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# Production and cost

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### Transcript

• 1. Overview
• I. Production Analysis
• Total Product, Marginal Product, Average Product
• Isoquants
• Isocosts
• Cost Minimization
• II. Cost Analysis
• Total Cost, Variable Cost, Fixed Costs
• Cubic Cost Function
• Cost Relations
• III. Multi-Product Cost Functions
• 2. Production Analysis
• Production Function
• Q = F(K,L)
• The maximum amount of output that can be produced with K units of capital and L units of labor.
• Short-Run vs. Long-Run Decisions
• Fixed vs. Variable Inputs
• 3. Total Product
• Cobb-Douglas Production Function
• Example: Q = F(K,L) = K .5 L .5
• K is fixed at 16 units.
• Short run production function:
• Q = (16) .5 L .5 = 4 L .5
• Production when 100 units of labor are used?
• Q = 4 (100) .5 = 4(10) = 40 units
• 4. Marginal Product of Labor
• MP L =  Q/  L
• Measures the output produced by the last worker.
• Slope of the production function
• 5. Average Product of Labor
• AP L = Q/L
• This is the accounting measure of productivity.
• 6. Q=F(K,L) Increasing Marginal Returns Diminishing Marginal Returns Negative Marginal Returns Stages of Production Q L MP AP
• 7. Isoquant
• The combinations of inputs (K, L) that yield the producer the same level of output.
• The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output.
L
• 8. Linear Isoquants
• Capital and labor are perfect substitutes
Q 3 Q 2 Q 1 Increasing Output L K
• 9. Leontief Isoquants
• Capital and labor are perfect complements
• Capital and labor are used in fixed-proportions
Q 3 Q 2 Q 1 K Increasing Output
• 10. Cobb-Douglas Isoquants
• Inputs are not perfectly substitutable
• Diminishing marginal rate of technical substitution
• Most production processes have isoquants of this shape
Q 1 Q 2 Q 3 K L Increasing Output
• 11. Isocost
• The combinations of inputs that cost the producer the same amount of money
• For given input prices, isocosts farther from the origin are associated with higher costs.
• Changes in input prices change the slope of the isocost line
K L C 1 C 0 L K New Isocost Line for a decrease in the wage (price of labor).
• 12. Cost Minimization
• Marginal product per dollar spent should be equal for all inputs:
• Expressed differently
• 13. Cost Minimization Q L K Point of Cost Minimization Slope of Isocost = Slope of Isoquant
• 14. Cost Analysis
• Types of Costs
• Fixed costs (FC)
• Variable costs (VC)
• Total costs (TC)
• Sunk costs
• 15. Total and Variable Costs
• C(Q): Minimum total cost of producing alternative levels of output:
• C(Q) = VC + FC
• VC(Q): Costs that vary with output
• FC: Costs that do not vary with output
\$ Q C(Q) = VC + FC VC(Q) FC
• 16. Fixed and Sunk Costs FC: Costs that do not change as output changes Sunk Cost: A cost that is forever lost after it has been paid \$ Q FC C(Q) = VC + FC VC(Q)
• 17. Some Definitions
• Average Total Cost
• ATC = AVC + AFC
• ATC = C(Q)/Q
• Average Variable Cost
• AVC = VC(Q)/Q
• Average Fixed Cost
• AFC = FC/Q
• Marginal Cost
• MC =  C/  Q
ATC AVC AFC \$ Q MC
• 18. Fixed Cost ATC AVC Q 0 AFC Fixed Cost Q 0  (ATC-AVC) = Q 0  AFC = Q 0  (FC/ Q 0 ) = FC \$ Q ATC AVC MC
• 19. Variable Cost AVC Variable Cost Q 0 Q 0  AVC = Q 0  [VC(Q 0 )/ Q 0 ] = VC(Q 0 ) \$ Q ATC AVC MC
• 20. ATC Total Cost Q 0 Q 0  ATC = Q 0  [C(Q 0 )/ Q 0 ] = C(Q 0 ) Total Cost \$ Q ATC AVC MC
• 21. Economies of Scale LRAC \$ Output Economies of Scale Diseconomies of Scale
• 22. An Example
• Total Cost: C(Q) = 10 + Q + Q 2
• Variable cost function:
• VC(Q) = Q + Q 2
• Variable cost of producing 2 units:
• VC(2) = 2 + (2) 2 = 6
• Fixed costs:
• FC = 10
• Marginal cost function:
• MC(Q) = 1 + 2Q
• Marginal cost of producing 2 units:
• MC(2) = 1 + 2(2) = 5
• 23. Multi-Product Cost Function
• C(Q 1 , Q 2 ): Cost of producing two outputs jointly
• 24. Economies of Scope
• C(Q 1 , Q 2 ) < C(Q 1 , 0) + C(0, Q 2 )
• It is cheaper to produce the two outputs jointly instead of separately.
• 25. Cost Complementarity
• The marginal cost of producing good 1 declines as more of good two is produced:
•  MC 1 /  Q 2 < 0.
• 26. Quadratic Multi-Product Cost Function
• C(Q 1 , Q 2 ) = f + aQ 1 Q 2 + (Q 1 ) 2 + (Q 2 ) 2
• MC 1 (Q 1 , Q 2 ) = aQ 2 + 2Q 1
• MC 2 (Q 1 , Q 2 ) = aQ 1 + 2Q 2
• Cost complementarity: a < 0
• Economies of scope: f > aQ 1 Q 2
• C(Q 1 ,0) + C(0, Q 2 ) = f + (Q 1 ) 2 + f + (Q 2 ) 2
• C(Q 1 , Q 2 ) = f + aQ 1 Q 2 + (Q 1 ) 2 + (Q 2 ) 2
• f > aQ 1 Q 2 : Joint production is cheaper
• 27. A Numerical Example:
• C(Q 1 , Q 2 ) = 90 - 2Q 1 Q 2 + (Q 1 ) 2 + (Q 2 ) 2
• Cost Complementarity?
• Yes, since a = -2 < 0
• MC 1 (Q 1 , Q 2 ) = -2Q 2 + 2Q 1
• Economies of Scope?
• Yes, since 90 > -2Q 1 Q 2
• Implications for Merger?