Welfare evaluations and price indices with path dependency problems

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.

Appendices

Appendix A: Proofs and explanations using manifold and differential forms

1.1 A.1 Definitions and Stokes’ theorem

Let M be a two dimensional smooth submanifold with boundary that is embedded in an N-dimensional Euclidean space $\mathbb {R}^{N}$, and let $\partial M$ be the boundary of the submanifold M. Define $T_{x}M$ as the tangent space of M at x, and define $T^{*}_{x}M$ as the cotangent space of M at x. Let $TM=\bigcup _{x \in M} T_{x}M$ be the tangent bundle, and let $T^{*}M=\bigcup _{x \in M}T^{*}_{x}M$ be the cotangent bundle.

Here, well known basic results on manifold and differential forms are explained and described. Let $\omega :M \rightarrow T^{*}M$ be a 1-form on M. Let $x_{1}, x_{2}, x_{3}, \ldots , x_{N}$ be the standard coordinates on $\mathbb {R}^{N}$. At each point $x\in M$, the value of the 1-form $\omega$ can be written as a linear combination:

$$\begin{aligned} \omega _{x}= \sum _{i=1}^{N}a_{i}(x)dx_{i}, \end{aligned}$$

where $dx_{1}, dx_{2}, dx_{3}, \ldots , dx_{N}$ are the basis for the cotangent space $T^{*}_{x}M$. The exterior derivative of $\omega$ is written as

$$\begin{aligned} d\omega =\sum _{i<j}\left( \frac{\partial a_{j}(x)}{\partial x_{i}}-\frac{\partial a_{i}(x)}{\partial x_{j}} \right) dx_{i}\wedge dx_{j}. \end{aligned}$$

From Stokes’ theorem, the following equation holds true:

$$\begin{aligned} \int _{M}d\omega =\int _{\partial M} \omega . \end{aligned}$$

(A-1)

We define smooth consumption surfaces by first defining smooth price surfaces because it is easy to define smooth price surfaces first and then define smooth consumption surfaces using smooth price surfaces in a roundabout way. Given a price pair $(p^{1, 1}, p^{2, 2})\in \mathbb {R}_{++}^{N} \times \mathbb {R}_{++}^{N}$ and the two price bundles $(p^{1, 2}, p^{2, 1})\in \mathbb {R}_{++}^{N} \times \mathbb {R}_{++}^{N}$ satisfying $p^{1, 2}=V^{p}(p^{2, 2})\cap R(p^{1, 1})$ and $p^{2, 1}=V^{p}(p^{1, 1})\cap R(p^{2, 2})$, we consider smooth price surfaces with boundary surrounded by the four prices $(p^{1, 1}, p^{1, 2}, p^{2, 1}, p^{2, 2})$. Let $S^{p}(p^{1, 1}, p^{2, 2})$ be a smooth price surface in an N-dimensional price space with boundary generated by a price pair $(p^{1, 1}, p^{2, 2})\in \mathbb {R}_{++}^{N} \times \mathbb {R}_{++}^{N}$ that satisfies the following three conditions:

• (1) $S^{p}(p^{1, 1}, p^{2, 2})$ is surrounded by the four prices $(p^{1, 1}, p^{1, 2}, p^{2, 1}, p^{2, 2})$ and accompanying four boundary curves.

• (2) The first and second boundary curves are two iso-price-ratio curves from $p^{1, 1}$ to $p^{1, 2}$ and $p^{2, 1}$ to $p^{2, 2}$:

  $$\begin{aligned} \{ p\in \mathbb {R}_{++}^{N}|p\in R(p^{1, 1}), U(x(p^{2, 2}))\ge U(x(p)) \ge U(x(p^{1, 1})) \} \end{aligned}$$

  and

  $$\begin{aligned} \{ p\in \mathbb {R}_{++}^{N}|p\in R(p^{2, 2}), U(x(p^{2, 2}))\ge U(x(p)) \ge U(x(p^{1, 1})) \}, \end{aligned}$$

  respectively.

