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week5-1_Econ2101_S1_2009

Course: ECON2101 2101, Spring 2012
School: UNSW
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and Technology Profit Maximization Instructor: Dr. Arghya Ghosh (a.ghosh@unsw.edu.au) Office Hours: Wed, 1pm 4pm, ASB 406 Weeks: 5 8, 11-12 Theme: Production, Firms, Market Structure Chapter 18: Technology Chapter 19: Profit Maximization Technology Technology Technology describes how inputs are converted output. Production Functions Isoquants Marginal Product Returns to Scale Technical rate of substitution...

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and Technology Profit Maximization Instructor: Dr. Arghya Ghosh (a.ghosh@unsw.edu.au) Office Hours: Wed, 1pm 4pm, ASB 406 Weeks: 5 8, 11-12 Theme: Production, Firms, Market Structure Chapter 18: Technology Chapter 19: Profit Maximization Technology Technology Technology describes how inputs are converted output. Production Functions Isoquants Marginal Product Returns to Scale Technical rate of substitution Long run and short run Lecture is being produced by using computer, projector, software, labour PRODUCTION FUNCTION: The maximum amount of output (y) one can produce using inputs x1, x2 , y = f(x1, x2 , ) WEEK 5-1 ECON2101 S1 2009 WEEK 5-1 One input, one output y = f(x) is the production function. y = f(x) y = f(x) is the maximal output level obtainable from x input units. x Input Level WEEK 5-1 ECON2101 S1 2009 S1 2009 Production Function with more than one input Production Function y ECON2101 Labour and Capital used for manufacturing production Labour and Land for agricultural production How are different factors/inputs combined to produce output? ISOQUANTS : combination of different inputs that produce same level of output x WEEK 5-1 ECON2101 S1 2009 1 Fixed Proportion Production Fn Fixed Proportion Isoquants Y = Digging holes , X1 = labourers, X2 = shovels Y = min (X1, X2) , X1 = 1, X2 = 1; Y = min (1,1) = 1 X1 = 2, X2 = 1; Y = min (2,1) = 1 X1 = 1, X2 = 2; Y = min (1,2) = 1 (1,1), (1,2), (2,1) . all these combinations produce 1 unit of output. Two inputs need to be used in fixed proportion. Adding more of one without having more of the other does not help WEEK 5-1 ECON2101 S1 2009 Perfect Substitutes WEEK 5-1 ECON2101 S1 2009 Perfect Substitutes Isoquants Y = X1 + X2 , X1 = 1, X2 = 1; Y =2 Y =2 X1 = 0, X2 = 2; Y =2 X1 = 2, X2 = 0; (1,1), (0,2), (2,0) . all these combinations produce 2 units of output. Two inputs are perfect substitutes Grading multiple choice exams: labour and capital (machine) are perfect substitutes WEEK 5-1 ECON2101 S1 2009 Cobb-Douglas Production Function b y = Ax1 x2 a Let a = , b = and A =1. Find input combinations such that y = 2. WEEK 5-1 ECON2101 S1 2009 Cobb-Douglas Isoquants Different combinations of (x1, x2) that yield y = 2 x2 x'2 1 2 = x1 x2 / 2 1/ 2 x1 = 1, x2 = 4; x1 = 4, x2 = 1; x1 = 2, x2 = 2; Note if x1 or x2 = 0, y = 0. Thus both inputs are essential WEEK 5-1 ECON2101 S1 2009 x" 2 y= 2 x' 1 x" 1 WEEK 5-1 ECON2101 x1 S1 2009 2 Marginal Product Computing Marginal Product (MP) The marginal product of any input i is the rate of change of the output level as the level of input i changes. 