Bidding and Sorting: The Theory of Local Public Finance ECN 741, Urban Economics Professor John Yinger, The Maxwell School, Syracuse University, 2019 The Theory of Local Public Finance Class Outline The U.S. Federal System The Consensus Model of Local Public Finance Deriving a Bid Function Residential Is

Sorting the U.S. Federal System Efficient? Introduction The Theory of Local Public Finance Class Outline The U.S. Federal System The Consensus Model of Local Public Finance Deriving a Bid Function Residential

Is Sorting the U.S. Federal System Efficient? Introduction The Theory of Local Public Finance The U.S. Federal System Constitutions and Politics Broad outlines defined by constitutions Details determined by politics Units Defined by U.S. Constitution The Federal Government State Governments

Units Defined by State Constitutions The State Government Counties and (usually) Townships Municipalities (Cities and Villages) School Districts Special Districts The U.S. Federal System The Theory of Local Public Finance County Township District Map of Hypothetical State The U.S. Federal System Municipality

School The Please Theory of Local Public Finance contact author for more information regarding this documents tables and figures 120000 100000 67355 12884 80000 37203 60000 12340 40000 20000 0

17202 16807 19522 3052 3031 1952 Counties Special Districts The U.S. Federal System 16364 2012 Municipalities Townships School Districts

Source: U.S. Census of Governments The Theory of Local Public Finance Class Outline The U.S. Federal System The Consensus Model of Local Public Finance Deriving a Bid Function Residential Is Sorting the U.S. Federal System Efficient?

Introduction The Theory of Local Public Finance Local Public Finance The literature on local public finance in a federal system is built around three questions: 1. How do housing markets allocate households to jurisdictions? = Bidding and sorting! 2. How do jurisdictions make decisions about the level of local public services and taxes? 3. Under what circumstances are the answers to the first two questions compatible? The Consensus Model The Theory of Local Public Finance The Role of Tiebout This

literature is often traced to a famous article by Charles Tiebout in the JPE (October 1956). Tiebout said people reveal their preferences for public services by selecting a community (thereby solving Samuelsons freerider problem). Tiebout said this choice is like any market choice so the outcome is efficient. But Tiebouts model is very simplistic. It has No housing market No property tax (just an entry fee) No public goods (just publically provided private goods) or voting No labor market or commuting (just dividend income) The Consensus Model

The Theory of Local Public Finance Key Assumptions This class focuses on a post-Tiebout consensus model for the first question based on 5 assumptions: 1. Household utility depends on a composite good (Z), housing (H), and public services (S). 2. Households differ in income, Y, and preferences, but fall into homogeneous income-taste classes. 3. Households are mobile, so utility is constant within a class. 4. All households in a jurisdiction receive the same S (and a household must live in a jurisdiction to receive its services). 5. A metropolitan area has many local jurisdictions with fixed boundaries and varying levels of S. The Consensus Model The Theory of Local Public Finance Additional Assumptions

Most models use 2 more assumptions: 6. Local public services are financed with a property tax with assessed value (A) equal to market value (V = PH/r, where r is the discount rate). Let m be the legal tax rate and the effective rate, then tax payment, T, is T mA V and T A m V V 7. All households are homeowners or households are renters and the property tax is fully shifted onto them. The Consensus Model

The Theory of Local Public Finance Class Outline The U.S. Federal System The Consensus Model of Local Public Finance Deriving a Bid Function Residential Is Sorting the U.S. Federal System Efficient? Introduction

The Theory of Local Public Finance The Household Problem The household budget constraint is Y Z PH V Z PH 1 Z PH (1 *) r where * is defined to be is is defined to be defined is defined to be to is defined to be be is defined to be /r. The household utility function is: The Consensus Model U {Z , H , S}

The Theory of Local Public Finance The Household Problem 2 The Lagrangian: U {Z , H , S } Y Z P{S , }H (1 *) The first-order conditions: U S PS H (1 *) 0 U Z 0 PH P H (1 *) 0 r The Consensus Model

The Theory of Local Public Finance The First-Order Conditions The 1st and 2nd conditions imply: US / UZ MBS PS H (1 *) H (1 *) The 3rd condition simplifies to: P P/r P (r ) (1 *) The Consensus Model

The Theory of Local Public Finance The First-Order Conditions, 2 The simplification in the first equation should be familiar from earlier topics in the class. Because Z has a price of unity, the marginal rate of substitution between S and Z, US/UZ, is the marginal benefit of S in dollar terms, or MBS. The Consensus Model The Theory of Local Public Finance The Market Interpretation These conditions indicate the values of S and that a household will

select. But all households cannot select the same S and ! Thus, these conditions must hold at all observed values of S and , that is, in all communities. As in an urban model, this is called, of course, locational equilibrium. No household has an incentive to move because lower housing prices exactly compensate them for relatively low values of S or relatively high values of . This is, of course, the issue that arises with commuting costs in a basic urban model. The Consensus Model The Theory of Local Public Finance