• (3) The third and fourth boundary curves are a pair of two indifference price paths, which are denoted as $C_{p}^{D}(i; p^{1, 1}, p^{2, 1})$ from $p^{1, 1}$ to $p^{2, 1}$ and $C_{p}^{D}(i; p^{1, 2}, p^{2, 2})$ from $p^{1, 2}$ to $p^{2, 2}$, that satisfy

  $$\begin{aligned} C_{p}^{D}(i; p^{1, 2}, p^{2, 2})\in R(q), \;\;q=C_{p}^{D}(i; p^{1, 1}, p^{2, 1}), \forall i. \end{aligned}$$

These four curves are chosen so that the integrations along the four curves are closely related to the compensating and equivalent variations.³³ Define $S^{x}(p^{1, 1}, p^{2, 2})=\{x\in \mathbb {R}_{+}^{N}| x=x(p), p\in S^{p}(p^{1, 1}, p^{2, 2}) \}$ as a smooth consumption surface in an N-dimensional consumption space with boundary generated by a price pair $(p^{1, 1}, p^{2, 2})\in \mathbb {R}_{++}^{N} \times \mathbb {R}_{++}^{N}$. Given a smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$, we consider a smooth consumption curve on the smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$:

$$\begin{aligned} C_{x}(i; x^{1, 1}, x^{2, 2})\in S^{x}(p^{1, 1}, p^{2, 2}),\;s\le i \le t, \end{aligned}$$

where $x^{1, 1}=x(p^{1, 1})$ and $x^{2, 2}=x(p^{2, 2})$.

1.2 A.2 Propositions and lemmas

Assume that the prices change from $p^{1, 1}$ to $p^{2, 2}$. Consider a smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$.³⁴ In order to calculate surface integrals, we need to choose two tangent vectors on the smooth consumption surface at each point. We take $x_{w}(p(x), w)$ as the first tangent vector on the smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$ at x.³⁵$x_{w}(p, w)$ is clearly one of the tangent vectors on the smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$, because, under no money illusion, a change in the wealth level is equivalent to the corresponding proportional price change, that is, a price change along an iso-price-ratio curve.

Let $c(x)=(c_{1}, c_{2}, c_{3}, \ldots , c_{N})\in \mathbb {R}^{N}$ be the second tangent vector that satisfies the following conditions:

• (1) At each consumption bundle x on $S^{x} (p^{1, 1}, p^{2, 2})$, there are two directions to which x can move while keeping it still on both $S^{x} (p^{1, 1}, p^{2, 2})$ and the indifference consumption hypersurface $V^{x} (x)$. Let $C_{x}(i; y, z)$ be a consumption curve from $y \in R(p^{1, 1})$ via x to $z \in R(p^{2, 2})$ each point of which is located on both $S^{x}(p^{1, 1}, p^{2, 2})$ and $V^{x}(x)$. I choose the direction of c(x) such that c(x) and $\partial C_{x}(i; y, z) / \partial i$ have the same direction, where where $x=C_{x}(i; y, z)$. Intuitively speaking, the direction of c(x) is chosen such that it points from $R(p^{1, 1})$ to $R(p^{2, 2})$ while keeping it on both $S^{x} (p^{1, 1}, p^{2, 2})$ and $V^{x} (x)$.

• (2) $|c(x)|=1$.

By construction, $c(x)\cdot p(x)=0$ holds true. In sum, we take $x_{w}(p(x), w)$ and c(x) as the two tangent vectors at each point on the smooth consumption surface $S^{x}(p^{1, 1},p^{2, 2})$.

We take the smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$ as a two dimensional smooth submanifold M with boundary.³⁶ The boundary $\partial M$ can be divided into two consumption curves from $x^{1, 1}=x(p^{1, 1})$ to $x^{2, 2}=x(p^{2, 2})$. Let $C_{x}^{1}(i; x^{1, 1}, x^{2, 2})$ be the consumption curve from $x^{1, 1}$ via $x^{2, 1}=x(p^{2, 1}), p^{2, 1}=V^{p}(p^{1, 1})\cap R(p^{2, 2})$ to $x^{2, 2}$ on the boundary $\partial M$, and let $C_{x}^{2}(i; x^{1, 1}, x^{2, 2})$ be the consumption curve from $x^{1, 1}$ via $x^{1, 2}=x(p^{1, 2}), p^{1, 2}=V^{p}(p^{2, 2})\cap R(p^{1, 1})$ to $x^{2, 2}$ on the boundary $\partial M$: $C^{j}_{x}(i; x^{1, 1}, x^{2, 2})\in \partial M, \forall i, j$.

