# Cost and Production Chapters 6 and 7.

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Cost and Production Chapters 6 and 7

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

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)

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

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

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*

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

Cost Exercise Q = 10L0.2K0.8 w=\$10, r=\$20 Suppose Q = 174
Find the cost of producing 174 units Cost = wL + rK

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

Long Run Cost Exercise Q = 10L0.2K0.8 w=\$5 r=\$20 Q = 200
Find Long Run Total Cost Find Long Run Average Cost

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)

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

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

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)

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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