Alternative Approach Solve the budget constraint for P and find the most a household is willing to pay per unit of H at a given utility level Y Z Maximize is defined to be P H (1 *) Subject to U {Z , H , S } U 0 Now PS and P can be found using the envelope theorem. The results are the same! The Consensus Model The Theory of Local Public Finance

Bidding for Property Tax Rates These two conditions are differential equations. The tax-rate equation can be written as P 1 P (r ) This is an exact differential equation which can be solved by integrating both sides to get: ln{P{ }} ln{r } C where C is a constant of integration. The Consensus Model The Theory of Local Public Finance Property Tax Rates, 2 We

can solve for C by introducing the notion of a before-tax bid, sometimes called the bid net of taxes and indicated with a hat: P{S , } P{S } when is defined to be 0 Substituting this condition into the above (after exponentiating) yields: rP{S } P{S } P{S , } (r ) (1 *) The Consensus Model The Theory of Local Public Finance Property Tax Rates, 3 Note

for future reference that we can differentiate this result with respect to S, which gives PS PS (1 *) This result makes it easy to switch back an forth from before-tax to after-tax bid-function slopes (with respect to S). The Consensus Model The Theory of Local Public Finance The House-Value Equation To test this theory, we want to estimate an equation of the following form: P{S , }H { X } P{S }H { X }

V r r The dependent variable is house value, V, or it could be apartment rent. The key explanatory variables are measures of public services, S, property tax rates, , and housing characteristics, X. The Consensus Model The Theory of Local Public Finance Capitalization In

this equation, the impact of on V is called property tax capitalization. The impact of S on V is called public service capitalization. These terms reflect the fact that these concepts involve the translation of an annual flow (T or S) into an asset or capital value (V). The Consensus Model The Theory of Local Public Finance Finding a Functional Form This house value equation {S } cannot be estimated

P without a form for . To derive a form we must solve the above differential equation for P{S}: MBS PS H (1 *) To solve this equation, we obviously need expressions for MBS and H. As in an urban model, these expressions require assumptions about the form of the utility function (which implies a demand function) or about the form of the demand function directly. Deriving a Bid Function

The Theory of Local Public Finance Finding a Functional Form 2 One possibility is to use constant elasticity forms: S K S Y W H K H Y P(1 *) K H Y P

where the Ks indicate vectors of demand determinants other than income and price, and W is the price of another unit of S. Deriving a Bid Function The Theory of Local Public Finance Finding a Functional Form 3 These 1. forms are appealing for three reasons: They have been successfully used in many empirical studies. Duncombe/Yinger (ITPF 2011), community demand for education Zabel (JHE 2004), demand for housing 2. They can be derived from a utility function.

The derivation assumes a composite good (=an incomplete demand system), zero cross-price elasticities, and modest restrictions on income elasticities [LaFrance (Journal of Agricultural Economics, August 1986)]. 3. They are tractable! Deriving a Bid Function The Theory of Local Public Finance Finding a Functional Form 4 Note that the demand function for S can be inverted to yield: S W K SY This

1/ MBS is, of course, the form in which it appears in earlier derivations. Deriving a Bid Function The Theory of Local Public Finance Finding a Functional Form 5 Now substituting the inverse demand function for S and the demand function for H into the differential equation yields: PS P

S 1/ KS 1/ KHY ( / ) KHY ( / ) S where KS Deriving a Bid Function

1/ 1 . 1/ , The Theory of Local Public Finance Finding a Functional Form 6 The solution to this differential equation is: ( 1 ) ( 2 ) P{S } C S

where C is a constant of integration and the parentheses indicate a Box-Cox form, or, X ( ) X 1 if 0 and ln{ X } if 0 and 1 1 1 and 2 Deriving a Bid Function The Theory of Local Public Finance Finding a Functional Form 7 This equation is, of course, a bid function.

It indicates how much a given type of household would pay for a unit of H in a location with a given level of S. It is analogous to the bid functions in a basic urban modelit indicates how much a household would pay at different locations (=levels of S) holding utility constant. Deriving a Bid Function The Theory of Local Public Finance Class Outline The U.S. Federal System The

Consensus Model of Local Public Finance Deriving a Bid Function Residential Is Sorting the U.S. Federal System Efficient? Introduction The Theory of Local Public Finance Sorting It is tempting to stop hereto plug this form into the house value equation and estimate.