Proposition 9

Given a smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$, the following holds true:

$$\begin{aligned} \varDelta CS_{1} - \varDelta CS_{2}=\int _{S}\tau (x)dS, \end{aligned}$$

(A-2)

where

$$\begin{aligned} \tau (x)= & {} -\frac{1}{\mu (c(x), x_{w}(p, w))} \sum _{i = 1}^{N} \sum _{j = i}^{N}c_{i} (x)\\&\times \left( \frac{\partial p_{i} (x, w)}{\partial x_{j}} - \frac{\partial p_{j} (x, w)}{\partial x_{i}} \right) \frac{\partial x_{j}(p, w)}{\partial w} \\ \varDelta CS_{1}= & {} \int _{C_{x}^{1}}\sum _{i=1}^{N}p_{i}(x)dx_{i},\;\varDelta CS_{2}=\int _{C_{x}^{2}}\sum _{i=1}^{N}p_{i}(x)dx_{i}, \\ \mu (c(x), x_{w}(p, w))= & {} \sqrt{|c(x)|^{2}|x_{w}(p, w)|^{2}-(c(x)\cdot x_{w}(p, w))^{2}}, \end{aligned}$$

and we integrate from $x^{1, 1}$ to $x^{2, 2}$.

Proof

When the value of a 1-form $\omega$ on $M=S^{x}(p^{1, 1}, p^{2, 2})$ at x is written as $\omega _{x}=\sum _{i=1}^{N}p_{i}(x)dx_{i}$,³⁷ the integration of the 1-form $\omega$ on the boundary $\partial M$ is written as:³⁸

$$\begin{aligned} \int _{\partial M} \omega =\varDelta CS_{1} - \varDelta CS_{2}, \end{aligned}$$

(A-3)

where we integrate from $x^{1, 1}$ to $x^{2, 2}$. $\varDelta CS_{1}$ and $\varDelta CS_{2}$ can be considered as the changes in consumer surplus along the respective integration paths. Thus, the right side of Eq. (A-3), $\varDelta CS_{1} - \varDelta CS_{2}$, can be considered as the difference between the change in consumer surplus along $C_{x}^{1}$ and that along $C_{x}^{2}$. From Appendix A.3, the integration of $d\omega$, denoted by $\int _{M}d\omega$, is written as

$$\begin{aligned}&\int _{S}\frac{1}{\mu (c(x), x_{w}(p, w))} \nonumber \\&\quad \sum _{i<j}\left( \frac{p_{j}(x, w)}{\partial x_{i}}- \frac{p_{i}(x, w)}{\partial x_{j}} \right) \left( \frac{\partial x_{j}(p, w)}{\partial w}c_{i}(x)- \frac{\partial x_{i}(p, w)}{\partial w}c_{j}(x) \right) dS. \end{aligned}$$

(A-4)

Arranging Eq. (A-4) and using Eq. (A-1), we get

$$\begin{aligned} \int _{M}d\omega = -\int _{S}\frac{1}{\mu (c(x), x_{w}(p, w))} \sum _{i = 1}^{N} \sum _{j = i}^{N} c_{i} (x) \left( \frac{\partial p_{i} (x, w)}{\partial x_{j}} - \frac{\partial p_{j} (x, w)}{\partial x_{i}} \right) \frac{\partial x_{j}(p, w)}{\partial w} d S \end{aligned}$$

Thus, we get Proposition 9. $\square$

Let $\mathscr {C}_{x}(S^{x}(p^{1, 1}, p^{2, 2}))$ be the set of all the smooth consumption curves from $x^{1, 1}$ to $x^{2, 2}$ on a smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 2})$. We can generalize Proposition 9 as follows.

Corollary 1

Consider a submanifold $M_{1} \subset M=S^{x}(p^{1, 1}, p^{2, 2})$ whose boundary $\partial M_{1}$ consists of two smooth consumption curves $C_{x}^{1, i}\in \mathscr {C}(S^{x}(p^{1, 1}, p^{2, 2})), i=1, 2$ that do not intersect each other. The difference between the change in consumer surplus along $C_{x}^{1, 1}$ and that along $C_{x}^{1, 2}$ is written as

$$\begin{aligned} \varDelta CS_{1, 1}-\varDelta CS_{1, 2}&=\int _{M_{1}}\tau (x)dS, \end{aligned}$$

where

$$\begin{aligned} \varDelta CS_{1, 1}=\int _{C_{x}^{1, 1}}\sum _{i=1}^{N}p_{i}(x)dx_{i},\quad \varDelta CS_{1, 2}=\int _{C_{x}^{1, 2}}\sum _{i=1}^{N}p_{i}(x)dx_{i}, \end{aligned}$$

and we integrate from $x^{1, 1}=x(p^{1, 1})$ to $x^{2, 2}=x(p^{2, 2})$.

Proof

This is proven in the same way as in the proof of Proposition 9. $\square$

Let $C_{x}^{0}(i; x^{1, 1}, x^{2, 2})$ be the observed consumption path. Along the consumption path $C_{x}^{0}(i; x^{1, 1}, x^{2, 2})$, the change in consumer surplus can be calculated. Consider the smooth consumption surface $S^{x}(p^{1, 1}, p^{2, 1}), p^{1, 1}=p(x^{1, 1}), p^{2, 2}=p(x^{2, 2})$ that contains all points of the consumption path $C_{x}^{0}(i; x^{1, 1}, x^{2, 2})$. By using Proposition 4 and Corollary 1, the compensating and equivalent variations are calculated from the observed change in consumer surplus if we know the Marshallian demand function.

