JOURNAL OF ECONOMIC THEORY 43, 348-354 (1987)

Non-connectedness of the Set of Equilibrium Money Prices: The Static Economy

JAMES PECK *

Department of Managerial Economics and Decision Sciences, .I. L. Kellogg Graduate School of Management.

Northwestern University. Evanston. Illinois 60201

Received December 16, 1986; revised May 15, 1987

A static economy in which nominal taxes and transfers are balanced. under cer- tain conditions, has a set of equilibrium money prices containing a proper interval. Two examples are given in which the entire set must consist of a proper interval. Then, an example is presented in which the set of equilibrium money prices is not connected. While the set contains a proper interval for balanced fiscal policies, the entire set is not in general a single interval. Journal of Economic Literature Classification Numbers: 021,022,023. ,i‘l 1987 Academic Press, Inc.

Balasko and Shell [l] consider a static exchange economy in which the government can levy taxes and distribute transfers in terms of the monetary unit. The price of money in terms of commodities, determined in the market, takes on any one of a range of values. For each balanced fiscal policy, the set of equilibrium money prices is shown to contain an interval [O, @“I, where p” is a positive scalar. Balasko and Shell indicate that the set might consist of two unconnected intervals or some more complicated structure, although in all the specific examples given so far the set of equilibrium money prices is an interval. An example is presented here, satisfying all the regularity assumptions of [l], in which the set of equilibrium money prices is not connected. While the set contains a proper interval for balanced fiscal policies, the entire set is not in general a single interval.

Each consumer in the example has preferences giving rise to a utility function that is strictly monotonic, differentiable, and strictly quasi- concave. Endowments are contained in the strictly positive orthant. We

*Support from the National Science Foundation under Grant SES-83-08450 is acknowledged. I would like to thank Karl Shell for his advice and encouragement. Remaining errors are my own.

348 0022-0531/87 $3.00 Copyright c 1987 by Academic Press, Inc. All rights of reproductmn in any form reserved.

SET OF EQUILIBRIUM 349

follow the notation of Balasko and Shell Cl]. In particular, the vector of commodity prices given by p = (p’, p’,..., p’) has commodity one as numeraire, so p1 = 1.

We start by recalling two examples in which the set of equilibrium money prices, Pm(u, t), is a proper interval. In Example 3, a non- degenerate example is presented in which PDm(a, r) is not an interval.

EXAMPLE 1. Let there be one commodity, l= 1. Consider the tax- transfer policy z = (pi , . . . . rh, . . . . r,) which is balanced i.e., C;: =, rh = 0, and nontrivial, i.e., T # 0. (Positive components of t represent taxes and negative components represent transfers.)

Claim 1. The set F(w, t) is a proper interval.

Proof of Claim I. The maximization problem for consumer h is

max uh(.xh)

s.t. p . .Yh 6 p . Wh - pn’th

Xh > 0.

Because there is one commodity, p is a scalar which can be normalized to one. Monotonicity of the utility function implies

Xh = Oh - pmTh (1)

Xh > 0. (2)

Summing the budget constraints of the consumers, and using the fact that z is balanced, we have

Therefore, markets always clear for any value of p”. Equilibrium obtains if and only if pm is consistent with a solution to (1) and (2) for all consumers. It follows that the set of equilibrium money prices is specified by

which is a proper interval. 1

EXAMPLE 2. Let there be I commodities but only two consumers, n = 2. Consider the tax-transfer policy T = (T,, s2), which is balanced and non- trivial.

Claim 2. The set F(e), T) is a proper interval.

