Study Guides (238,472)
York University (9,811)
Economics (609)
ECON 2300 (24)

# AP ECON 2300 F2012 Session 10.doc

15 Pages
123 Views

School
York University
Department
Economics
Course
ECON 2300
Professor
Wai Ming Ho
Semester
Fall

Description
AP ECON 2300 F2012 Session 10 Topic: Profit Maximization and Cost Minimization Reading: Chapters 19 & 20 Ch 19: Profit Maximization: Economic Profit • A firm uses inputs j = 1…,m to make products i = 1,…n; Output levels are y ,…,y 1 n • Input levels are x1,…,x m; Product prices are p ,1,p . n • Input prices are w 1…,w m. Economic Profit • The economic profit generated by the production plan (x ,…,x1,y ,…my 1 is n Π = p y +  + p y − w x −  w x . 1 1 n n 1 1 m m • x1might be the number of labor units used per hour. • And y 3ight be the number of cars produced per hour. How do we value a firm? • Suppose the firm’s stream of periodic economic profits is P , P 0 P1, …2and r is the rate of interest. • Then the present-value of the firm’s economic profit stream is Π Π PV = Π + 1 + 2 + 0 1+r (1+r) 2 • A competitive firm seeks to maximize its present-value. • How? Page 1 of 15 • Suppose the firm is in ~ short-run circumstance in which its short-run production function is y = f (x1, x2). • The firm’s fixed cost is ~ FC = w x 2 2 • and its profit function is ~ Π = py − w x1−1w x 2 2 Short-Run Iso-Profit Lines • A \$ Π iso-profit line contains all the production plans that provide a profit levΠl \$. ~ • A \$ Π iso-profit line’s equation Π ≡ py − w x1−1w x 2 2 ~ ~ y = w 1x + Π + w 2 2. has a slope of + w 1 and a vertical intercept of Π + w 2 2. p 1 p p p Short - Run Iso - Profit Lines y x 1 © 2010 W. W. Norton & Company, Inc. 13 Short-Run Profit-Maximization • The firm’s problem is to locate the production plan that attains the highest possible iso-profit line, given the firm’s constraint on choices of production plans. • Q: What is this constraint? • A: The production function. Short -Run Profit -Maximization Given p, w and the short -run y 1 profit -maximizing plan is And the maximum possible profit is x 1 © 2010 W. W. Norton & Company, Inc. 20 Page 2 of 15 Short-Run Profit -Maximization At the short -run profit -maximizing plan, y the slopes of the short -run production function and the maximal iso-profit line are equal. x 1 © 2010 W. W. Norton & Company, Inc. 22 w 1 MP =1 ⇔ p × MP = 1 1 p • p *MP 1 is the marginal revenue product of input 1, the rate at which revenue increases with the amount used of input 1. • If p *MP >1w the1 profit increases with x . 1 • If p *MP < w then profit decreases with x . 1 1 1 Short-Run Profit-Maximization; A Cobb-Douglas Example 1/3~ 1/3 • Suppose the short-run production function is y = x 1 x 2 . MP = ∂ y = 1 x − 2 x1/ . 1 ∂ x1 3 1 2 p * −2 /~1/3 p MRP =1p × MP = 1 (x1) x2 = w 1 (x 1 −2 /x32/ = w 1 3 3 ~1/3 * −2/3 3w 1 * 2/3 px 2 (x1) = ~ 1/3. (x1) = px2 3w 1 ~1/3 3/2 3/2 *  px 2   p  ~1/2 x 1   =   x2 .  3w 1  3w 1   p 3/ 2 x 1   x2/ 2 is the firm’s short-run demand for input 1 when the level of input 2 is fixed 3w 1  at ~ units. x2 x 2 The firm’s short-run output level is thus 1/ 2 y = (x )* 1/3x1/ 3=  p  x1/ . 1 2 3w  2  1 Comparative Statics of Short-Run Profit-Maximization Page 3 of 15 • What happens to the short-run profit-maximizing production plan as the output price p changes? • The equation of a short-run iso-profit line is w Π + w x ~ y = 1 x1+ 2 2 p p • so an increase in p causes -- a reduction in the slope, and -- a reduction in the vertical intercept. Comparative Statics of Short - y Run Profit -Maximization x 1 © 2010 W. W. Norton & Company, Inc. 34 • An increase in p, the price of the firm’s output, causes - an increase in the firm’s output level (the firm’s supply curve slopes upward), and - an increase in the level of the firm’s variable input (the firm’s demand curve for its variable input shifts outward). 1/3~1/3 The Cobb-Douglas example: When y = x 1 x2 then the firm’s short-run demand for its variable input 1 is 3 / 2 1/ 2 x =  p  x 1/ 2y =  p  x1/ . 