The scalar field $\tau (x)$ can be expressed in different ways.

Lemma 1

The scalar field $\tau (x)$ can be rewritten as

$$\begin{aligned} \tau (x)&= \frac{1}{\mu (c(x), x_{w}(p, w))}\sum _{i = 1}^{N} \sum _{j = 1}^{N} c_{i} (x) \frac{\partial p_{j} (x, w)}{\partial x_{i}} \frac{\partial x_{j} (p, w)}{\partial w}\\&=-\frac{1}{\mu (c(x), x_{w}(p, w))} \sum _{i = 1}^{N} \sum _{j = 1}^{N} c_{i} (x) \frac{\partial p_{i} (x, w)}{\partial x_{j}} x_{j}. \end{aligned}$$

Proof

From Mas-Colell et al. (1995),

$$\begin{aligned} p_{i} + \sum _{j = 1}^{N} \frac{\partial p_{i} (x, w)}{\partial x_{j}}\frac{\partial x_{j} (p, w)}{\partial w} = 0, \quad p_{i} + \sum _{j = 1}^{N} \frac{\partial p_{j} (x, w)}{\partial x_{i}} x_{j} =0 \end{aligned}$$

hold true.³⁹ Observe that

$$\begin{aligned} \sum _{i = 1}^{N} \sum _{j = 1}^{N} c_{i} (x) \frac{\partial p_{i} (x, w)}{\partial x_{j}} \frac{\partial x_{j} (p, w)}{\partial w} = 0, \quad \sum _{i = 1}^{N} \sum _{j = 1}^{N} c_{i} (x) \frac{\partial p_{j} (x, w)}{\partial x_{i}} x_{j} = 0 \end{aligned}$$

hold true. Here, if

$$\begin{aligned} \sum _{i = 1}^{N} \sum _{j = 1}^{N} c_{i} (x) \left( \frac{\partial p_{i} (x, w)}{\partial x_{j}} -\frac{\partial p_{j} (x, w)}{\partial x_{i}} \right) \left( \frac{\partial x_{j} (p, w)}{\partial \ w} - x_{j} \right) =0 \end{aligned}$$

(A-5)

hold true, then Lemma 1 must hold true. Thus, we prove Eq. (A-5) instead. When $x_{w}(p, w)-x=0$, it is obvious. The other cases are proven as follows: $x_{w}(p, w)$ can be written as $x_{w}(p, w)=x+c$, where c is a column vector and orthogonal to p because the inner product of p and both sides of $x_{w}(p, w)=x+c$ are written as $p\cdot x_{w}(p, w)=p\cdot (x+c)\Longrightarrow 1=1+0=1$. Thus, $x_{w}(p, w)-x$ is orthogonal to p unless x and $x_{w}(p, w)$ are the same. Consider a smooth surface S with boundary on an indifference consumption hypersurface. S can be considered a two-dimensional smooth submanifold M. Assume that the boundary of M, $\partial M$, consists of a simple closed curve C. When the value of a 1-form $\omega$ on M at x is written as $\omega _{x}=\sum _{i=1}^{N}p_{i}(x)dx_{i}$, the integration of the 1-form $\omega$ on the boundary $\partial M$ is written as $\int _{\partial M}\omega =0$ because the smooth surface M is on an indifference consumption hypersurface; thus, the change in consumer surplus is zero. The integration of the exterior derivative of $\omega$ is written as⁴⁰

$$\begin{aligned} \int _{M}d\omega = \int _{S}A \sum _{i = 1}^{N }\sum _{j = 1}^{N} c_{i}^{1} (x) \left( \frac{\partial p_{i} (x, w)}{\partial x_{j}} -\frac{\partial p_{j} (x, w)}{\partial x_{i}} \right) c_{j}^{2} (x) dS=0, \end{aligned}$$

(A-6)

where A is a scalar, and $c^{1}(x)$ and $c^{2}(x)$ are tangent vectors at x on the smooth surface M and are not parallel. By substituting c(x) for $c^{1}(x)$ and $x_{w}(p, w)-x$ for $c^{2}(x)$ and factoring in the fact that all derivatives of the inverse demand function are continuous, Lemma 1 is proven. $\square$