As we will see, many studies proceed, incorrectly, in exactly this manner. But To we have left out something important: sorting. put it another way, we have not recognized that households are heterogeneous and compete with each other for entry into desirable locations. Sorting The Theory of Local Public Finance Sorting 2 Sorting in this context is the separation of different household types into different jurisdictions. As

in an urban model, the key conceptual step to analyze sorting is to focus on P, the price per unit of H, not on V, the total bid. In the long run, the amount of H can be altered to fit a households preferences. A seller wants to make as much as possible on each unit of H that it supplies. Sorting The Theory of Local Public Finance Sorting 3 This framing leads to a standard picture in which P{S } is on the vertical axis and S is on the horizontal axis.

Each household type {Shas P } its own bid function; that is, its own . The household that wins the competition for housing in a given jurisdiction is the one that bids the most there. Sorting The Theory of Local Public Finance Sorting 4 Yinger

(JPE, September 1982) was an early user of this picture (although not the inventor). His version (with E instead of S): P(E,t*) Sorting The Theory of Local Public Finance Sorting 5 sorting picture must distinguish between bid functions and envelopes. Here is another is defined to be example: The envelope must slope upwards, but its second derivative, which reflects the balance between bidding and sorting, could be positive or

negative . Bid is defined to be per is defined to be Unit is defined to be of is defined to be Housing is defined to be Services is defined to be (Log) Any 10 Hedonic is defined to be Envelope Bid is defined to be Functions 20 30 40 50 60

70 Amenity is defined to be (e.g. is defined to be School is defined to be Test is defined to be Passing is defined to be Rate) Sorting 80 90 100 The Theory of Local Public Finance Sorting 6 The logic of this picture leads to several key theorems. 1.

Household types with steeper bid function end up in higher-S jurisdictions. This important theorem indicates that sorting is determined by the slopes of bid functions. It is illustrated in the following figure. Sorting The Theory of Local Public Finance Consensus Bidding and Sorting P P3 P2 P1 Group 2 lives in jurisdictions with this range of S. Sorting S1

S2 S2 S The Theory of Local Public Finance Sorting 6 This theorem depends on a single crossing assumption, namely, that if a household types bid function is steeper at one value of S, it is also steeper at other values of S. This is a type of regularity condition on utility functions. Sorting

The Theory of Local Public Finance Sorting 7 2. Some jurisdictions may be very homogeneous in their demand for the amenity. Sorting tends to separate households with different amenity demands. This is clear in the above figure. Sorting The Theory of Local Public Finance Sorting 8 3. But other jurisdictions may be very heterogeneous in their demand for the amenity. In the standard picture, these jurisdictions may include those at

the intersections between bid functions. Or large cities may contain many household types. See the following figure. Households can be heterogeneous in income and other factors in the demand function for H or the latent demand function for S. Sorting The Theory of Local Public Finance A Heterogeneous Jurisdiction P P3 P2 Four is defined to be household is defined to be types is defined to be live is defined to be in is defined to be the is defined to be jurisdiction is defined to be where is defined to be S is defined to be = is defined to be S1. P1 4 3

Sorting 2 1 S1 S2 S2 S The Theory of Local Public Finance Sorting 9 4. Sorting does not depend on the property tax rate. As shown above, P

1 P (r ) Nothing on the right side depends on Y (or any other household trait); starting from a given P, the percentage change in P with respect to is the same regardless of Y. Sorting The Theory of Local Public Finance Sorting 10 5. In contrast, income, Y, (or any other demand trait) can affect sorting. Because

is defined to be does not affect sorting, we can focus on before-tax bids. We will also focus on what is called normal sorting, defined to be sorting in which S increases with Y. Sorting The Theory of Local Public Finance Sorting 11 Normal sorting occurs if the slope of household bid functions increases with Y, that is, if PS MBS 1 MBS H

0 2 Y Y H H Y This Sorting condition is assumed in Yingers JPE picture. The Theory of Local Public Finance Sorting 12 After some rearranging, we find that PS MBS 1 MBS H 0 if Y Y H

H 2 Y or MBS Y H Y Y MBS Y H Normal sorting occurs if the income elasticity of MBS exceeds the income elasticity of H. Sorting The Theory of Local Public Finance Sorting 13 The constant elasticity form for S implies that MBS Y

Y MBS Hence, the slope, so long as: PS / ,Y will increase with Y Sorting The Theory of Local Public Finance Sorting 14 The available evidence suggests that and are

approximately equal in absolute value and that is defined to be 0.7. It is reasonable to suppose, therefore, that this condition usually holds. Competition, not zoning, appears to be the main reason that highY people live in high-S jurisdictions (although zoning may preserve existing patterns and prevent adjustments when conditions change). No study yet estimates the impact of zoning on the hedonic equilibrium. Sorting The Theory of Local Public Finance Sorting 15 6. This analysis of bidding and sorting applies to any public service or amenity that is linked to a location.

