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Social Choice and Welfare

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02.07.2018 | Original Paper

Welfare evaluations and price indices with path dependency problems

verfasst von: Tsuyoshi Sasaki

Erschienen in: Social Choice and Welfare | Ausgabe 1/2019

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Abstract

In cases where multiple prices change, this paper develops methods for calculating compensating and equivalent variations from the prices, the consumption bundle, and the wealth elasticities of demand for goods. The methods are a natural extension and generalization of the method in Willig (Am Econ Rev 66(4):589–597, 1976), and have the ability to provide second-order approximations of compensating and equivalent variations. In addition, this paper considers two types of price paths: indifference price paths, along which utility levels are kept constant, and iso-price-ratio paths, along which prices change proportionally. Along these two paths, changes in consumer surplus, compensating variation, and equivalent variation are easily calculated, and that there are interesting and precise relationships among them. Furthermore, the methods and relationships above correspond to those of the Laspeyres–Konüs and Paasche–Konüs true cost of living price indices and the Divisia indices.

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The concept of consumer surplus was introduced by Dupuit (1844). Hicks (1941, 1946) and Marshall (1961) advanced and analyzed the concept of consumer surplus. Harberger (1971) showed that consumer surplus has some desirable properties and that it is superior to gross national income in many ways. Hicks (1946) introduced compensating and equivalent variations.

In recent years, various authors have advanced the research on consumer surplus. Blackorby and Donaldson (1999) showed that the area to the left of market demand curves in directly affected markets is equal to the sum of consumer and producer surpluses in all markets in general equilibrium settings. Vives (1987), Miyake (2006) and Hayashi (2013) examined under what conditions income effects on commodities under consideration vanish.

McKenzie (1976) and McKenzie and Pearce (1976, 1982) developed the method for expressing compensating and equivalent variations as the Taylor series of the indirect utility function. Using Hurwicz and Uzawa (1971) results, Hausman (1981) derived the compensating and equivalent variations from an estimate of the demand curve.

The Divisia price index is closely related to changes in consumer surplus. As Hulten (1973) pointed out, the Divisia price index has the same path dependency property as consumer surplus.

Price indices are also used for comparing standards of living between countries. For example, see Deaton and Heston (2010).

Diewert (1988) is a great summary of the early history of price index research. When it comes to recent developments in price index theory, see Moulton (1996), Nordhaus (1998), Pollak (1998), Persky (1998), Boskin et al. (1998), Abraham et al. (1998), Diewert (1998), Schultze (2003), Hausman (2003), Abraham (2003) and Lebow and Rudd (2003).

When the utility function is homothetic, the superlative indices proposed by Diewert (1976) provide close approximations of true cost of living indices.

As will be shown in Appendix C.1, the Laspeyres–Konüs and Paasche–Konüs true cost of living indices are closely related to compensating and equivalent variations, and the Divisia price index is closely related to changes in consumer surplus.

In this paragraph, I focus on compensating variation, equivalent variation, and consumer surplus. However, similar local and global properties about price indices also hold true because there are close relationships between these three measures of economic welfare and the corresponding price and quantity indices, as explained in Appendix C.

We regard vectors as column vectors.

When there is no money illusion, without loss of generality, we can normalize the income level to one by replacing p with $p / ( p\cdot x )$. When the consumer’s income level changes, we can adjust the prices for the changes.

When income level w is normalized to one, and there is no need to express the Marshallian demand function as x(p, w) and the inverse demand function as p(x, w), we denote the Marshallian demand function by x(p) and the inverse demand function by p(x), respectively. Since the consumer has a rational, strongly monotone, locally nonsatiated, and strictly convex preference relation $\succsim $, x(p) and p(x) are bijections.

In general, we use $C_{p}(i, p^{1}, p^{2})$ and $C_{p}(i)$ as both a curve and a point on a curve interchangeably. $C_{p}$ is used as a curve: $C_{p}=\{ p\in \mathbb {R}_{++}^{N}| p=C_{p}(i; p^{1}, p^{2}), s\le i \le t\}$.

The Divisia price index, which will be explained later, also has a path dependency problem.

As mentioned before, compensating and equivalent variations are path independent.

