Inverse problems of demand analysis and their applications to computation of positively-homogeneous Konüs–Divisia indices and forecasting

Nikolay I. Klemashev; Alexander A. Shananin

doi:10.1515/jiip-2015-0015

Article

Inverse problems of demand analysis and their applications to computation of positively-homogeneous Konüs–Divisia indices and forecasting

Nikolay I. Klemashev and Alexander A. Shananin

Published/Copyright: May 21, 2015

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From the journal Journal of Inverse and Ill-posed Problems Volume 24 Issue 4

Abstract

According to Pareto’s theory of consumer demand a rational representative consumer should choose their consumption bundle as the solution of a mathematical programming problem of maximization of the utility function under their budget constraint. The inverse problem of demand analysis is to recover the utility function from the demand functions. The answer to the question of solvability of this problem is based on the revealed preference theory. If the problem is unsolvable, one should apply regularization procedure by introducing irrationality indices. When recovering the utility function one puts a priori requirements on it. In this paper, we suggest including positively-homogeneity property into these requirements. We compare the setups with and without this requirement both theoretically and empirically and provide evidence in favor of requiring this property when computing economic indices for consumer representing consumption behavior of large number of various households for large time intervals.

Keywords: Inverse problem; mathematical programming; revealed preference; economic indices; forecasting

MSC 2010: 91B16

Proof of Corollary 7.

Relations (2.10)–(2.11) imply that for all X∈int⁡ℝ+m and P∈ℝ+m such that Q⁢(P)>0,

1Q⁢(P⁢(X))⁢〈P⁢(X),X〉⩽1Q⁢(P)⁢〈P,X〉.

The division by Q⁢(P⁢(X)) is correct because if X∈int⁡ℝ+m, then F⁢(X)>0 and 〈P⁢(X),X〉>0. Therefore, (2.11) implies Q⁢(P⁢(X))>0. Substituting P with P⁢(Y) leads to

1Q⁢(P⁢(X))⁢〈P⁢(X),X〉⩽1Q⁢(P⁢(Y))⁢〈P⁢(Y),X〉 for all ⁢X,Y∈int⁡ℝ+m.

Therefore, λ⁢(X)=1Q⁢(P⁢(X)) satisfies (2.9). Since Q⁢(P⁢(X)) is continuous, so is λ⁢(X). ∎

Proof of Corollary 8.

Define F⁢(X) and Q⁢(P) by

F⁢(X)=inf⁡{λ⁢(Y)⁢〈P⁢(Y),X〉:Y∈ℝ+m},

Q⁢(P)=inf⁡{〈P,Y〉F⁢(Y):Y⩾0,F⁢(Y)>0},

where λ⁢(Y) satisfies (2.9). Then the functions F⁢(X) and Q⁢(P) satisfy (2.10). Since λ⁢(Y) satisfies (2.9), we have

F⁢(X)=λ⁢(X)⁢〈P⁢(X),X〉

and

Q⁢(P⁢(X))=inf⁡{〈P⁢(X),Y〉λ⁢(Y)⁢〈P⁢(Y),Y〉:Y⩾0,λ⁢(Y)⁢〈P⁢(Y),Y〉>0}

=1λ⁢(X)⁢inf⁡{λ⁢(X)⁢〈P⁢(X),Y〉λ⁢(Y)⁢〈P⁢(Y),Y〉:Y⩾0,λ⁢(Y)⁢〈P⁢(Y),Y〉>0}=1λ⁢(X).∎

Proof of Corollary 10.

We show that the functions F⁢(X) and Q⁢(P) defined as

F⁢(X)=F0⁢(F1⁢(X1),…,FK⁢(XK),XK+1),

Q⁢(P)=Q0⁢(Q1⁢(P1),…,QK⁢(PK),PK+1)

are from ΦH and satisfy (2.10) and (2.11). The fact that these functions are from ΦH is implied from the definition of the functions Qk and Fk (k=1,…,K+1). Inequality (2.10) follows from (2.12) and (2.14):

Q⁢(P)⁢F⁢(X)=Q0⁢(Q1⁢(P1),…,QK⁢(PK),PK+1)⁢F0⁢(F1⁢(X1),…,FK⁢(XK),XK+1)

⩽∑k=1KQk⁢(Pk)⁢Fk⁢(Xk)+〈PK+1,XK+1〉

⩽〈P,X〉

Identity (2.11) follows from (2.15). Theorem 6 implies that the inverse demand functions P⁢(X) are rationalizable in ΦH. Identities (2.16) and (2.17) are satisfied by the definition of the functions F⁢(X) and Q⁢(P). ∎

Proof of Theorem 11.

(1) ⇒ (2). This part represents a modified version of the part of the proof of Theorem 1 given in [44]. Let max⁡(I) be the index of maximal element with respect to the binary relation R*⁢(ω) which is the transitive closure of the binary relation R⁢(ω) defined on {Xt}t=1T×{Xt}t=1T as

Xt⁢R⁢(ω)⁢Xτ⇔〈Pt,Xt〉⩾ω⁢〈Pt,Xτ〉.

In other words,

for all ⁢t∈I,Xt⁢R*⁢(ω)⁢Xmax⁡(I)⟹Xmax⁡(I)⁢R*⁢(ω)⁢Xt.

Consider the following algorithm.

Algorithm

Output: A set of numbers Ut, λt>0 (t=1,…,T).

