Comparative statics of short run profit maximization

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Comparative Statics of Short-Run Profit-Maximization The Cobb-Douglas example: The firm’s short-run demand for its variable input 1 is y x x 1 1/3 2 1/3 ~ y * increases as p increases. and the firm’s short-run supply is x 1 * increases as p increases. x p w x 1 1 3 2 2 1/2 3 * / ~ § © ¨ · ¹ ¸ y p w x * ~ . § © ¨ · ¹ ¸ 3 1 1/2 2 1/2 Comparative Statics of Short-Run Profit-Maximization What happens to the short-run profit- maximizing production plan as the variable input price w 1 changes?
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2/21/2017 9 Comparative Statics of Short-Run Profit-Maximization y w p x w x p ± ± 1 1 2 2 3 ~ The equation of a short-run iso-profit line is so an increase in w 1 causes -- an increase in the slope, and -- no change to the vertical intercept. Comparative Statics of Short-Run Profit-Maximization x 1 Slopes w p ± 1 y y f x x ( , ~ ) 1 2 x 1 * y * 3 3 { c 3 3 { cc 3 3 { ccc Comparative Statics of Short-Run Profit-Maximization x 1 Slopes w p ± 1 y y f x x ( , ~ ) 1 2 x 1 * y * 3 3 { c 3 3 { cc 3 3 { ccc
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2/21/2017 10 Comparative Statics of Short-Run Profit-Maximization An increase in w 1 , the price of the firm’s variable input, causes a decrease in the firm’s output level (the firm’s supply curve shifts inward), and a decrease in the level of the firm’s variable input (the firm’s demand curve for its variable input slopes downward). Comparative Statics of Short-Run Profit-Maximization The Cobb-Douglas example: The firm’s short-run demand for its variable input 1 is y x x 1 1/3 2 1/3 ~ y * and the firm’s short-run supply is x 1 * x p w x 1 1 3 2 2 1/2 3 * / ~ § © ¨ · ¹ ¸ y p w x * ~ . § © ¨ · ¹ ¸ 3 1 1/2 2 1/2 decreases as w 1 increases. decreases as w 1 increases. Short-Run Profit-Maximization Chose x 1 to maximize … Solve for . Then ൌ ࢌሺ࢞ , ࢞ . , ࢞ ൌ ࡼ ∙ ࢌ , ࢞ െ ࢝ െ ࢝ ࣔ࣊ , ࢞ ࣔ࢞ ൌ ࡼ ∙ ࣔࢌ , ࢞ ࣔ࢞ െ ࢝ ൌ ૙ set ሺ࢞
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2/21/2017 11 Long-Run Profit-Maximization Now allow the firm to vary both input levels. Since no input level is fixed, there are no fixed costs. Both x 1 and x 2 are variable. Long-Run Profit-Maximization Chose x 1 and x 2 to maximize … Solve simultaneously for and . , ࢞ ൌ ࡼ ∙ ࢌ , ࢞ െ ࢝ െ ࢝ ࣔ࣊ , ࢞ ࣔ࢞ ൌ ࡼ ∙ ࣔࢌ , ࢞ ࣔ࢞ െ ࢝ ൌ ૙ set ࣔ࣊ , ࢞ ࣔ࢞ ൌ ࡼ ∙ ࣔࢌ , ࢞ ࣔ࢞ െ ࢝ ൌ ૙ set Long-Run Profit-Maximization For Cobb-Douglas ࢟ ൌ ࢞ ૚/૜ ૚/૜ ࢟ ൌ ࢞ ૚/૜ ૚/૜ The optimal factor demands are The output level will be and
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