1 3w  2 3w  2  1   1  x* increases as p increases 1 * y increases as p increases. What happens to the short-run profit-maximizing production plan as the variable input price w changes? 1 The equation of a short-run iso-profit line is ~ w 1 Π + w x2 2 y = p x1+ p so an increase in w ca1ses -- an increase in the slope, and -- no change to the vertical intercept. Page 4 of 15 Comparative Statics of Short - Run Profit -Maximization y x 1 © 2010 W. W. Norton & Company, Inc. 43 • An increase in w , 1he 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). • Example: 3 / 2 1/ 2 1/3~ 1/3 *  p  ~1/ 2 *  p  ~1/ 2 PF: y = x 1 x 2 D: x1=  3w  x2 S: y = 3w  x2 .  1   1  x1* decreases as w in1reases, y* decreases as w increase1 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 1nd x ar2 variable. • Think of the firm as choosing the production plan that maximizes profits for a given value of x , and the2 varying x2to find the largest possible profit level. • The equation of a long-run iso-profit line is w Π + w x y = 1x1+ 2 2 p p • so an increase in x c2uses -- no change to the slope, and -- an increase in the vertical intercept. Page 5 of 15 Long -Run Profit -Maximization y The marginal product of input 2 is diminishing. Larger levels of input 2 increase the x 1 © 2010 W. W. Norton & Company, Inc. 53 Long -Run Profit -Maximization y for each short -run production plan. the marginal profit of input 2 is diminishing. x 1 © 2010 W. W. Norton & Company, Inc. 57 • Profit will increase as x2increases so long as the marginal profit of input 2 p × MP 2 w > 2. • The profit-maximizing level of input 2 therefore satisfies p × MP 2 w =20. p × MP − w = 0 1 1 • That is, marginal revenue equals marginal cost for all inputs. The Cobb-Douglas example: When y = x 1/3x1/3 1 2 • then the firm’s short-run demand for its variable input 1 is 3 / 2 1/ 2 x =  p  x 1/ 2 y =  p  x 1/.2 1  3w 1 2 3w 1  2 Page 6 of 15 Π = py − w x 1 1 x 2~ 2 1/ 2 3/ 2  p  1/ 2  p  1/ 2 = p   x2 − w   x2 − w 2~2 3w 1   3w 1  1/ 2 1/ 2  p  ~ 1/ 2 p  p  ~ = p   x2 − w 1   − w 2 2 3w 1  3w 1 3w 1  1/ 2 2 p  p  ~ 1/ 2 ~ =   x 2 − w 2 2 3  3w 1 3 1/ 2  4 p  ~1/ 2 ~ =   x2 − w 2 .2  27w 1   4 p3 1/ 2 Π =   x 1/ − w x .  27 w 1  2 2 2 What is the long-run profit-maximizing level of input 2? Solve 3 1/ 2 ∂ Π 1  4 p  ~ −1/ 2 0 = ~ =   x2 − w 2 ∂ x 2 2  27 w 1  3 ~ * p x2= x =2 2. 27w w1 2 3 / 2 *  p  ~ 1/ 2 x1=   x 2  3w 1  3/ 2 3 1/ 2 3 x =  p   p  = p . 1  3w   27w w 2  27w w2  1   1 2  1 2  1/  3 1/ 2 2 y =  p   p  = p .  3w   27 w w 2  9w w  1   1 2  1 2 y = x 1/3x1/3 1 2  p 3 p 3 p 2  (x , x , y ) =  , , . 1 2  27w w 2 27w w 2 9w w   1 2 1 2 1 2  Returns-to-Scale and Profit-Maximization • If a competitive firm’s technology exhibits decreasing returns-to-scale then the firm has a single long-run profit- maximizing production plan. • If a competitive firm’s technology exhibits exhibits increasing returns-to-scale then the firm does not have a profit- maximizing plan. • So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive. What if the competitive firm’s technology exhibits constant returns-to-scale? • So if any production plan earns a positive profit, the firm can double up all inputs to produce twice the original output and earn twice the original profit. Page 7 of 15 • Therefore, when a firm’s technology exhibits constant returns-to-scale, earning a positive economic profit is inconsistent with firms being perfectly competitive. • Hence constant returns-to-scale requires that competitive firms earn economic profits of zero. Revealed Profitability: Not included: Page 356 Chapter 19.11. Ch 20: Cost Minimization • A firm is a cost-minimizer if it produces any given output level y ≥ 0 at smallest possible total cost. • c(y) denotes the firm’s smallest possible total cost for producing y units of output. • c(y) is the firm’s total cost function. • When the firm faces given input prices w = (w ,w ,…1w 2 the tntal cost function will be written as c(w 1…,w ,yn. The Cost-Minimization Problem • Consider a firm using two inputs to m
More Less

Related notes for ECON 2300

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.