The approach in Willig (1976) and mine differ a bit. Willig approximated the lower and upper bounds of compensating and equivalent variations with changes in consumer surplus, using the lower and upper bounds of the wealth elasticity of demand. On the other hand, given a price change, I calculate the approximations of compensating and equivalent variations.

$C_{x}^{j}$ in the following equations is defined from $C_{x}^{j}(i; x^{1}, x^{2})$ in the same way.

In Appendix A, using manifolds and differential forms, the differences among the changes in consumer surplus along different paths are analyzed in a more general way. In Appendix A.4, the proof of Proposition 7 is given.

Of course, some of the following equations in this and following subsections are approximations. Thus, there would be slightly different ways to express these equations.

The corresponding Laspeyres–Konüs and Paasche–Konüs true cost of living price indices are stated in Appendix C.2.

In Eqs. (10), (11), (16), and (17), we ignore small terms that contain three or more $\varDelta p$ or $\varDelta x$, like $(\varDelta p \cdot x^{1, 1})(\varDelta p \cdot \varDelta x)$.

In response to McKenzie (1979), Willig (1979) also provided the following methodology for calculating the upper and lower bounds on the errors of the approximations of compensating variations with changes in consumer surplus when multiple prices change:

$$\begin{aligned}&C-A \ge \frac{\underline{\mu }_{u}A^{2}_{u}}{2m^{0}}+\frac{\underline{\mu }_{d}A^{2}_{d}}{2m^{0}}+\frac{\overline{\mu }_{d}A_{u}A_{d}}{m^{0}}\left( 1+\frac{\overline{\mu }_{u}A_{u}}{2m^{0}} \right) +0.01A_{d}-0.005A_{u} \end{aligned}$$

(18)

$$\begin{aligned}&C-A \le \frac{\overline{\mu }_{u}A^{2}_{u}}{2m^{0}}+\frac{\overline{\mu }_{d}A^{2}_{d}}{2m^{0}}+\frac{\underline{\mu }_{d}A_{u}A_{d}}{m^{0}}\left( 1+\frac{\underline{\mu }_{u}A_{u}}{2m^{0}} \right) -0.015A_{d}+0.005A_{u}, \end{aligned}$$

(19)

where $A_{i}, i=u, d$ is the change in consumer surplus caused by price changes; $\overline{\mu }_{i}, i=u, d$ and $\underline{\mu }_{i}, i=u, d$ are the upper and lower bounds of the income elasticity of demand; and the subscripts u and d stand for “prices up” and “prices down.” In this estimation, Willig divided all goods into the “prices up” and the “prices down” categories and estimated the upper and lower bounds of the errors.

However, in comparison to Eqs. (16) and (17), Eqs. (18) and (19) have flaws. First, since Willig divided all goods into two types, the error margins of the approximations might be very large when more than two prices change. Second, the approximation of the utility function causes unnecessary errors, such as $0.01A_{d}$ and $0.005A_{u}$. Substituting zero into $A_{u}$ or $A_{d}$ in Eqs. (18) and (19), it is clear that (18) and (19) are not consistent with (9).

A more rigorous definition of consumption surface is given in Appendix A.1.

If $\varDelta x^{i, i}$ is small enough, the area enclosed by four points $(x^{i, i}, x^{i+1, i}, x^{i, i+1}, x^{i+1, i+1})$ is assumed to be a parallelogram. It is assumed that $\varDelta x^{i, i}, \forall i$ divides the parallelogram with four points $(x^{i, i}, x^{i+1, i}, x^{i, i+1}, x^{i+1, i+1})$ into two parts whose areas are the same.

The following equation corresponds to Corollary 2 in Appendix A.2.

In a more general form, the following equations also hold true:

$$\begin{aligned}&\log ( r(p^{k+n, l}, p^{k+n, l+m}) )-\log ( r(p^{k, l}, p^{k, l+m}) ) =-\sum _{j=1}^{m}\sum _{i=1}^{n}\psi ^{k+i-1, l+j-1} \\&\quad \frac{r(p^{k+n, l}, p^{k+n, l+m}) }{ r(p^{k, l}, p^{k, l+m})}=\exp \left( -\sum _{j=1}^{m}\sum _{i=1}^{n}\psi ^{k+i-1, l+j-1} \right) . \end{aligned}$$

See Appendix B.

