transport lecture12

7/27/2019 Transport Lecture12

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Economics 190/290 Lecture 12

Transportation Economics:

Production and Costs I


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Demand and Supply

Chapters from Essays textbook:

Q0

P

P0

Q

Supply (Chapter 3)

Pricing (Chapter 4)

Demand (Chap. 2)


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How to obtain supply?

1)What is the production technology?

2)Therefore, what are production costs?

3)Given demand, what will the firm (orgovernment) choose to supply?

We will focus here on production and

costs.


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Production function:

)x(f)x,...,x(fy n1

Given inputs x=(x1,x2,,xn), write outputy as,

Properties:Increasing:

Quasi-concave:

if and

then

0x/f

)x(f)x(f 10

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

10


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Production Function

Illustrate with iso-quants:

f(x)=y0x0+(1-)x1x0

x1

X2

X1

f(x)=y1>y0


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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


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Input prices

Prices for inputs xi are wi

E.g. labor wage

Capital rental price

Fuel cost of oil;

Total costs are,

n

1iiixwC


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The firms problem:

n

1iiixwmin subject to f(x) > y

A

B

C

x2

x1

Slope=-w1/w2

f(x)=y


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The firms problem (contd):

Point A is the lowest cost method ofproducing y

B and C are more expensive to get y

Write solution as cost function:

with:

- input demands

)w,y(xw)w,y(C

n

1iii

)w,y(xi


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Change in Price:

Suppose price falls from w1 to w1:

f(x)=yB

A

x2

x1

Slope = -w1/w2

Slope = -w1/w2


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Change in Price (contd):

Fall in w1 will increase demand for x1,and reduce demand for x2, moving from

A to B.

- pure substitution effect0

w

x

i

i


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Demand Curve

This gives us downward sloping demand:

D

wi

Xi


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Derivative of costs:

Because

(substitution effects all cancel out)

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

)w,y(xw

xw)w,y(x

w

Ci

n

1j i

jji

i

0w

xw

n

1j i

jj


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Returns to Scale:

If,

then, =1 doubling output will double costs;

>1 Increasing returns to scale.

doubling output will less than double costs;


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Average costs:

Hold the (single) input price w fixed:

Define average costs,

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

,wy

y

wy

y

)w,y(CAC

1/1


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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


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Returns to Scale:

so,

which is a measure of returns to scale!

,wyAC

1

1

wy1

MC

,MCy

CostsTotal

MC

AC


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Constant Returns to Scale:

$

=1

y

AC=MC

Constant returns


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Increasing Returns to Scale:

$

y

AC

MC

Increasing

returns

>1


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Decreasing Returns to Scale:


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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

111


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Cobb-Douglas Cost Function

Use a Lagrangian,

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

yxxsubject toxwxwmin 212211

)xx-(yxwxwL 212211

)x(fxw0x/)x(fwxL 11111

)x(fxw0x/)x(fw

x

L2222

2


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Cobb-Douglas Cost Function (contd)

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


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Cobb-Douglas Cost Function (contd)

Solving for, costs then are,

where,

is a constant.

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

1

221121 xwxw)w,w,y(C

)(A


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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


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Cobb-Douglas Input Demands:

Demands for inputs are obtained bydifferentiating costs:

2

1

1

1

1211 wwAy

wC)w,w,y(x

1

21

1

2212 wwAy

w

C)w,w,y(x


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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


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Elasticity of Substitution

For the Cobb-Douglas function:

So the elasticity of substitution is,

This may not be a good description of actualsubstitution between inputs! So consider..

1)w/wln(

)x/xln(

21

21

1

2

2

1

w

w

x

x


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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

wlnwlnb

2

1ylnylna

2

1

wlnbylnaa)w,y(Cln

)y,,y(y m1

)w,,w(w n1


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Translog Cost Function (contd)

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 substitutionbetween inputs and outputs.

n

1iii

m

1iii0 wlnbylnaa)w,y(Cln


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Translog Cost Shares:

Differentiating the cost function,

so that,

this allows for a wide pattern of substitutionbetween inputs.

j

m

1i ij

n

1j jiji

ii

i

ylngwlnbbC

wx

wln

Cln

0b

wlnwln

Clnij

ji

2


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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

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