The Theory of Production
• The production process
– One variable input
– Two or more variable inputs
• Optimal combination of inputs
• Economic region of production
Production Functions
• Transformation of inputs to output
• Basis for all cost analysis
• Short-run (at least one fixed input) vs. longrun (all variable inputs)
• Q = Q(X1, X2, …, Xn)
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
Production with Two Variable
Inputs
• Production isoquants
• Economic region of production
• MRTSYX = MPX/MPY = -dY/dX
Production elasticities
• For any input: EX = %Q/%X
• MPL > APL, EL > 1
Three Stages of Production
• Stage 1: AP rising
• Stage 2: AP falling, but MP positive
• Stage 3: MP negative and TP falling
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
Returns to Scale
• Q = Q(K, L)
• zQ = Q(hK, hL)
– CRS when z = h
– IRS when z > h
– DRS when z < h
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 = 
Optimal Combination of Inputs
• Objective: minimize cost for a given Q
• Isocost: combination of inputs for a given
cost
• Equimarginal principle
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
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%?
Long-run production with CobbDouglas
• 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
Appendix 8A: Lagrangians
• Maximize Q s.t. cost constraint
• r is the cost of capital; w is the wage rate
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.