(a) Let f: R→ R+ be a a firm's differentiable production function satisfying theusual assumptions. Suppose that x = R¹ is a vector of inputs and that w € R¹are the factor prices. Let p E R+ denote the output price. The profit of the firmis given by:π(p, w) = max [pf (x) - wx].Derive and explain Shephard's lemma:ƏTθω;(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 supplyfunction.(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 whatoutput level is x₁ = 1 the optimum plant size?(d) Suppose that p = 4 and that the government imposes a lump sum tax T = 2.50on the firm if it produces positive output. What will the firm's output be i) inthe long-run and ii) in the short-run where x₁ is fixed at 1? (A lump-sum tax isdeducted from pre-tax profits, 7 (y) – T).

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(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)

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter2: Mathematics For Microeconomics

Section: Chapter Questions

Problem 2.8P

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