1. Cost and Production
Chapters 6 and 7

2. 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

3. 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)

4. 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

5. 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

6. 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*

7. 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

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

9. 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

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

11. 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)

12. 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

13. 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

14. 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)

15. 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