ECN 145 Lecture 12

Transportation Economics:

Production and Costs I

Demand and Supply

• Chapters from Essays textbook:

Q0

P

P0

Q

Supply (Chapter 3)

Pricing (Chapter 4)

Demand (Chap. 2)

How to obtain supply?

• 1)What is the production technology?

• 2)Therefore, what are production costs?

• 3)Given demand, what will the firm (or government) choose to supply?

• We will focus here on production and costs.

Production function:

)x(f)x,...,x(fy n1 • Given inputs x=(x1,x2,…,xn), write output

y as,

• Properties: – Increasing:

– Quasi-concave:

– if and

– then

0x/f

)x(f)x(f 10

)x(f)x)1(x(f 010 10

Production Function

• Illustrate with “iso-quants”:

f(x)=y0x0+(1-)x1

x0

x1

X2

X1

f(x)=y1>y0

Returns to Scale

• Suppose that there is one input x, and,

=1 constant returns to scale• doubling all inputs will just double output >1 Increasing returns to scale• doubling all inputs more than doubles output <1 decreasing returns to scale.• doubling all inputs will less than double output

x)x(fy

Input prices

• Prices for inputs xi are wi

• E.g. labor – wage

• Capital – rental price

• Fuel – cost of oil;

• Total costs are,

n

1iiixwC

The firm’s problem:

n

1iiixwmin subject to f(x) > y

A

B

C

x2

x1

Slope=-w1/w2

f(x)=y

The firm’s problem (cont’d):

• Point A is the lowest cost method of producing y – B and C are more expensive to get y

• Write solution as cost function:

• with:

• - input demands

)w,y(xw)w,y(Cn

1iii

)w,y(xi

Change in Price:

• Suppose price falls from w1 to w1’:

f(x)=yB

A

x2

x1

Slope = -w1/w2

Slope = -w1/w2

Change in Price (cont’d):

• Fall in w1 will increase demand for x1, and reduce demand for x2, moving from A to B.

• - pure “substitution” effect0w

x

i

i

Demand Curve

• This gives us downward sloping demand:

D

wi

Xi

Derivative of costs:

• Because

– (substitution effects all cancel out)

• Thus, the derivative of costs w.r.t. factor prices equals factor demands

)w,y(xw

xw)w,y(x

w

Ci

n

1j i

jji

i

0w

xw

n

1j i

jj

Returns to Scale:

• If,

• then,

=1 doubling output will double costs; >1 Increasing returns to scale.• doubling output will less than double costs; <1 Decreasing returns to scale.• doubling output will more than double costs.

wy)w,y(C /1

x)x(fy

Average costs:

• Hold the (single) input price w fixed:

• Define average costs,

• E.g. Total costs =$100, y=5, so AC=$20

,wyy

wy

y

)w,y(CAC

1/1

Marginal costs

• Hold the (single) input price w fixed:

• Define marginal costs,

• E.g. Total costs=$100 when y=5, $115 when y=6• So marginal costs are $15.

1

yw

y

)w,y(CMC

Returns to Scale:

• so,

• which is a measure of returns to scale!

,wyAC1

1

wy1

MC

,MCy

CostsTotal

MC

AC

Constant Returns to Scale:

$

=1

y

AC=MC

Constant returns

Increasing Returns to Scale:

$

y

AC

MC

Increasingreturns

>1

Decreasing Returns to Scale:

<1(Say, =1/2)

$MC

AC

Decreasingreturns

y

Eg: Cobb-Douglas Production Function

• Notice that doubling both x1 and x2 :

• So,• constant returns to scale• increasing returns to scale• decreasing returns to scale

0 , ,xx)x(fy 21

)x(f2)x2()x2()x2(f 21

1 1 1

Cobb-Douglas Cost Function

• Use a Lagrangian,

• Find first-order condition w.r.t x1, x2:

yxx subject to xwxwmin 212211

)xx -(yxwxwL 212211

)x(fxw0xxwx

L112

111

1

)x(fxw0xxwx

L22

1212

2

Cobb-Douglas Cost Function (cont’d)

• From these two conditions, we solve for,

• We can solve for from the FOC,

)x(f)(xwxw 2211

,)x(fxw 11 )x(fxw 22

)x(f)()x(f)ww( 21

Cobb-Douglas Cost Function (cont’d)

• Solving for , costs then are,

• where,

• is a constant.

)ww(Ay)()x(f)( 21

1

221121 xwxw)w,w,y(C

)(A

Returns to Scale:

• First write the log of costs as:

• Differentiating this w.r.t. y, we see that,

• measures the returns to scale!

21 wlnwlnyln1

BCln

)(yln

Cln

)y/C(y

Costs

MC

AC1

Cobb-Douglas Input Demands:

• Demands for inputs are obtained by differentiating costs:

2

1

1

1

1211 wwAy

w

C)w,w,y(x

1

21

1

2212 wwAy

w

C)w,w,y(x

Costs Shares:

• Comparing x1 and x2 with total costs:

• we see that,

• which are constant!

C

xw 11

C

xw 22

)ww(Ay)(C 21

1

Elasticity of Substitution

• For the Cobb-Douglas function:

• So the elasticity of substitution is,

• This may not be a good description of actual substitution between inputs! So consider…..

1)w/wln(

)x/xln(

21

21

1

2

2

1

w

w

x

x

Translog Cost Function

• Many outputs (joint prod.)

• And inputs

m

1i

n

1jjiij

m

1i

n

1jjiij

m

1i

n

1jjiij

n

1iii

m

1iii0

wlnylng

wlnwlnb2

1ylnylna

2

1

wlnbylnaa)w,y(Cln

)y,,y(y m1 )w,,w(w n1

Translog Cost Function (cont’d)

• Note that the first line is just Cobb-Douglas, • (in logs, with multiple inputs and outputs):

• The “extra” terms on the second and third lines allow for more general substitution between inputs and outputs.

n

1iii

m

1iii0 wlnbylnaa)w,y(Cln

Translog Cost Shares:

• Differentiating the cost function,

• so that,

• this allows for a wide pattern of substitution between inputs.

j

m

1iij

n

1jjiji

ii

iylng wlnbb

C

wx

wln

Cln

0bwlnwln

Clnij

ji

2

Returns to Scale:

• so if aij=0, then,

• is a measure of returns to scale!

1

iiyln/Cln

)y/C(y

Costs

MC

AC

1

iia

MC

AC