1 f (L, K) = L / 3K2/ 3 Production function: y = f(L,K) y Marginal product of labour (MPL) is L = Marginal product of capital (MPK) is f ( L, K ) L y f ( L, K ) = K K Properties (that typically hold) Marginal product of L depends on K and similarly marginal product of K depends on L Diminishing Marginal product --- MPL declines as L increases. MPK declines as K increases WEEK 5-1 ECON2101 S1 2009 Returns to Scale ECON2101 S1 2009 Returns to scale: an example Consider Note, for Cobb-Douglas technology MPL increases as K increases MPL diminishes as L increases 2 MP f (L, K ) 2 f ( L, K ) L = = L5/ 3 K 2/ 3 < 0 = L2 9 L L L WEEK 5-1 ECON2101 S1 2009 Returns to Scale (contd.) Marginal product describe the change in output level as a single input level changes Returns-to-scale describes how the output level changes as all input levels change in direct proportion (e.g. all input levels or halved). WEEK doubled, 5-1 f (L, K) 1 2 / 3 2 / 3 = L K >0 L 3 MPL is f ( L, K ) = La K b Double L and K. Then Consider the production function y = F(L,K) Increase t fold where t > 1 Constant returns to scale (CRS) if F(tL, tK) = tF(L,K) Increasing returns to scale (IRS) if F(tL, tK) > tF(L,K) Decreasing returns to scale (DRS) if F(tL, tK) < tF(L,K) WEEK 5-1 ECON2101 S1 2009 Technical Rate of Substitution At what rate can a firm substitute one input for another without changing its output level F(2L,2K)= ( 2L)a( 2K)b = 2a+b La K b = 2a+b F(L, K) Constant returns to scale if a + b = 1 Increasing returns to scale if a + b > 1 Decreasing returns to scale if a + b <1 WEEK 5-1 ECON2101 S1 2009 WEEK 5-1 ECON2101 S1 2009 3 Technical rate of substitution(contd.) To compute TRS output level must remain unchanged. Thus dy = 0 This implies Technical rate of substitution The slope is the rate at which input 2 must be given up as input 1s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution. x2 f ( L, K ) f ( L, K ) dL + dK = 0 L K Rearranging we get the TRS TRS = f ( L, K ) dK MPL = L = f ( L, K ) dL MPK K WEEK 5-1 ECON2101 S1 2009 Technical rate of substitution Compute TRS at K = 1, L = 4 for a CobbDouglas production function f(L,K) L / 3K 2 / 3 =1 MPL = MPK = 1 f(L,K) = L 2 / 3 K 3 L 2 f(L,K) = L1 / 3 K 3 K 2/3 1 / 3 TRS = - (MPL/MPK) = - (K/2L); At L = 4, K = 2, the TRS is 1/8 WEEK 5-1 ECON2101 S1 2009 Long run and the short run (contd.) Suppose y = f(x1, x2) Consider a short run where x1 is variable x2 is fixed. Note, depending on the level of x2, the short run production function changes. WEEK 5-1 ECON2101 S1 2009 x'2 y x' 1 WEEK 5-1 x1 ECON2101 S1 2009 Long Run and the Short run The long-run is the circumstance in which a firm is unrestricted in its choice of all input levels. A short-run is a circumstance in which a firm is restricted in some way in its choice of at least one input level. In a manufacturing firm that employs capital and labour, capital (K) might be fixed in the short run but variable in the long run WEEK 5-1 ECON2101 S1 2009 Short run production functions y = f ( x1 , x2 ) = x1/ 3 x2 / 3 WEEK 5-1 ECON2101 S1 2009 4 Profit maximization Preliminaries Short run profit maximization Long run profit maximization Factor demand curves WEEK 5-1 ECON2101 S1 2009 Short run profit-maximization Suppose x2 is fixed in the short run What is the level of x1 that maximize profits? Example: A manufacturing firm uses labour (L) and capital (K) to produce its own output. Suppose capital is fixed in the short run. How much labour should it employ? To answer that first we need to find out the benefit and cost from employing an extra unit of labour. WEEK 5-1 ECON2101 S1 2009 Preliminaries Output level y = f(x1,x2) Product price p Revenue = py = pf(x1,x2) Inputs: x1,x2. Input prices: w1,w2. Costs = w1x1 + w2x2 Profits = Revenue Costs = pf(x1,x2) - w1x1 - w2x2 Taking p, w1 and w2 as given each firm chooses x1,x2.to maximize Profits WEEK 5-1 ECON2101 S1 2009 Thinking clearly Production function y = f(L,K) where K is fixed in the short run. As a owner you are contemplating whether to employ a bit more L. Let w denotes the wage rate. Increase in output: dy = (MPL) X dL Increase in revenue: pdy = p X MPL X dL Increase in cost: w X dL Optimum: p X MPL = w WEEK 5-1 ECON2101 S1 2009 Next Profit maximization (contd.) Pages: 339 345, Chapter 19, Varian 5
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