350 JAMES PECK

Proof of Claim 2. The set of tax-transfer policies consistent with equilibrium and p” = 1, F&,, , is called the set of normalized bona fide tax- transfer policies. The set I$;,, must be contained in the one-dimensional subspace defined by rr + TV = 0, since only balanced policies can be nor- malized bona fide. By Balasko and Shell [ 1, Proposition 3.61, we know that Z& is arc-connected (and hence connected) and contains zero in its relative interior. A subset of R consisting of more than one point is connec- ted if and only if it is a proper interval, so F&“, is a proper interval. It follows from [ 1, Proposition 2.51 that :Y(w, z) is also a proper interval. 1

EXAMPLE 3 (in which Y”(o, r) is not an interval). In view of Example 1 and Example 2, we must have at least two commodities and at least three consumers, so let I = 2 and n = 3. Also, let endowments be

w, = (CD;, of)= (4, 8),

oz=(w;,o;)=(20,4),

03 = to:+ 0:) = (4,8),

and let the tax-transfer policy be given by T = (T, , TV, z3) = (20,0, - 20).

The preferences of the three consumers are depicted in Figs. 1-3. Figure 1 shows two of consumer l’s indifference curves, labelled Lii and tif . The marginal rate of substitution is 1 at the point (8,4) and 0.25 at the point (4, 3). To complete the preference map, indifference curves below ii: are for- med from convex combinations of iif and (0,O) along rays from the origin. Form convex combinations of ~7: and iit along rays from the origin to con- struct the indifference curves between iif and 6;. Form radial projections of ~7; to get the indifference curves above U ;. This procedure guarantees that preferences satisfy the required regularity conditions.

Figure 2 shows three of consumer 2’s indifference curves, labelled ii:, iis, and ii:. The marginal rates of substitution at the points (16,8), (4,8), and (16, 9) are 1, t, and f, respectively. Below ii:, construct the missing indif- ference curves by taking convex combinations of Ui and (0,O) along rays from the origin. Form convex combinations of iii and tii along rays from the origin between those curves, and form radial projections of ii: above &. Between tii and zI$, the indifference curves are found by forming convex combinations of the two curves along uertical lines. Preferences are smooth and convex, and the offer curve between tii and iii is a vertical segment.’

’ This method of filling in the indifference curves to generate vertical or horizontal segments of the offer curve was inspired by Cass. Okuno, and Zilcha 131.

SET OF EQUILIBRIUM 351

x:

pm=0

FIGURE I

This follows from the fact that the line tangent to Uf at x1 = 16 and the line tangent to tii at x’ = 16 intersect at the endowment point.

Figure 3 shows four of consumer 3’s indifference curves, labelled ii:, zi:, ii:, and ii;. The marginal rates of substitution at the points (4, 8), (20,9), (17,4), and (30,4) are I, $, j, and f, respectively. To fill in the mis- sing indifference curves below u3, -4 form convex combinations of tit: and (0,O) along rays from the origin. Between 11: and ti:, form convex com- binations of the two curves along rays from the origin. Above U:, form radial projections of il,. ’ Between ti: and ti:, form convex combinations of the two curves along horizontal fines. Between U: and Ui, form convex com- binations of the two curves along vertical lines. Preferences are smooth and convex, and the offer curve between ti: and 63 is a horizontal segment.’ Also, when p”=O.5 and consumer 3’s “translated” endowment point is (14,8), he will demand at least 20 units of commodity one whenever we have p2 > 4.

For a given value of pm, nominal taxes and transfers are equivalent to taxes and transfers of commodity one. Thus, when we have pm = 0.5, Mr. l’s “translated endowment” point, G,,, is given by c.G~ = (4 - 20[0.5], 8) =

’ Since different values of pm lead to different translated endowment points, the offer curve varies with pm. The offer curve between U: and zi: is a horizontal segment for the translated endowment ~5~ = (14.8) associated with pm = 0.5.

352 JAMES PECK

x: FIGURE 2

FIGURE 3

SET OF EQUILIBRIUM 353

(-6,8),andwhenp”=l holdswehave(3,=(4-20,8)=(-16,8).Since r1 = 0, Mr 2’s translated endowment always equals his untranslated endow- ment, 5, = II+ = (20,4). For pm = 0.5, Mr. 3’s translated endowment point is G3, = (4 + 20[0.5], 8) = (14,8), and for pM = 1 we have CZ~ = (4 + 20,8) = (24, 8). The translated endowments are not all in the positive orthant, so it does not necessarily follow that a competitive equilibrium price vector p = (1, p’) will exist for every positive pm.