Corollary 2

Given a smooth surface $S=S^{x}(p^{1, 1}, p^{2, 2})$, the following holds true:

$$\begin{aligned} \frac{r(p^{2, 1}, p^{2, 2})}{r(p^{1, 1}, p^{1, 2})}&=\exp \left[ \int _{S}\tau (x)dS \right] . \end{aligned}$$

Proof

From Propositions 1 and 2, the changes in consumer surplus, $\varDelta CS_{1}$ and $\varDelta CS_{2}$, are written as $\varDelta CS_{1}=\log (r(p^{2, 1}, p^{2, 2}))$ and $\varDelta CS_{2}=\log (r(p^{1, 1}, p^{1, 2}))$. Exponentiating the both sides of Proposition 9, we get Corollary 2. $\square$

1.3 A.3 Calculation of Eq. (A-4)

When a basic two form $dx_{i}\wedge dx_{j}$ accepts as input two vectors $v^{1}\in \mathbb {R^{N}}$ and $v^{2}\in \mathbb {R^{N}}$, the output is written as

$$\begin{aligned} dx_{i}\wedge dx_{j}(v^{1}, v^{2})=\det \left[ \begin{array}{cc} dx_{i}(v^{1})&{} dx_{i}(v^{2}) \\ dx_{j}(v^{1})&{} dx_{j}(v^{2}) \end{array} \right] =\det \left[ \begin{array}{cc} v^{1}_{i}&{}v^{2}_{i} \\ v^{1}_{j}&{}v^{2}_{j} \end{array} \right] =v^{1}_{i}v^{2}_{j}-v^{1}_{j}v^{2}_{i}, \end{aligned}$$

where $v^{1}_{i}$ and $v^{2}_{i}$ are the i-th elements of $v^{1}$ and $v^{2}$ and $v^{1}_{j}$ and $v^{2}_{j}$ are the j-th elements of $v^{1}$ and $v^{2}$, respectively. Thus, the integration of $d\omega$ is written as

$$\begin{aligned} \int _{M}d\omega= & {} \int _{S}\frac{1}{\mu (c(x), x_{w}(p, w))}d\omega (c(x), x_{w}(p, w)) \\= & {} \int _{S}\frac{1}{\mu (c(x), x_{w}(p, w))}\sum _{i<j}\left( \frac{\partial p_{j}(x, w)}{\partial x_{i}}-\frac{\partial p_{i}(x, w)}{\partial x_{j}} \right) dx_{i}\wedge dx_{j}(c(x), x_{w}(p, w))\\= & {} \int _{S}\frac{1}{\mu (c(x), x_{w}(p, w))}\sum _{i<j}\left( \frac{\partial p_{j}(x, w)}{\partial x_{i}}-\frac{\partial p_{i}(x, w)}{\partial x_{j}} \right) \det \left[ \begin{array}{cc} c_{i}(x)&{}\frac{\partial x_{i}(p, w)}{\partial w} \\ c_{j}(x)&{}\frac{\partial x_{j}(p, w)}{\partial w} \end{array} \right] dS \\= & {} \int _{S}\frac{1}{\mu (c(x), x_{w}(p, w))}\sum _{i<j}\left( \frac{p_{j}(x, w)}{\partial x_{i}}- \frac{p_{i}(x, w)}{\partial x_{j}} \right) \\&\times \left( \frac{\partial x_{j}(p, w)}{\partial w}c_{i}(x)- \frac{\partial x_{i}(p, w)}{\partial w}c_{j}(x) \right) dS, \end{aligned}$$

where $\mu (c(x), x_{w}(p, w))$ is the normalization parameter that normalizes the area of a parallelogram formed by c(x) and $x_{w}(p, w)$ to one, that is, $\mu (c(x), x_{w}(p, w))$ is the reciprocal of the area of a parallelogram formed by c(x) and $x_{w}(p, w)$.

1.4 A.4 Proof of Proposition 7

We need to know $\tau (x)$ at $x^{1, 1}$ and the area of $S^{x}(p^{1, 1}, p^{2, 2})$ to prove Eq. (6). First, $c(x^{1, 1})$ can be written as a linear combination of $x_{w}^{1, 1}$ and $\varDelta x$: $c(x^{1, 1})=a_{1}x_{w}^{1, 1}+a_{2}\varDelta x$, where $a_{1}$ and $a_{2}$ are scalars. The inner product of $c(x^{1, 1})$ and $p^{1, 1}$ is $p^{1, 1}\cdot c(x^{1, 1})=a_{1}+a_{2}p^{1, 1}\cdot \varDelta x=0$. Thus, we get $a_{1}=-a_{2}(p^{1, 1}\cdot \varDelta x)$. Then, part of $\tau (x^{1, 1})$ is calculated as⁴¹

