# Profit and expenditure functions in public finance: a correction.

PROFIT AND EXPENDITURE FUNCTIONS IN PUBLIC FINANCE: A CORRECTION

Greenberg and Denzau's 1988 paper in this journal uses duality to demonstrate lucidly many important theorems about deadweight losses from taxes and subsidies. However, an error in their Figure 5, "Market and Welfare Effects of a Tax on Output," may be worth correcting.

Following Kay [1980!, they base their measure of deadweight loss on the equivalent variation. As they state, one method of graphing this deadweight loss uses the Hicksian demand curve that holds utility constant at the level obtained when the unit tax is paid. However, their Figure 5 shows this curve intersecting the Marshallian demand curve at the original price, P.sub.O, which is not correct. The intersection should be at the gross price, P.sub.G. (1) In addition, they should state that they are considering only a partial equilibrium analysis of one market. I clarify these points below using a single-consumer general equilibrium model.

Following Greenberg and Denzau, suppose a unit tax is levied on the final output of an industry, and the seller is legally liable for the tax. To keep the analysis focused, assume there is only one other good, the numeraire, through which all compensation is paid, and all factors are inelastically supplied. There is now a consumer price of P.sub.G and a producer price of [P.sub.n!, where [P.sub.n! = [P.sub.g! - t. The pre-tax supply functions is S and the post-tax supply function is S'. Then the single-consumer general equilibrium measure of deadweight loss based on equivalent variation is (2)

(1) * = [pi!([P.sub.o! - e([P.sub.o!,*) - tax.

The first term in the equation is initial aggregate profits in the economy. In other words, it represents the single consumer's initial income. When the unit tax is paid the individual faces price P.sub.G and earns income based on P.sub.N, giving utility level *:

(2) * = U[[P.sub.g!,[pi!([P.sub.n!)!.

The second term in equation (1) is the least amount of money the individual would need to attain * if prices were unchanged.

The difference between terms one and two is the amount of money one must take from the individual to make his utility at the original price equivalent to his utility at the after-tax price. To see that this difference is positive, recall the identity (3)

(3) [pi!([P.sub.N!)*e{y,U[y,[pi!([P.sub.N!)[}.

Set Y=P.sub.O, use equation (2), and use the fact that the profit and expenditure functions are increasing in prices and the indirect utility function is decreasing in prices, to obtain.

[pi!([P.sub.o!) [is greater =! [pi!([P.sub.n!) =

e{[P.sub.O!,U[[P.sub.O!,[pi!([P.sub.N!)!} [is greater than or equal to! e([P.sub.o!*).

In the last term of equation (1) we deduct the tax revenue actually raised. In partial equilibrium this equals the tax rate multiplied by the Marshallian demand. However, Marshallian demand is misleading in general equilibrium, since prices and endowments are not fixed but are given by the production possibilities frontier. (4) We do not need an explicit expression for the tax revenue to complete the arguments.

To transform (1) into an area between supply and Hicksian demand curves, set Y = [P.sub.g! in (3) to obtain [pi!([P.sub.n') = e([P.sub.g!,*). Therefore

* = [[pi!([P.sub.o!) - [pi!([P.sub.n!)!

+ [e([P.sub.g!,*) - e([P.sub.o!,*)! - tax.

By duality, the term in the first bracket is the area left of the supply curve between [P.sub.N! and [P.sub.o!. The term is the second bracket is the area left of the Hicksian demand curve (that holds utility at *) between [P.sub.g' and [P.sub.o!. The key point is that this Hicksian demand at P.sub.O must be less than the original market demand at P.sub.O. (5) The figure is below.

In terpreting this measure is straightforward in the single-consumer case in both partial and general equilibrium, provided in the latter we use the correct demand curve. In the many-consumer, partial equilibrium case, the area represents aggregate consumer welfare loss provided that each individual's welfare loss is assigned equal weight. (6)

(1*) Department of Economics, Washington University in St. Louis, 63130. I thank Art Denzau and Ed Greenberg for helpful conversations.

