Outline 1. Production Function II 2. Returns to Scale 3. Two-Step Cost
Economics 101A (Lecture 14)
Stefano DellaVigna
March 10, 2015 Outline
1. Production Function II
3. Two-step Cost Minimization 1 Production Function II
Convex production function if convex isoquants •
Mathematically, convex isoquants if 220 •
Solution: • 2 2 00 0 2 00 0 + 00 0 0 = − 2 − 2 ³ ´ 0 ³ ´
Hence, 220 if 0 (inputs are com- • 00 plements in production) 2 Returns to Scale
Nicholson, Ch. 9, pp. 310-313 •
Effect of increase in labor: • 0
Increase of all inputs: (z) with scalar, 1 •
How much does output increase? • — Decreasing returns to scale: for all z and 1
(z) (z) — Constant returns to scale: for all z and 1
(z)= (z)
— Increasing returns to scale: for all z and 1
(z) (z) Example: = ( )= •
Marginal product of labor: = • 0
Decreasing marginal product of labor: = • 00
= •
Convex isoquant? •
Returns to scale: ( )= () () = • + = + ( ) 3 Two-step Cost minimization
Nicholson, Ch. 10, pp. 333-341 •
Objective of firm: Produce output that generates • maximal profit.
Decompose problem in two: • — Given production level choose cost-minimizing combinations of inputs
— Choose optimal level of
First step. Cost-Minimizing choice of inputs • Two-input case: Labor, Capital •
Input prices: • — Wage is price of
— Interest rate is rental price of capital
Expenditure on inputs: + •
Firm objective function: • min + ( ) ≥ Equality in constraint holds if: • 1. 00;
2. strictly increasing in at least or
Counterexample if ass. 1 is not satisfied •
Counterexample if ass. 2 is not satisfied • Compare with expenditure minimization for consumers •
First order conditions: • =0 − 0 and =0 − 0
Rewrite as • ( ) 0 ∗ ∗ = 0 (∗∗)
MRTS (slope of isoquant) equals ratio of input prices • Graphical interpretation • Derived demand for inputs: •
— = ∗ ( )
— = ∗ ( )
Value function at optimum is cost function: • ( )=∗ ( )+∗ ( ) Second step. Given cost function, choose optimal • quantity of as well
Price of output is •
Firm’s objective: • max ( ) −
First order condition: • ( )=0 − 0
Price equals marginal cost — very important! • Second order condition: • ( ) 0 − 00 ∗
For maximum, need increasing marginal cost curve. • 4 Cost Minimization: Example I
Continue example above: = ( )= •
Cost minimization: • min + =
What is the return to scale for this example? •
Increase of all inputs: (z) with scalar, 1 •
How much does input increase? • — Decreasing returns to scale: for all z and 1
(z) (z) — Constant returns to scale: for all z and 1
(z)= (z)
— Increasing returns to scale: for all z and 1
(z) (z)
Returns to scale depend on + 1: ( )= • ≶ () () = + = + ( ) Solutions: • — Optimal amount of labor:
1 + −+ ∗ ( )= µ¶ à ! — Optimal amount of capital:
1 + −+ ∗ ( )= = µ¶ à ! 1 + + = µ¶ à !
Check various comparative statics: •
— ∗ 0 (technological progress and unem- ployment)
— ∗ 0 (more workers needed to produce more output) — ∗ 0∗ 0 (substitute away from more expensive inputs)
Parallel comparative statics for • ∗ Cost function • ( )=∗ ( )+∗ ( )= 1 + + − + = ⎡ ³ ´ ⎤ µ¶ + ⎢ + ⎥ ⎢ ⎥ ⎣ ³ ´ ⎦
Define := −+ + + • ³ ´ ³ ´
Cost-minimizing output choice: • 1 + max − µ¶ 5 Next Lecture
Solve an Example •
Cases in which s.o.c. are not satisfied •
Start Profit Maximization •