The corresponding Laspeyres–Konüs and Paasche–Konüs true cost of living price indices are stated in Appendix C.2.

Since it is apparent that if a property about consumer surplus, compensating variation, or equivalent variation holds true, a similar property about the corresponding price and quantity indices holds true from the discussion in Appendix C.1. Hereafter in this subsection, I focus on the properties of consumer surplus, compensating variations, and equivalent variations and their calculation methods.

See Appendix A.

For the same reason, this paper’s methods for calculating true cost of living indices are able to give superior estimates to Diewert’s superlative indices, even though they need more information about the demand and inverse demand function than Diewert’s, as explained below.

It is also important to note that since this example has only one parameter, $\theta $, the specification is really restrictive; thus, unless you are confident about the functional form of the underlying utility function, this CES functional form might cause large misestimation of compensating and equivalent variations regardless of whether price changes are small or large.

As Proposition 4 shows, when the prices change along an iso-price-ratio path, the change in consumer surplus, the compensating variation, and the equivalent variation are easily convertible to each other. And as Proposition 1 implies, the change in consumer surplus along an indifference price path is zero. Thus, these definitions make it easier to compare the change in consumer surplus, the compensating variation, and the equivalent variation when the prices change from $p^{1, 1}$ to $p^{2, 2}$.

As the definition of $S^{x}(p^{1, 1}, p^{2, 2})$ implies, there could be infinitely many smooth consumption surfaces generated by the price change from $p^{1, 1}$ to $p^{2, 2}$. Thus, we can choose any one of them. However, if the prices change along a price path, the smooth consumption surface that contains the price path should be chosen so that the relationships among the change in consumer surplus, the compensating variation, and the equivalent variation are easily analyzed.

x in $x_{w}(p(x), w)$ will be omitted when unnecessary.

We use the word ‘submanifold’ because a smooth consumption surface could be considered as a subset of the consumption space, which is a manifold.

The equation is the same as Eq. (1) so that the equation calculates changes in consumer surplus along consumption paths.

We choose the sign of Eq. (A-3) such that it is consistent with the sign of Eq. (A-4).

As I have said, we assume that the income level is normalized to one: $w=1$.

Equation (A-6) implies $p (x, w) \parallel q^{k} (x, w)$ for $k = 1, 2$, where $q^{k}(x, w)$ for $k= 1, 2$ is an N-dimensional vector and defined by

$$\begin{aligned} q_{i}^{k} (x, w) = \sum _{j = 1}^{N} \left( \frac{\partial p_{i} (x, w)}{\partial x_{j}} - \frac{\partial p_{j} (x, w)}{\partial x_{i}} \right) c_{j}^{k},\quad k=1, 2. \end{aligned}$$

Since we consider infinitely small change cases now, we assume that $p^{1, 1}\cdot \varDelta x=- \varDelta p \cdot x^{1, 1}$.

As M increases, $( 1+ \alpha )^{(1 / z)}-1$ approaches $(1 / M)\log (1+\alpha )$.

Since the income level along the price path is always one, $\sum _{j=1}^{N} x_{j}(q(i), w )q_{j}(i)=1, \forall i$ holds true.

Abraham KG (2003) Toward a cost-of-living index: progress and prospects. J Econ Perspect 17(1):45–58CrossRef

Abraham KG, Greenlees JS, Moulton BR (1998) Working to improve the consumer price index. J Econ Perspect 12(1):27–36CrossRef

Blackorby C, Donaldson D (1999) Market demand curves and Dupuit–Marshall consumers’ surpluses: a general equilibrium analysis. Math Soc Sci 37(2):139–163CrossRef

Boskin MJ, Dulberger ER, Gordon RJ, Griliches Z, Jorgenson DW (1998) Consumer prices, the consumer price index, and the cost of living. J Econ Perspect 12(1):3–26CrossRef

Bruce N (1977) A note on consumer’s surplus, the divisia index, and the measurement of welfare changes. Econometr J Econometr Soc 45(4):1033–1038CrossRef