(1.) The correct picture may be found in Stiglitz [1988, 452!.

(2.) General equilibrium compensation measures are developed in detail in Tresch [1981, 75!. I use the negative of his equivalent variation measure to obtain positive values for a unit tax.

(3.) See, for example, Varian [1984!, whose uses y for [pi!(P.sub.N).

(4.) Broadway and Bruce [1984, 238!. However, their approach is somewhat different.

(5.) General equilibrium demand for the untaxed good must exceed the pre-tax demand. Therefore, if Hicksian demand at P.sub.O for the taxed good, h*(P.sub.O,), equals for exceeds the pre-tax equilibrium market demand, then equals or exceeds before-tax utility, a contradiction.

(6.) Broadway and Bruce [1984, 280!. Auerbach [1985, 75-83! discusses conditions for this interpretation to hold with many consumers.

REFERENCES

Auerbach, A. J. "The Theory of Excess Burden and Optimal Taxation," In Handbook of Public Economics, vol. 1, edited by A. J. Auerbach and M. Feldstein. New York: Elsevier, 1985, 61-127.

Broadway, R. W. and N. Bruce. Welfare Economics. London: Basil Blackwell, 1984.

Greenberg, E. and A. T. Denzau. "Profit and Expenditure Functions in Basic Public Finance: An Expository Note." Economic Inquiry, January 1988, 145-58.

Kay, J. A. "The Deadweight Loss from a Tax System." Journal of Public Economics, February 1980, 111-19.

Stiglitz, J. E. Economics of the Public Sector, 2nd ed. New York: W. W. Norton and Co., 1988.

Tresch, Richard W. Public Finance: A Normative Theory. Texas: Business Publications Inc., 1981.

Varian Hal R. Microeconomic Analysis, 2nd ed. New York: W. W. Norton & Company, 1984.

Greenberg and Denzau's 1988 paper in this journal uses duality to demonstrate lucidly many important theorems about deadweight losses from taxes and subsidies. However, an error in their Figure 5, "Market and Welfare Effects of a Tax on Output," may be worth correcting.

Following Kay [1980!, they base their measure of deadweight loss on the equivalent variation. As they state, one method of graphing this deadweight loss uses the Hicksian demand curve that holds utility constant at the level obtained when the unit tax is paid. However, their Figure 5 shows this curve intersecting the Marshallian demand curve at the original price, P.sub.O, which is not correct. The intersection should be at the gross price, P.sub.G. (1) In addition, they should state that they are considering only a partial equilibrium analysis of one market. I clarify these points below using a single-consumer general equilibrium model.

Following Greenberg and Denzau, suppose a unit tax is levied on the final output of an industry, and the seller is legally liable for the tax. To keep the analysis focused, assume there is only one other good, the numeraire, through which all compensation is paid, and all factors are inelastically supplied. There is now a consumer price of P.sub.G and a producer price of [P.sub.n!, where [P.sub.n! = [P.sub.g! - t. The pre-tax supply functions is S and the post-tax supply function is S'. Then the single-consumer general equilibrium measure of deadweight loss based on equivalent variation is (2)

(1) * = [pi!([P.sub.o! - e([P.sub.o!,*) - tax.

The first term in the equation is initial aggregate profits in the economy. In other words, it represents the single consumer's initial income. When the unit tax is paid the individual faces price P.sub.G and earns income based on P.sub.N, giving utility level *:

(2) * = U[[P.sub.g!,[pi!([P.sub.n!)!.

The second term in equation (1) is the least amount of money the individual would need to attain * if prices were unchanged.

The difference between terms one and two is the amount of money one must take from the individual to make his utility at the original price equivalent to his utility at the after-tax price. To see that this difference is positive, recall the identity (3)

(3) [pi!([P.sub.N!)*e{y,U[y,[pi!([P.sub.N!)[}.

