A Problem with Euclidean Preferences in Spatial Models of Politics

A Problem with Euclidean Preferences in Spatial Models of Politics Jeffrey Milyo Department of Economics Tufts University Discussion Paper 99-20 Department of Economics Tufts University

Department of Economics

Tufts University

Medford, MA 02155

(617) 627-3560

Department of Economics, Braker Hall, Tufts University, Medford, MA

1

02155. Phone: (617) 627-3662. Fax: (617) 627-3917. Email: [email protected]. JEL code: D70

Keywords: public sector preferences, spatial models

A Problem with Euclidean Preferences in Spatial Models of Politics Jeffrey Milyo1

Tufts University

Abstract:

Euclidean public sector preferences can not be induced from a strictly quasiconcave primitive utility function and a linear constraint.

Introduction

In spatial models of collective choice, individual preferences are often represented as induced preferences in policy space (or “public sector preferences”). These induced preferences are typically understood to be derived from an

underlying constrained maximization involving a primitive utility function of the conventional sort (e.g., Denzau and Parks 1977, 1979, Diba and Feldman 1984, and Slutsky 1975, 1977). Given the usual assumptions of individual utility maximization, it is well-known that the induced preferences over policy exhibit a unique satiation point (corresponding to the constrained maximum of the

underlying primitive utility function) and have strictly convex level sets. In many applications, public sector preferences also are assumed to be Euclidean; that is, the public sector preferences are not just separable, but utility declines monotonically in distance from the ideal point. This assumption drives many important results in collective choice (e.g., McKelvey 1976, Laver and Shepsle 1996, Ferejohn and Krehbiel 1984 and Koford 1989; also, see Milyo 1999). Below, I demonstrate below that well-behaved public sector preferences are never Euclidean.

The canonical derivation of public sector preferences involves a collective of individuals in an economy consisting of two publicly provided goods (z , z ) and_{1 2} one composite private good (x). For simplicity, let the publicly provided goods be pure public goods; therefore, any individual citizen in the collective has three arguments in his utility function: (x, z , z ). From the perspective of this citizen,_{1 2} the private good is a free choice variable and the publicly provided goods (i.e., “policies”) are strictly rationed at a level chosen by the collective.

Assume a linear budget constraint and normalize prices relative to the price of x. The individual’s utility maximization problem is then:

Maximize U(x, z , z ) subject to B = p z + p z + x._{1 2 1 1 2 2} (1)

Let U be strictly quasiconcave and continuously differentiable, so that there exits a unique constrained maximum to this problem and well-behaved demand functions. The solution to (1) is trivial when both publicly provided goods are rationed:

x* = B - p z - p z .1 1 2 2 (2)

of (1) when no goods are rationed. Denote this as (x*, z *, z *)._{1 2}

In spatial models of politics, it is the convention to work with the individual’s induced preferences over policy (or “public sector preferences”). These may be represented by substituting (2) into the utility function. Denote the resulting mixed indirect utility function by V(z , z ). The individual’s preferences_{1 2} over policy are then described by the level sets of V(z , z ) and the ideal point is_{1 2} simply the maximum of this function.

Separable and Euclidean public sector preferences

It is common in spatial models of politics to assume that V(z , z ) is _{1 2} separable; that is, the individual’s most preferred level of z is independent of any₁ fixed level of z , and vice versa. Clearly, this property can not hold for all feasible₂ policies (e.g., when p z + p z = B), so throughout this analysis I restrict attention_{1 1 2 2} to regular and interior solutions to (1).

There are several caveats and criticisms in the economics literature concerning the properties of public sector preferences (for a review, see Milyo 1999). For example, Slutsky (1975) shows that V(z , z ) is separable when the1 2 demands for the publicly provided goods are independent of income. Diba and Feldman (1984) demonstrate for the canonical three good case presented here, that

a quasilinear utility function can generate separable public sector preferences. Milyo (1999) extends Diba and Feldman’s result to show that given a strictly quasiconcave primitive utility function and a linear constraint, separability can be a general property of the public sector preferences only for the case of one private good. Consequently, the example presented in this exercise is fairly general. This is because any number of publicly provided goods may be imagined to be fixed at their ideal level in this example.

Euclidean preferences are a subset of separable preferences for which utility declines monotonically in distance from the ideal point. This implies that the level sets of V(z , z ) are concentric circles about the ideal point._{1 2}

Public sector preferences are not Euclidean

Suppose an individual’s public sector preferences are Euclidean with an ideal point located at (z *, z *). Consider a particular level set, V(z , z )=c, which_{1 2 1 2} is away from the ideal point. This implies that:

V(z * + dcosè , z * + dsinè ) = c,1 2 (3)

Making use of (2) and the definition of public sector preferences, (3) may be re-written as:

U(B - p z * - p dcosè - p z * - p dsinè , z * + dcosè , z * + dsinè ) = c_{1 1 1 2 2 2 1 2} (4)

for all d > 0 and 0 < è < 2ð

Differentiate (4) with respect to è and re-arrange terms to show that:

dcosè*(p U - U ) + dsinè*(p U - U ) = 0._{2 x z2 1 x z1} (5)

It will now be shown that this condition can be satisfied for only four points in any given level set (other than the ideal point).

