# Ols Problem Set Essay

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Economics 522 Econometrics Problem Set 2

Professor Engelhardt

(25 points) An important topic of interest in public economics is tax incidence. That is, who (or what) bears the burden of government taxation. The incidence of a tax on capital can be reflected sometimes in changes in asset prices. Consider a tax on housing, for example. Housing is both a consumption good and an asset. As an asset, it has a price at which it sells in the market. This price is determined as the present discounted value of the flow of rental payments over the life of the structure. In a world with no government taxation, this can be formalized algebraically as PH = ∑ Rt t , t =0 (1 + r)
T

(1)

where P is the price of the house today before any taxes, Rt is the rent in period t and r is the interest rate, also known as the market capitalization rate. [Note: it is implicitly assumed that the interest rate does not change over time in this example.] T is the last period in which the house produces rental income. Now consider the effect of property taxes. Property taxes are levied on the value of property in a locality, and then the revenues are used to fund various local public goods and services such as police and fire protection, sanitation, schools, parks, etc., from which property owners benefit. Therefore, the price of housing is now the present discounted value of the sum of the stream of future rental income and future benefits, less future taxes: PH = ∑ t =0 T

Rt + bt − τ t , (1 + r)t

(2)

where bt is the benefit (in dollar terms) to the owner and  is property taxes (in dollar terms). Equation (2) can be separated algebraically into three components:
PH = ∑
T T Rt bt τt +∑ −∑ t t t . t =0 (1 + r) t= 0 (1 + r) t = 0 (1 + r ) T

(3)

The first term is the pre-tax price of the house. The second term is the present discounted value of the future stream of benefits from the tax. The third term is the present discounted value of the future stream of tax payments. Now consider a decrease in property taxes---to the level of τ * --- that leaves the level of benefits unchanged. The price of housing in this case is P* = ∑ H
T T Rt bt τt* +∑ −∑ t . t t t = 0 (1 + r ) t =0 (1 + r) t= 0 (1 + r) T

(4)

The effect of this decrease in property taxes on house prices can be seen by subtracting equation (4) from equation (3) to yield

* PH − P H = ∑ t =0

T

τ* t
(1 + r) t −∑ t=0 T

τt
(1+ r )t

(5)

or,
∆P = ∑ −∆ τ t , t =0 (1 + r)
T

(6)

* where ∆P = PH − PH and ∆ τ = τ − τ * . If the useful life of the structure, T, is sufficiently long (specifically, infinite), then equation (6) can be written more simply as

∆P =

−∆ τ . r

(7)

Equation (7) illustrates what is known as the tax capitalization effect: changes in taxes are reflected in changes in house prices. That is, the change in price due to the change in tax is the difference in the present discounted value of tax payments in the two tax regimes. An increase in taxes will be reflected in lower prices (because buyers of housing require a lower price because they have to pay higher taxes), and vice versa. The rate at which changes in taxes are translated into changes in prices is 1/r, where r is the market capitalization rate. In a classic study, Kenneth Rosen examined this relationship between changes in house prices and changes in property taxes. [If you would like more background on this topic, you can read Kenneth T. Rosen, "The Impact of Proposition 13 on House Prices in Northern California: A Test of the Interjurisdictional Capitalization Hypothesis," Journal of Political Economy 90 (1982): 191-200]. In particular, he examined the effect of the adoption of Proposition 13 in California on house prices. Proposition 13, enacted in June, 1978, required communities to reduce their property taxes, but guaranteed that the level of local public goods and services provided would be held constant (i.e., left the level of benefits