Public Finance, 2nd Edition

the subtleties associated with the hard case decreasing cost services. Their

belief in proWtability as the proper guide for the use of scarce resources is

deeply held, and understandably so. They see the proWtability test applied

every day in the marketplace.

REFERENCES

Bator, F. M., The Question of Government Spending, Harper Brothers, New York, 1960.

Diamond, P. and McFadden, D., ``Some Uses of the Expenditure Function in Public Finance,''

Journal of Public Economics, February 1974.

Hammond, P., ``Theoretical Progress in Public Economics: A Provocative Assessment,'' Oxford

Economic Papers, January 1990.

Hausman, J., ``The EVect of Taxes on Labor Supply,'' published as a chapter entitled ``Labor

Supply'' in H. Aaron and J. Pechman (Eds.), How Taxes AVect Economic Behavior,

Brookings, Washington, D.C., 1981.

Peck, M. and Scherer, F., The Weapons Acquisition Process: An Economic Analysis, (Division of

Research, Graduate School of Business Administration, Harvard University, Boston, MA,

1962).

Scherer, F., The Weapons Acquisition Process: Economic Incentives, (Division of Research,

Graduate School of Business Administration, Harvard University, Boston, MA, 1964).

Willig, R., ``Consumer's Surplus Without Apology,'' American Economic Review, September

1976.

APPENDIX: RETURNS TO SCALE, HOMOGENEITY, AND

DECREASING COST

Since increasing returns to scale imply decreasing average cost, the two terms

are used interchangeably in the chapter. To see that the former implies the

latter, consider the homogeneous production function:

Y  fX

1

, ...,X

N

fX

i

 (9A:1)

where X

i

 input i, i  1, ..., N, and Y  output. By the deWnition of

homogeneous functions,

l

b

Y  f l  X

i

 (9A:2)

Increasing returns to scale implies that b > 1, or a scalar increase (decrease)

in each of the factors generates a more-than-proportionate increase (de-

crease) in output. Furthermore, if the production function is homogeneous

of degree b, then the marginal product functions, qY/qX

k

 f

K

X

i

 are

homogeneous of degree b  1. This follows immediately by diVerentiating

l

b

Y  l

b

fX

i

flX

i

 with respect to X

k

:

l

b

f

k

X

i



q f lX

i



q X

K

 lf

K

lX

i

 (9A:3)

304 APPENDIX: RETURNS TO SCALE, HOMOGENEITY, AND DECREASING COST

Hence:

l

b1

f

K

 f

K

lX

i

 k  1, ..., N (9A:4)

To minimize production costs for any given output, the Wrm solves the

following problem:

min

X

i



P

P

i

X

i

s:t: Y  fX

i



The Wrst-order conditions imply:

P

i

P

1



f

i

X

i



f

1

X

i



i  2, ..., N (9A:5)

The ratio of factor prices equals the marginal rate of technical substitution of

the factors in production. Suppose the Wrm increases (decreases) its use of all

factors X

i

by the scalar l. Since f

i

lX

i

l

b1

f

i

X

i

, this scalar increase

(decrease) continues to satisfy the Wrst-order conditions:

f

i

lX

i



f

1

lX

i





l

b1

f

i

X

i



l

b1

f

1

X

i





f

i

X

i



f

1

X

i





P

i

P

1

(9A:6)

Hence, if a vector of inputs

~

X

i

*

minimizes cost, so too will any vector l

~

X

i

*

.

But, if all inputs are increased by the scalar l, total costs increase by l and

output increases by a factor l

b

. Thus, the total cost function must be of the

form:

TC  kY

1/b

(9A:7)

since k l

b

Y

1/b

 l  k  Y

1/b

 l  TC:

Finally,

AC  TC/Y  k  Y

1/b1

 k  Y

1b/b

(9A:8)

DiVerentiating,

q AC

q Y



1  b

b



k  Y

1b/b1

< 0, for b > 1 (9A:9)

Hence, average cost declines continuously as output increases with increasing

returns to scale.

9. THE THEORY OF DECREASING COST PRODUCTION 305

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