Chipman JS, Moore JC (1976) The scope of consumer’s surplus arguments. In: Tang AM, Westfield FM, Worley JS (eds) Welfare and time in economics: essays in honor of Nicholas Georgescu-Roegen. Lexington Books, Lexington, pp 69–123

Deaton A, Heston A (2010) Understanding PPPs and PPP-based national accounts. Am Econ J Macroecon 2(4):1–35CrossRef

Diewert WE (1976) Exact and superlative index numbers. J Econometr 4(2):115–145CrossRef

Diewert WE (1988) The early history of price index research. https://doi.org/10.3386/w2713

Diewert WE (1998) Index number issues in the consumer price index. J Econ Perspect 12(1):47–58CrossRef

Dupuit J (1844) On the measurement of the utility of public works. Int Econ Pap 2:83–110 [translated and reprinted in Readings in Welfare Economics (Kenneth J. Arrow and Tibor Scitovsky, eds.), 1969, 255–283, Homewood]

Harberger AC (1971) Three basic postulates for applied welfare economics: an interpretive essay. J Econ Lit 9(3):785–797

Hausman JA (1981) Exact consumer’s surplus and deadweight loss. Am Econ Rev 71(4):662–676

Hausman JA (2003) Sources of bias and solutions to bias in the consumer price index. J Econ Perspect 17(1):23–44CrossRef

Hayashi T (2013) Smallness of a commodity and partial equilibrium analysis. J Econ Theory 148(1):279–305CrossRef

Hicks JR (1941) The rehabilitation of consumers’ surplus. Rev Econ Stud 8(2):108–116CrossRef

Hicks JR (1946) Value and capital: an inquiry into some fundamental principles of economic theory. Clarendon Press, Oxford

Hulten CR (1973) Divisia index numbers. Econometr J Econometr Soc 41(6):1017–1025CrossRef

Hurwicz L, Uzawa H (1971) On the integrability of demand functions. In: Chipman JS (ed) Preferences, utility, and demand. Harcourt Brace Jovanovich, New York, pp 114–148

Lebow DE, Rudd JB (2003) Measurement error in the consumer price index: where do we stand? J Econ Lit 41(1):159–201CrossRef

Marshall A (1961) Principles of economics, 9th edn. Macmillan, New York (first published in 1890)

Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory, vol 1. Oxford University Press, New York

McKenzie GW (1976) Measuring gains and losses. J Polit Econ 84(3):641–646CrossRef

McKenzie GW (1979) Consumer’s surplus without apology: comment. Am Econ Rev 69(3):465–468

McKenzie GW, Pearce IF (1976) Exact measures of welfare and the cost of living. Rev Econ Stud 43(3):465–468CrossRef

McKenzie GW, Pearce IF (1982) Welfare measurement: a synthesis. Am Econ Rev 72(4):669–682

Miyake M (2006) On the applicability of Marshallian partial-equilibrium analysis. Math Soc Sci 52(2):176–196CrossRef

Moulton BR (1996) Bias in the consumer price index: what is the evidence? J Econ Perspect 10(4):159–177CrossRef

Nordhaus WD (1998) Quality change in price indexes. J Econ Perspect 12(1):59–68CrossRef

Persky J (1998) Retrospectives: price indexes and general exchange values. J Econ Perspect 12(1):197–205CrossRef

Pollak RA (1998) The consumer price index: a research agenda and three proposals. J Econ Perspect 12(1):69–78CrossRef

Schultze CL (2003) The consumer price index: conceptual issues and practical suggestions. J Econ Perspect 17(1):3–22CrossRef

Silberberg E (1972) Duality and the many consumer’s surpluses. Am Econ Rev 62(5):942–952

Vives X (1987) Small income effects: a Marshallian theory of consumer surplus and downward sloping demand. Rev Econ Stud 54(1):87–103CrossRef

Willig RD (1976) Consumer’s surplus without apology. Am Econ Rev 66(4):589–597

Willig RD (1979) Consumer’s surplus without apology: reply. Am Econ Rev 69(3):469–474

Titel: Welfare evaluations and price indices with path dependency problems
verfasst von: Tsuyoshi Sasaki
Publikationsdatum: 02.07.2018
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 1/2019
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1142-4

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