Set Y=P.sub.O, use equation (2), and use the fact that the profit and expenditure functions are increasing in prices and the indirect utility function is decreasing in prices, to obtain.

[pi!([P.sub.o!) [is greater =! [pi!([P.sub.n!) =

e{[P.sub.O!,U[[P.sub.O!,[pi!([P.sub.N!)!} [is greater than or equal to! e([P.sub.o!*).

In the last term of equation (1) we deduct the tax revenue actually raised. In partial equilibrium this equals the tax rate multiplied by the Marshallian demand. However, Marshallian demand is misleading in general equilibrium, since prices and endowments are not fixed but are given by the production possibilities frontier. (4) We do not need an explicit expression for the tax revenue to complete the arguments.

To transform (1) into an area between supply and Hicksian demand curves, set Y = [P.sub.g! in (3) to obtain [pi!([P.sub.n') = e([P.sub.g!,*). Therefore

* = [[pi!([P.sub.o!) - [pi!([P.sub.n!)!

+ [e([P.sub.g!,*) - e([P.sub.o!,*)! - tax.

By duality, the term in the first bracket is the area left of the supply curve between [P.sub.N! and [P.sub.o!. The term is the second bracket is the area left of the Hicksian demand curve (that holds utility at *) between [P.sub.g' and [P.sub.o!. The key point is that this Hicksian demand at P.sub.O must be less than the original market demand at P.sub.O. (5) The figure is below.

In terpreting this measure is straightforward in the single-consumer case in both partial and general equilibrium, provided in the latter we use the correct demand curve. In the many-consumer, partial equilibrium case, the area represents aggregate consumer welfare loss provided that each individual's welfare loss is assigned equal weight. (6)

(1*) Department of Economics, Washington University in St. Louis, 63130. I thank Art Denzau and Ed Greenberg for helpful conversations.

(1.) The correct picture may be found in Stiglitz [1988, 452!.

(2.) General equilibrium compensation measures are developed in detail in Tresch [1981, 75!. I use the negative of his equivalent variation measure to obtain positive values for a unit tax.

(3.) See, for example, Varian [1984!, whose uses y for [pi!(P.sub.N).

(4.) Broadway and Bruce [1984, 238!. However, their approach is somewhat different.

(5.) General equilibrium demand for the untaxed good must exceed the pre-tax demand. Therefore, if Hicksian demand at P.sub.O for the taxed good, h*(P.sub.O,), equals for exceeds the pre-tax equilibrium market demand, then equals or exceeds before-tax utility, a contradiction.

(6.) Broadway and Bruce [1984, 280!. Auerbach [1985, 75-83! discusses conditions for this interpretation to hold with many consumers.

REFERENCES

Auerbach, A. J. "The Theory of Excess Burden and Optimal Taxation," In Handbook of Public Economics, vol. 1, edited by A. J. Auerbach and M. Feldstein. New York: Elsevier, 1985, 61-127.

Broadway, R. W. and N. Bruce. Welfare Economics. London: Basil Blackwell, 1984.

Greenberg, E. and A. T. Denzau. "Profit and Expenditure Functions in Basic Public Finance: An Expository Note." Economic Inquiry, January 1988, 145-58.

Kay, J. A. "The Deadweight Loss from a Tax System." Journal of Public Economics, February 1980, 111-19.

Stiglitz, J. E. Economics of the Public Sector, 2nd ed. New York: W. W. Norton and Co., 1988.

Tresch, Richard W. Public Finance: A Normative Theory. Texas: Business Publications Inc., 1981.

Varian Hal R. Microeconomic Analysis, 2nd ed. New York: W. W. Norton & Company, 1984.

Printer friendly Cite/link Email Feedback | |

Author: | Rothstein, Paul |
---|---|

Publication: | Economic Inquiry |

Date: | Jan 1, 1991 |

Words: | 997 |

Previous Article: | Premium bundling. |

Next Article: | In search of Giffen behavior. |

Topics: |