Examples include: The perceived quality of local elementary schools; Distance from a pollution source; Access to parks or other neighborhood amenities. As we will see, this framework also links nicely with the largely empirical literature on so-called hedonic regressions. Sorting The Theory of Local Public Finance Sorting 16 7. Finally, the logic of bidding and sorting does not apply only to the highly decentralized federal system in the U.S.

It also applies to any country in which A location-based public service or neighborhood amenity varies across locations, Housing markets are competitive and households can decide where to live, and Access (or the cost of access) to the service or amenity depends on residential location. Sorting The Theory of Local Public Finance Preview In the next set of classes, we will bring in the complementary literature on housing hedonics, which builds on Rosens famous article in the JPE (Jan./Feb. 1974). The

Rosen article provides some more theory to think about as well as the framework used by most empirical work on the capitalization of public service and neighborhood amenities into house values. We will also cover a new approach to hedonics that draws on the theory we have reviewed today. Preview The Theory of Local Public Finance Class Outline The U.S. Federal System The Consensus Model of Local Public Finance

Deriving a Bid Function Residential Is Sorting the U.S. Federal System Efficient? Introduction The Theory of Local Public Finance Bids Including Property Taxes One well-known approach to bidding and sorting comes from Hamilton (Urban Studies, June 1975). To

understand his approach, let us begin by pointing out that even though property taxes do not affect sorting, they can be incorporated into bid functions. With property taxes included, bid functions eventually slope downward, as the increment in service quality is not worth the added property-tax cost. Hamilton The Theory of Local Public Finance Bids Including Property Taxes, 2 The problem is that these bid functions are not just a function of S, but also depend on . One

approach is to assume that community income, Y*, or some other variable provides an index of the quality of service-tax packages, not just of services. Then we can plot bids as a function of the services and taxes associated with Y*. Hamilton The Theory of Local Public Finance Consensus Bidding and Sorting Net of Taxes P P3 P2 P1 Hamilton Y*1

Y*2 Y*3 Y* The Theory of Local Public Finance The Hamilton Approach Hamiltons model adds three assumptions: Housing supply is elastic (through building on open space in a jurisdiction or through expansion of a jurisdictions boundaries). Zoning is set at exactly the optimal level of housing for the residents of a jurisdiction. Government services are produced at a constant cost per household. Hamilton The Theory of Local Public Finance

Hamilton, 2 The striking implication of the Hamilton assumptions is that capitalization disappears. See the following figure (or the proof in the Ross/Yinger, 1999 Handbook chapter). Everyone ends up in their most-preferred jurisdiction, so nobody bids up the price in any other jurisdiction. This prediction (no capitalization) is rejected by all the evidence. Hamilton The Theory of Local Public Finance

Hamilton Bidding and Sorting P _ P Hamilton Y1 Y2 Y3 Y The Theory of Local Public Finance Is a Federal System Efficient? Tiebout (with no housing or property tax) said

picking a community is like shopping for a shirt and is therefore efficient (in the allocative sense). This result is replicated with a housing market and property tax under the Hamilton assumptions. These assumptions are extreme and the model is rejected by the evidence. Hamilton The Theory of Local Public Finance Sources of Inefficiency 1. Heterogeneity Heterogeneity leads to inefficient levels of local public services. Outcomes are determined by the median voter; voters

with different preferences experience dead-weight losses compared with having their own jurisdiction. See the next figure. Hamilton The Theory of Local Public Finance Inefficiency Due to Heterogeneity (When Community Selects Median Service Level S3) Hamilton The Theory of Local Public Finance Sources of Inefficiency, 2 This figure misses the role of the property tax. Some households have a higher tax price than others. The tax price is marginal resource cost (MC) multiplied by tax share (=a households assessed value relative to the average).

If higher tax shares are associated with higher demand for services, as seems reasonable, some or all of the inefficiency in this figure may disappear. Tax share is discussed in the readings on public service demand. Hamilton The Theory of Local Public Finance Sources of Inefficiency, 3 2. The Property Tax Producers of housing base their decisions on P, whereas buyers respond to P(1 + /rr). This introduces a so-called tax wedge between producer and consumer decisions, which leads to underconsumption of housing. Hamilton

The Theory of Local Public Finance Hamilton and Inefficiency These two sources of inefficiency disappear in Hamiltons model: Boundaries adjust so that all jurisdictions are homogeneous (despite extensive evidence that boundaries do not change for this reason). Zoning is set so that housing consumption is optimal (despite the incentives of residents to manipulate zoning and the bluntness of zoning tools). Hamilton The Theory of Local Public Finance Conclusions on Efficiency The

Hamilton model shows how extreme the assumptions must be to generate an efficient outcome. Nevertheless, many scholars defend our federal system as almost efficient, Instead of identifying policies to improve its efficiency. Hamilton