First, consider the case when only z is fixed at some arbitrary z , while x_{2 2}0 and z are left as choice variables. In this case, and for a well-behaved primitive₁ utility function, the first order conditions for the constrained maximum of

U(x,z ,z ) imply that (p U - U ) = 0 (a.k.a., the “equimarginal” condition). Note_{1 2}0 _{1 x z1} also that the definition of Euclidean preferences implies that constrained optimal value of z is z *. Consequently, dcosè = 0 and condition (5) is satisfied at the_{1 1} constrained maximum, (B-p z *-p ,z , z *,z ). There will be two points on any1 1 2 2 1 2

level set of V(z , z ) which are constrained optima when z is a free choice variable_{1 2 1} (when z = z * + dsinè); at these two points, condition (5) is satisfied. Similarly,₂0 ₂ when z is fixed at an arbitrary level, condition (5) will be satisfied at two₁

analogous points on the given level set (these four points correspond to the four “compass points” on any level set).

Now consider the case when both z and z are fixed at arbitrary points_{1 2}

(z ,z ) not equal to z * or z *. It follows that neither dcosè nor dsinè is equal to₁0 ₂0 _{1 2}

zero. It is also the case that each of the publicly provided goods is either being under-consumed or over-consumed relative to consumption when either or both of the z’s are free variables. Consequently, for a well-behaved primitive utility

fuinction, the equimarginal condition does not hold and condition (5) is not

satisfied at (B-p z -p z ,z ,z ). This demonstrates that not only are well behaved_{1 1}0 _{2 2}0 ₁0 ₂0 public sector preferences never Euclidean, but that (for d>0) no level set of

References

Denzau, A. and R. Parks, 1977, A problem with public sector preferences, Journal of Economic Theory, 14, 454-457.

Denzau, A. and R. Parks, 1979, Deriving public sector preferences, Journal of Public Economics, 11, 335-352.

Diba, B. and A. Feldman, 1984, Utility functions for public outputs and majority voting, Journal of Public Economics, 25, 235-243.

Ferejohn, J. and K. Krehbiel, 1987, The budget process and the size of the budget, American Journal of Political Science, 31, 296-320.

Koford, K., 1989, Dimensions in congressional voting, American Political Science Review, 83, 949-964.

Laver, M. and K. Shepsle, 1990, Coalitions and cabinet government, American Political Science Review, 84, 873-890.

McKelvey, R., 1976, Intransitivities in multidimensional voting models, Journal of Economic Theory, 12, 472-482.

Milyo, J., 1999, Logical deficiencies in spatial models: a constructive critique, Public Choice (forthcoming).

Slutsky, S., 1975, Majority voting and the allocation of public goods, ph.d. dissertation (Yale University).

Slutsky, S., 1977, A voting model for the allocation of public goods: existence of an equilibrium, Journal of Economic Theory, 14, 299-325.

ECONOMICS DEPARTMENT DISCUSSION PAPERS SERIES 1999

99-20 Jeffrey MILYO; A Problem with Euclidean Preferences in Spatial Models of Politics. 99-19 Drusilla K. BROWN; Can Consumer Product Labels Deter Foreign Child Labor

Exploitation?

99-18 Alexander M. BRILL, Kevin A. HASSETT, and Gilbert E. METCALF; Household Energy Conservation Investment and the Uninformed Consumer Hypothesis. 99-17 Yannis M. IOANNIDES and Kambon KAN; The Nature of Two-Directional

Intergenerational Transfers of Money and Time: An Empirical Analysis.

99-16 Linda HARRIS DOBKINS and Yannis M. IOANNIDES; Dynamic Evolution of the U.S. City Size Distribution.

99-15 Anna M. HARDMAN and Yannis M. IOANNIDES; Residential Mobility and the Housing Market in a Two-sector Neoclassical Growth Model.

99-14 Yannis M. IOANNIDES; Why Are There Rich and Poor Countries? Symmetry Breaking in the World Economy: A Note.

99-13 Linda HARRIS DOBKINS and Yannis M. IOANNIDES; Spatial Interactions Among U.S. Cities: 1900-1990.

99-12 Yannis M. IOANNIDES; Residential Neighborhood Effects.

99-11 Yannis M. IOANNIDES; Neighborhood Interactions in Local Communities and Intergenerational Transmission of Human Capital.

99-10 Yannis M. IOANNIDES; Economic Geography and the Spatial Evolution of Wages in the United States.

99-09 Edward KUTSOATI and Dan BERNHARDT; Can Relative Performance Compensation Explain Analysts' Forecasts of Earnings?

Polluting Industry is Not a Tax on Pollution: The Importance of Hitting the Target. 99-07 Marcelo BIANCONI; A Dynamic Monetary Model with Costly Foreign Currency. 99-06 Marcelo BIANCONI; The Effects of Alternative Fiscal Policies on the Intertemporal

Government Budget Constraint.

99-05 Lynne PEPALL and Dan RICHARDS; The Simple Economics of "Brand-Stretching." 99-04 Drusilla K. BROWN, Alan V. DEARDORFF, and Robert M. STERN; U.S. Trade and

Other Policy Options and Programs to Deter Foreign Exploitation of Child Labor. 99-03 Margaret S. MCMILLAN and William A. MASTERS; A Political Economy Model of

Agricultural Taxation, R&D, and Growth in Africa.

99-02 MASTERS, William A. and Margaret S. MCMILLAN; Ethnolinguistic Diversity, Government Expenditure and Economic Growth Across Countries.

99-01 MCMILLAN, Margaret S.; Foreign Direct Investment: Leader or Follower?

Discussion Papers are available on-line at