm's supply
function.
(c) Suppose now that x₁ is fixed at x₁ = 1. Derive the short run supply function.
Which is more elastic, the short-run supply or the long-run supply? For what
output level is x₁ = 1 the optimum plant size?
(d) Suppose that p = 4 and that the government imposes a lump sum tax T = 2.50
on the firm if it produces positive output. What will the firm's output be i) in
the long-run and ii) in the short-run where x₁ is fixed at 1? (A lump-sum tax is
deducted from pre-tax profits, 7 (y) – T).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F190d5eb1-ccff-4979-b204-db939efe7803%2Fa8d0e85f-1fea-4b89-908d-22e1fee2341f%2Fnm5ugyn_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let f: R→ R+ be a a firm's differentiable production function satisfying the
usual assumptions. Suppose that x = R¹ is a vector of inputs and that w € R¹
are the factor prices. Let p E R+ denote the output price. The profit of the firm
is given by:
π(p, w) = max [pf (x) - wx].
Derive and explain Shephard's lemma:
ƏT
θω;
(b) Now let f: R² → R₁ be defined by:
:-xi (p, w).
==
ƒ (x) = x1 x3.
Suppose this firm is a price taker in the output market and in the factor market.
The output price is p and the factor prices are w = (1,1). Derive the firm's supply
function.
(c) Suppose now that x₁ is fixed at x₁ = 1. Derive the short run supply function.
Which is more elastic, the short-run supply or the long-run supply? For what
output level is x₁ = 1 the optimum plant size?
(d) Suppose that p = 4 and that the government imposes a lump sum tax T = 2.50
on the firm if it produces positive output. What will the firm's output be i) in
the long-run and ii) in the short-run where x₁ is fixed at 1? (A lump-sum tax is
deducted from pre-tax profits, 7 (y) – T).
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