$$\begin{aligned}&\sum _{i = i}^{N}\sum _{j = 1}^{N} \frac{\partial x_{i}^{1, 1} }{\partial w}\frac{\partial p_{i}^{1, 1} }{\partial x_{j}} c_{j} = \sum _{i = i}^{N}\sum _{j = 1}^{N} \frac{\partial x_{i}^{1, 1} }{\partial w}\frac{\partial p_{i}^{1, 1} }{\partial x_{j}} \left( a_{1} \frac{\partial x_{j}^{1, 1} }{\partial w} + a_{2} \varDelta x_{j} \right) \\&\quad = \sum _{i = 1}^{N} \frac{\partial x_{i}^{1, 1}}{\partial w} ( - a_{1} p_{i}^{1, 1} + a_{2} \varDelta p_{i} ) = a_{2} (p^{1, 1}\cdot \varDelta x+\varDelta p \cdot x_{w}^{1, 1} ) \\&\quad =a_{2}\varDelta p\cdot (x_{w}^{1, 1}-x^{1, 1} ), \end{aligned}$$

where $\partial x_{i}^{1, 1}/ \partial w =\partial x_{i} (p^{1, 1}, w) / \partial w$, $\partial p_{i}^{1, 1} / \partial x_{j} = \partial p_{i} (x^{1, 1}, w) / \partial x_{j}$, and

$$\begin{aligned} \sum _{j = 1}^{N} \frac{\partial p_{i}^{1, 1} }{\partial x_{j}} \frac{\partial x_{j}^{1, 1} }{\partial w} = -p_{i}^{1, 1},\quad \sum _{j = 1}^{N} \frac{\partial p_{i}^{1, 1} }{\partial x_{j}} \varDelta x_{j} = \varDelta p_{i}, \quad \sum _{i = 1} \frac{\partial x_{i}^{1, 1}}{\partial w} p_{i}^{1, 1} = 1. \end{aligned}$$

Next, $\varDelta x$ can be written as a linear combination of $x_{w}(p, w)$ and c(x):

$$\begin{aligned} \varDelta x=b_{1}x_{w}^{1, 1}+b_{2}c(x^{1, 1})=(b_{1}-a_{2}b_{2}(p^{1, 1}\cdot \varDelta x))x_{w}^{1, 1}+a_{2}b_{2}\varDelta x, \end{aligned}$$

where $b_{1}$ and $b_{2}$ are scalars. Thus, $a_{2}b_{2}=1$ and $b_{1}=(p^{1, 1}\cdot \varDelta x)$ must hold true. This means that $b_{1}=p^{1, 1}\cdot \varDelta x$ and $b_{2}= 1 / a_{2}$. The area of $S^{x}(p^{1, 1}, p^{2, 2})$ is written as

$$\begin{aligned} S=b_{1}b_{2}\sqrt{|c(x^{1, 1})|^{2}|x_{w}^{1, 1}|^{2}-(c(x^{1, 1})\cdot x_{w}^{1, 1})^{2}}=b_{1}b_{2}\mu (c(x^{1, 1}), x_{w}^{1, 1}). \end{aligned}$$

Thus, the difference between the change in consumer surplus along $C_{x}^{1}(i; x^{1, 1}, x^{2, 2})$ and that along $C_{x}^{2}(i; x^{1, 1}, x^{2, 2})$ is written as

$$\begin{aligned} \int _{S}\tau (x)dS&=b_{1}b_{2}\mu (c(x^{1, 1}), x_{w}^{1, 1})\times \frac{1}{\mu (c(x^{1, 1}), x_{w}^{1, 1})}a_{2}\varDelta p\cdot (x_{w}^{1, 1}-x^{1, 1})\\&=-\ (\varDelta p\cdot x^{1, 1})\{ \varDelta p\cdot (x_{w}^{1, 1}-x^{1, 1}) \}. \end{aligned}$$

When $\varDelta p$ is small enough, the area surrounded by $C_{x}^{1}(i; x^{1, 1}, x^{2, 2})$ and $C_{x}^{2}(i; x^{1, 1}, x^{2, 2})$ can be considered as a parallelogram. Since the straight line $C_{0}(i; x^{1, 1}, x^{2, 2})$ from $x^{1, 1}$ to $x^{2, 2}$ divides the parallelogram into two triangles whose areas are the same, Eqs. (7) and (8) hold true.

