IE4229 Special Topics in Logistics

Transportation Economics

Instructor: Dr. Liu Yang

3. Cost and Supply(2)

1

Review: Production Theory

Production Function

(Q is dependent variable)

Isoquants

(Q is constant)

Marginal Product MRTS=MPL/MPK

Average Product Elasticity of Substitution

(sigma)

Return to Scale

Productivity Substitutability

2

Average Product and Marginal Product

1. When the marginal product of an input is positive,

Q=f(L,K) is increasing in that input.

2. When the average product is decreasing, the

corresponding marginal product must be smaller than

the average product.

Can you convince yourself of the conclusion above?

Hints:

1. Q is increasing in L if MPL>0;

2. dAPL/dL<0 is equivalent to MPL<APL

3

Source:

McCarthy 2001Figure: Productivity Curves

Average product

maximized

Q maximized

where marginal

product is zero.Marginal

product

maximized

4

Average Product and Marginal Product

Q and MP

1. Q is increasing when MPL>0 ;

2. Q is decreasing when MPL<0 ;

3. Q is maximized when MPL=0 ;

MP and AP

1. APL is decreasing (dAPL/dL<0) when MPL<APL;

2. APL is increasing (dAPL/dL>0) when MPL>APL;

3. APL is maximized (dAPL/dL=0) when MPL=APL;

5

Special Production Functions

Linear Production Function (perfect substitutes)

6

Special Production Functions

Fixed Proportions Production Function (perfect

complements)

7

Special Production Functions

Cobb-Douglas Technology Production Function

8

Fixed and Variable Costs

A cost is fixed when it can not be (easily) adjusted to the

size/scale of production.

In contrast, a cost is variable when it depends on Q.

Example.

The purchase of a lot to use to operate a restaurant is a

fixed cost.

The purchase of the different ingredients used to prepare

the meals is variable cost.

9

Short-Run Cost

In short term, some inputs are clearly fixed which result

in fixed cost

E.g., road infrastructure, rail tracks, seaport

Some inputs are variable which result in variable cost

E.g., labor, fuel, maintenance

10

Long-Run Cost

In long run, all the costs are variable

All inputs can vary to get the optimal cost

Including the infrastructure and equipment

Because of time delays and high costs of changing

transportation infrastructure, e.g., seaport infrastructure,

this may be a rather idealized concept in many systems

11

The (Long Run) Cost Minimization Problem

Suppose that a firm’s manager wishes to minimize costs

Let the desired output level be Q0

Production function (Technology): Q=f(L,K)

12

The (Long Run) Cost Minimization Problem

Manager’s decision problem

Min TC(K,L)=rK+wL

s.t. Q0=f(L,K)

Decision variables: optimal L and K

TC: total cost

r: price of K

w: wage of L

In long run, both K and L are variable

13

The (Long Run) Cost Minimization Problem

14

Optimal L* and K*

Where isocost curve

is tangent of isoquant

at tangency point E

Isocost curve rK+wL=C2

The (Long Run) Cost Minimization Problem

15

How to find out the optimal

K* and L* for the production

function=Q1?