1. The Theory of Production
The production process
One variable input
Two or more variable inputs
Optimal combination of inputs
Economic region of production

2. Production Functions Transformation of inputs to output
Basis for all cost analysis
Short-run (at least one fixed input) vs. long-run (all variable inputs)
Q = Q(X1, X2, …, Xn)

3. Production with One Variable Input
Refer to production Table 8.1 on p. 267
Total, average, and marginal product
Law of diminishing returns
Stages of production

4. Production with Two Variable Inputs
Production isoquants
Economic region of production
MRTSYX = MPX/MPY = -dY/dX

5. Production elasticities
For any input: EX = %Q/%X
MPL > APL, EL > 1

6. Three Stages of Production
Stage 1: AP rising
Stage 2: AP falling, but MP positive
Stage 3: MP negative and TP falling

7. Optimal Input Use in Perfect Competition
Hire workers, if:
addition to revenue > addition to cost
TR/ L > TC/ L
MRPL > MRCL
At optimum, MRPL = wage

8. Returns to Scale Q = Q(K, L) zQ = Q(hK, hL) CRS when z = h
IRS when z > h
DRS when z < h

9. Economic Region of Production
Slope of isoquant = MPL/MPK
Define region by ridge lines
MPL = 0; slope of isoquant = 0
MPK = 0; slope of isoquant = 

10. Optimal Combination of Inputs
Objective: minimize cost for a given Q
Isocost: combination of inputs for a given cost
Equimarginal principle

11. Cobb-Douglas Production Function
Q = AKaLb
a + b = 1, then CRS
a + b > 1, then IRS
a + b < 1, then DRS
MPs depend on both inputs
Exponents represent output elasticities
Estimated by using log transformation
log Q = logA + a logK + b logL

12. Interpreting Cobb-Douglas
Using Q = 100L0.5 K0.5
What is degree of homogeneity?
What about returns to scale?
What are labor and capital elasticities?
What happens to Q, if L increases by 4% and K increases by 2%?

13. Long-run production with Cobb-Douglas
Impact on Q (= h) of proportionate increase in inputs (z)
hQ = A(zK)a (zL)b
= zazb(AKa Lb)
= za+b(AKa Lb)
since Q = (AKa Lb), then h = za+b
when a+b = 1, h = z => double inputs, double output

14. Appendix 8A: Lagrangians
Maximize Q s.t. cost constraint
r is the cost of capital; w is the wage rate

15. Appendix 8B: Linear programming
Manufacturers have alternative production processes, some involving mostly labor, others using machinery more intensively.
The objective is to maximize output from these production processes, given constraints on input availability, such as plant capacity or labor constraints.
We will discuss linear programming techniques more extensively in Chapter 11.