Appendix B: Calculation of Eq. (23)

Choose $p^{i, i}, \forall i$ such that $\varDelta CS_{0, i}=\varDelta CS_{0, j}$ holds true for all i and j. When M is large enough, $\log \left( r(p^{i+1, i}, p^{i+1, i+1})\right)$ can be approximated by $\varDelta CS_{0, i}$. Thus, $\varDelta CS_{1}$ is approximately written as

$$\begin{aligned} \varDelta CS_{1}\approx \sum _{i=1}^{M-1} \varDelta CS_{0, i} \prod _{j=1}^{M-i-1} (1-\eta ^{i+j, i})= \sum _{i=1}^{M-1} \frac{\varDelta CS_{0}}{M-1} \prod _{j=1}^{M-i-1} (1-\eta ^{i+j, i}). \end{aligned}$$

When $\eta ^{i, j}, \forall i, j$ are constant and all the same, the above equation is rewritten as

$$\begin{aligned} \varDelta CS_{1}\approx \sum _{i=1}^{M-1}(1-\eta )^{i}\frac{\varDelta CS_{0}}{M-1}, \end{aligned}$$

(A-7)

where $\eta =\eta ^{i, j}, \forall i, j$. Since $\varDelta p^{i+1, i} \approx \varDelta p^{j, j+1}, \forall i, j$ holds true, $\eta$ is approximated by $( 1+ \alpha )^{M-1}-1$, where $\alpha =\varDelta p \cdot (x_{w}^{1, 1}-x^{1, 1})$. Thus, when M increases, Equation (A-7) is calculated as⁴²

$$\begin{aligned} \varDelta CS_{1}\approx \lim _{z \rightarrow \infty } \sum _{i=1}^{z}\left\{ 2-( 1+ \alpha )^{\frac{1}{z}} \right\} ^{i}\frac{\varDelta CS_{0}}{z}=\frac{\alpha }{\alpha +1}\frac{\varDelta CS_{0}}{\log (1+\alpha )}. \end{aligned}$$

Using the same reasoning, $\varDelta CS_{2}$ is calculated as $\alpha (\varDelta CS_{0} / \log (1+\alpha ))$.

Appendix C: Application to price indices

1.1 C.1 Relationship between welfare evaluations and price indices

This subsection shows the relationships between price indices and measures of consumer welfare change. The following relationships have been already indicated or pointed out by Hicks (1941), Hulten (1973), Bruce (1977), and others. However, it is important to formally state the following relationships between price indices and measures of welfare change to make it easier to understand the main results in this paper. Assume that the prices and the consumption bundle change from $(p^{1}, x^{1})$ at time 1 to $(p^{2}, x^{2})$ at time 2, where $p^{1}\cdot x^{1}=w^{1}$ and $p^{2}\cdot x^{2}=w^{2}$ hold. When these changes are caused by a policy change, the change in economic welfare can be evaluated by using consumer surplus, compensating variation, or equivalent variation. Also, the price and quantity indices at time 1 and 2 can be calculated.

Let $P_{i}(p^{1}, p^{2}, x^{1}, x^{2})$ be the true cost of living price index, and let $Q_{i}(p^{1}, p^{2}, x^{1}, x^{2})$ be the corresponding implicit quantity index. All the price and quantity indices at time 1 are assumed to be normalized to one. The Laspeyres–Konüs true cost of living price index is written as

$$\begin{aligned} P_{L}(p^{1}, p^{2}, x^{1}, x^{2})= \frac{e(p^{2}, v(p^{1}, w^{1}))}{e(p^{1}, v(p^{1}, w^{1}))}, \end{aligned}$$

(A-8)

and the Paasche–Konüs true cost of living price index is written as

$$\begin{aligned} P_{P}(p^{1}, p^{2}, x^{1}, x^{2})= \frac{e(p^{2}, v(p^{2}, w^{2}))}{e(p^{1}, v(p^{2}, w^{2}))}. \end{aligned}$$

(A-9)

The Laspeyres–Konüs true cost of living price index and its corresponding implicit quantity index, denoted by $Q_{L}(p^{1}, p^{2}, x^{1}, x^{2})$, satisfy the following equation:

$$\begin{aligned} P_{L}(p^{1}, p^{2}, x^{1}, x^{2})Q_{L}(p^{1}, p^{2}, x^{1}, x^{2})=\frac{w^{2}}{w^{1}}. \end{aligned}$$

(A-10)

The Paasche–Konüs true cost of living price index and its corresponding implicit quantity index, denoted by $Q_{P}(p^{1}, p^{2}, x^{1}, x^{2})$, satisfy the following equation:

$$\begin{aligned} P_{P}(p^{1}, p^{2}, x^{1}, x^{2})Q_{P}(p^{1}, p^{2}, x^{1}, x^{2})=\frac{w^{2}}{w^{1}}. \end{aligned}$$

