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Cost and Production Chapters 6 and 7

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**Production Functions Describe the technology available to the firm**

Q = f(K, L) where K and L are input quantities K is capital, L is labor Represented graphically as Isoquants Combinations of K and L that produce the same quantity of output Convex Downward sloping: K and L are substitutes Diminishing marginal product Marginal Rate of Technical Substitution MRTS = -∆K/∆L = negative of slope of isoquant

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**Costs TC = rK + wL Represented graphically by a cost line**

Where r is “price” of capital And w is the wage, or price of labor Prices should reflect opportunity costs Represented graphically by a cost line Combinations of K and L that cost the same K = (TC/r) – (w/r)L A straight line connects the end points K = (TC/r) and L = (TC/w)

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Producer’s Problem Choose input quantities to minimize the cost of producing a given quantity of output Min TC = rK + wL s.t. Q* = f(K, L) Same solution as max Q, s.t. TC (∆Q/∆L)/(∆Q/∆K) = MPL/MPK = w/r Ratio of marginal products equals price ratio Cost line is tangent to an isoquant

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**Solution Characteristics**

MPL/MPK = w/r = MRTS = -∆K/∆L Slope of cost line = slope of isoquant -MPL/MPK = -w/r MPL/w = MPK/r Marginal product per dollar equal across inputs w/MPL = r/MPK Cost of additional output equal across inputs

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**Long Run vs. Short Run Long Run: All Inputs Variable**

Expansion path implies long run cost function Combinations of K & L that minimize cost as output increases All costs are variable Returns to Scale: relative proportional changes in inputs and output Short Run: at least one input fixed S-R Expansion path implies short run cost f’n Combinations of L and K = K* as output increases Fixed and variable costs Marginal and Average Product of labor MPL = ∆Q/∆L, K = K*; APL = Q/L, K=K*

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**Finding Costs from Production Functions**

Suppose Q = 10L0.2K0.8, w=$10, r=$20 What is the cost of producing 100 units? Use MPL/MPK = w/r MPL/MPK = [(0.2)/(0.8)](K/L) = ¼ (K/L) w/r = $10/$20 = ½ ¼ (K/L) = ½ or K = 2L Substitute into Production Function 100 = 10(L0.2)(2L)0.8 = 10L(2)0.8 = 10L(1.74) L = 100/17.4 = 5.75, K = 2L = 11.5 Cost = wL + rK = $10(5.75) + $20(11.5) = $287.40

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**Cost Exercise Q = 10L0.2K0.8 w=$10, r=$20 Suppose Q = 174**

Find the cost of producing 174 units Cost = wL + rK

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**Expansion Path We now have two points on the expansion path for**

Q = 10L0.2K0.8 w = $10, r = $20 Q1 = 100, Q2 = 174 Calculate Average Costs for each Q What is the implied relationship between cost and quantity produced? Returns to Scale

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**Long Run Cost Exercise Q = 10L0.2K0.8 w=$5 r=$20 Q = 200**

Find Long Run Total Cost Find Long Run Average Cost

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**Short Run Cost Suppose K is fixed: K* = 20 Q = 10L0.2K0.8 w=$5 r=$20**

Find Short Run Total Cost for Q = 200 What are fixed costs? Variable costs? Calculate Average Total Cost (ATC) Average Variable Cost (AVC) Average Fixed Cost (AFC)

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**Short Run Cost Exercise**

Q = 10L0.2K0.8 , w=$5 r=$20, K* = 20 Find Short Run Total Cost for Q = 500 What are fixed cost and variable cost? ATC, AVC, AFC? Marginal Product of Labor (MPL) Average Product of Labor (APL) Marginal Cost, MC = w/ MPL Draw graphs to illustrate

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**Short Run Production and Cost**

General Relationships Short Run Production Average and Marginal Products of Labor Short Run Costs ATC, AFC, AVC SRMC Short Run Production and Cost APL = MPL AVC = MC Short Run Cost and Long Run Cost

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**Simple Profit Maximization**

π = PQ – C, for a price-taker At a maximum ∆π/∆Q = 0 = P - MC MC = ∆C/∆Q P = MC In the short run MC = w/MPL So P = MC implies P = w/MPL OR w = P(MPL) π = PQ – C = Q(P – ATC)

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**Example: Profit Max Q = 10L0.2K0.8, w= $5, r = $20, K* = 20**

Price = $10.00 P = MC = w/MPL MPL = ∆Q/∆L = 10(0.2)L(0.2-1)K0.8 MPL = 10(0.2)L(-0.8)(20)0.8 = 2(10.986)L-0.8 MPL = 22L-0.8 $10.00 = $5.00/(22L-0.8) 2 = 1/(22L-0.8) or 44 = L0.8 L = (44)1.25 = 113 Q = 10(113)0.2(20)0.8 = 283 π = PQ – C = 10(283)-5(113)-20(20)=$1865

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