The economy we have specified exhibits the following two equilibria:

Equilibrium 1 Equilibrium 2

p”=O, p=(l, 1) pm= 1, p=(l,4) x, = (xf , XT) = (8,4) x, = (xi, x:, = (4, 3) ~~=(.~;,x;)=(16,8) x2 = (x;, x;, = (4, 8) x3 = (xi, xi, = (4, 8) x3 = (xi, x:, = (20, 9).

Claim 3. For this economy which satisfies the regularity assumptions of [ 11, there is no competitive equilibrium with p”’ = 0.5.

Proof of Claim 3. For p’ < 0.75, Mr. 1 has a negative level of wealth, so he cannot afford anything in his consumption set. This is inconsistent with equilibrium.

For p2 E CO.75, 31, we have .x: = 16 and X: b 17, so there is excess demand for commodity one. When p2 E (3,4] occurs, ~4 > 4 and xl 3 26 hold, so there is still excess demand for commodity one. Finally, for p2 > 4, we have X: 2 4 and xi > 20. Once again, there is excess demand for the numeraire commodity.

Since market clearing fails whenever pz 30.75 holds and Mr. 1 goes bankrupt whenever p2 ~0.75 holds, pm =0.5 is inconsistent with equilibrium. 1

However, by construction there is a competitive equilibrium with p”’ = 1 and with pm = 0. Hence we have

0 E 9yf3, t),

0.5 $2 Fyc!l, T),

and

1 E glrn(CO, 5).

This example thus establishes the existence of a “nice” economy in which Prn(~, T) is not a connected set.

354 JAMES PECK

Concluding Remarks. Example 3 provides a nonpathological economy in which the set of equilibrium money prices is not an interval. We used preferences which generate vertical or horizontal segments of offer curves, merely as a simple way of guaranteeing that excess demand exists for certain prices. Any other preferences yielding excess demand for the appropriate range of prices would work just as well. However, Example 3 could be considered somewhat “unusual” in that it depends on powerful income effects. Increasing pm causes a redistribution of commodity one from consumer 1 to consumer 3. Eventually, consumer 1 cannot meet his tax obligations and there is no equilibrium. However, as pm is increased further, income effects cause relative prices to change, increasing the value of consumer l’s endowment by more than the additional tax. Equilibrium is restored when consumer I’s wealth becomes positive once again. This broken interval property is not limited to static models; see [4] for exam- ples of infinite horizon overlapping generations economies (with and without balanced tax-transfer policies)3 for which the set of equilibrium money prices is not an interval.

REFERENCES

1. Y. BALASKO AND K. SHELL, “Lump-Sum Taxation: The Static Economy,” CARESS Work- ing Paper No. 83-08RR, University of Pennsylvania, 1983.

2. Y. BALASKO AND K. SHELL, “Lump-Sum Taxes and Transfers: The Overlapping- Generations Model with Money,” IMSSS Technical Report 458, Stanford University, Palo Alto, CA, March 1985, also appearing in “Essays in Honor of Kenneth J. Arrow, Volume II: Equilibrium Analysis” Chap. 5, pp. 121-153 (W. Heller, R. Starr, and D. Starett, Eds.), Cambridge Univ. Press, London/New York, 1986.

3. D. CASS, M. OKUNO, AND I. ZILCHA, The role of money in supporting the Pareto optimality of competitive equilibrium in consumption-loan type models, J. Econ. Theory 20 (1979) 41-80.

4. J. PECK, Non-connectedness of the set of equilibrium money prices: The overlapping- generations economy, Northwestern University mimeo, April 1986; J. Econ. Theory 43 (1987), 355-363.

3 See Balasko and Shell [2] for an analysis of the overlapping generations model with nominal taxes and transfers.