(A-11)

When income level w is normalized to one, the followings hold:

$$\begin{aligned}&CV=1-e(p^{2}, v(p^{1}, w)),\quad EV=e(p^{1}, v(p^{2}, w))-1 \\&P_{L}(p^{1}, p^{2}, x^{1}, x^{2})= e(p^{2}, v(p^{1}, w)), \quad P_{P}(p^{1}, p^{2}, x^{1}, x^{2})= \frac{1}{e(p^{1}, v(p^{2}, w))}\\&P_{L}(p^{1}, p^{2}, x^{1}, x^{2})Q_{L}(p^{1}, p^{2}, x^{1}, x^{2})= P_{P}(p^{1}, p^{2}, x^{1}, x^{2})Q_{P}(p^{1}, p^{2}, x^{1}, x^{2})=1. \end{aligned}$$

Thus, rearranging the above equations, we get the following equations:

$$\begin{aligned} CV&=1-P_{L}(p^{1}, p^{2}, x^{1}, x^{2})=1-\frac{1}{Q_{L}(p^{1}, p^{2}, x^{1}, x^{2})} \end{aligned}$$

(A-12)

$$\begin{aligned} EV&= \frac{1}{P_{P}(p^{1}, p^{2}, x^{1}, x^{2})} -1=Q_{P}(p^{1}, p^{2}, x^{1}, x^{2})-1. \end{aligned}$$

(A-13)

Next, we consider the relationship between the Divisia price index and the change in consumer surplus. Let $P_{D}(i)$ be the Divisia price index at time i. Assume that the prices and the consumption bundle continuously change from $(p^{1}, x^{1})$ at time $t=1$ to $(p^{2}, x^{2})$ at time $s=2$ along a price path $C_{p}(i, p^{1}, p^{2}), s\le i \le t$, and that income level w is normalized to one at each point along $C_{p}(i, p^{1}, p^{2})$. The Divisia price index at time 2 is written as⁴³

$$\begin{aligned} P_{D}(2)=\exp \left[ \int _{s}^{t}\frac{\sum _{j=1}^{N} x_{j}(q(i), w )dq_{j}(i)}{\sum _{j=1}^{N} x_{j}(q(i), w )q_{j}(i)} \right] =\exp \left[ \int _{s}^{t}\sum _{j=1}^{N} x_{j}(q(i), w )dq_{j}(i) \right] . \end{aligned}$$

(A-14)

From Eqs. (1) and (A-14), we get

$$\begin{aligned} P_{D}(2)=\exp (-\varDelta CS). \end{aligned}$$

(A-15)

Thus, when income level w is normalized to one at each time along a price path, the compensating variation is convertible with the Laspeyres–Konüs true cost of living price index and its corresponding implicit quantity index by using Eq. (A-12), the equivalent variation is convertible with the Paasche–Konüs true cost of living price index and its corresponding implicit quantity index by using Eq. (A-13), and the change in consumer surplus and the Divisia price index along the price path are convertible with each other by using Eq. (A-15). Therefore, even though we mainly deal with consumer surplus, compensating variation, and equivalent variation in the main text, the results provided in the main text are also applicable to these price and quantity indices.

1.2 C.2 Price indices

Under the assumption of Proposition 8, the Laspeyres–Konüs true cost of living price index and the Paasche–Konüs true cost of living price index are written as

$$\begin{aligned} P_{L}(p^{1, 1}, p^{2, 2}, x^{1, 1}, x^{2, 2})=\frac{1}{\exp \left( \varDelta CS_{1} \right) },\quad P_{P}(p^{1, 1}, p^{2, 2}, x^{1, 1}, x^{2, 2})=\frac{1}{\exp \left( \varDelta CS_{2} \right) }. \end{aligned}$$

In large price change cases in Sect. 2.5, the Laspeyres–Konüs true cost of living price index and the Paasche–Konüs true cost of living price index are written as

$$\begin{aligned}&P_{L}(p^{1, 1}, p^{M, M}, x^{1, 1}, x^{M, M})\approx \frac{1}{\exp \left( \frac{\alpha }{\alpha +1}\frac{\varDelta CS_{0}}{\log (1+\alpha )} \right) } \end{aligned}$$

(A-16)

$$\begin{aligned}&P_{P}(p^{1, 1}, p^{M, M}, x^{1, 1}, x^{M, M})\approx \frac{1}{\exp \left( \alpha \frac{\varDelta CS_{0}}{\log (1+\alpha )} \right) }. \end{aligned}$$

(A-17)