I={1,…,T}, B=∅.
Let m=max⁡(I).
Set E={t∈I:Xt⁢R*⁢(ω)⁢Xm}. If B=∅, set Um=λm=1 and go to (6). Otherwise go to (4).
Set
Um=mint∈E⁡minτ∈B⁡min⁡{Uτ+λτ⁢(ω⁢〈Pτ,Xt〉-〈Pτ,Xτ〉),Uτ}.
Set
λm=maxt∈E⁡maxτ∈B⁡max⁡{Uτ-Umω⁢〈Pt,Xτ〉-〈Pt,Xt〉,1}.
Set Ut=Um, λt=λm for all t∈E.
Set I=I∖E, B=B∪E. If I=∅, stop. Otherwise, go to (2).

Let us prove that if the trade statistics satisfies GARP⁢(ω), then this algorithm provides the solution to (2.18). The algorithm is an iterative process. We show that after each iteration of step (6) the constructed functions U and λ satisfy the corresponding inequalities of (2.18). Namely, we show that

Ut⩽Uτ+λτ⁢(ω⁢〈Pτ,Xt〉-〈Pτ,Xτ〉) for all τ∈B, t∈E,
Uτ⩽Ut+λt⁢(ω⁢〈Pt,Xτ〉-〈Pt,Xt〉) for all τ∈B, t∈E,
Ut⩽Uτ+λτ⁢(ω⁢〈Pτ,Xt〉-〈Pτ,Xτ〉) for all t,τ∈E, t≠τ.

Proof of (a). By step (4) of the algorithm,

Ut=Um⩽Uτ+λτ⁢(ω⁢〈Pτ,Pt〉-〈Pτ,Xτ〉) for all ⁢τ∈B,t∈E.

Proof of (b). For this we need to use step (5). Notice that ω⁢〈Pt,Xτ〉>〈Pt,Xt〉 for all τ∈B. If not, then Xt⁢R*⁢(ω)⁢Xτ for some τ∈B. But then t would have been moved into B before τ was and we have contradiction with t∈E. Hence, the division is well defined and

λt=λm⩾Uτ-Utω⁢〈Pt,Xτ〉-〈Pt,Xt〉 for all ⁢τ∈B,t∈E.

Cross multiplying,

λt⁢(ω⁢〈Pt,Xτ〉-〈Pt,Xt〉)⩾Uτ-Ut for all ⁢τ∈B,t∈E,

which proves (b).

Proof of (c). Note that if t,τ∈E, then ω⁢〈Pτ,Xt〉⩾〈Pτ,Xτ〉. If not, then 〈Pτ,Xτ〉>ω⁢〈Pτ,Xt〉, which contradicts GARP⁢(ω). Indeed, t∈E implies Xt⁢R*⁢(ω)⁢Xm, τ∈E implies Xτ⁢R*⁢(ω)⁢Xm which (by definition of m) implies Xm⁢R*⁢(ω)⁢Xτ, so Xt⁢R*⁢(ω)⁢Xτ and GARP(ω) implies 〈Pτ,Xτ〉⩽ω⁢〈Pt,Xτ〉. Now for all t,τ∈E,

Ut=Uτ,λτ=λm>0,

Ut⩽Uτ+λτ⁢(ω⁢〈Pτ,Xt〉-〈Pτ,Xτ〉).

(2) ⇒ (1). Let {Ut,λt}t=1T with λt>0 for all t∈{1,…,T} satisfy (2.18), and

〈Pt,Xt〉⩾ω⁢〈Pt,Xt1〉,〈Pt1,Xt1〉⩾ω⁢〈Pt1,Xt2〉,…,〈Ptk,Xtk〉⩾ω⁢〈Ptk,Xs〉.

Taking into account that λt>0 for all t∈{1,…,T}, these inequalities imply that Ut⩽Us. This implies 〈Ps,Xs〉⩽ω⁢〈Ps,Xt〉 which completes the proof of this part. ∎

Proof of Theorem 13.

(1) ⇒ (2). Note that HARP⁢(ω) is invariant with respect to change of scales. Therefore if the trade statistics {(Pt,Xt)}t=1T satisfies HARP⁢(ω), then the trade statistics {(Pt,μt⁢Xt)}t=1T also satisfies HARP⁢(ω) for any {μt}t=1T. This implies that {(Pt,μt⁢Xt)}t=1T satisfies GARP⁢(ω).

(2) ⇒ (1). Fix some subset of indices (t1,…,tk)⊂{1,…,T} such that there are no two identical indices. Select μt1,…,μtk so that

〈Pt1,μt1⁢Xt1〉=ω⁢〈Pt1,μt2⁢Xt2〉,

〈Pt2,μt2⁢Xt2〉=ω⁢〈Pt2,μt3⁢Xt3〉,

⋮

〈Ptk-1,μtk-1⁢Xtk-1〉=ω⁢〈Ptk-1,μtk⁢Xtk〉.

It is possible to do that because there are k-1 equalities to satisfy by choosing k numbers. Then GARP⁢(ω) implies

〈Ptk,μtk⁢Xtk〉⩽ω⁢〈Ptk,μt1⁢Xt1〉.

Therefore,

〈Pt1,Xt2〉〈Pt1,Xt1〉⁢〈Pt2,Xt3〉〈Pt2,Xt2〉⁢⋯⁢〈Ptk,Xt1〉〈Ptk,Xtk〉⩾1ωk.∎

Consider a group of three goods the demand on which is given by the three Engel curves

q1⁢(x)=(1,ε,ε)′⁢x,

q2⁢(x)=(ε,1,ε)′⁢x,

q3⁢(x)=(ε,ε,1)′⁢x,

which correspond to the following price vectors:

P1=(2 1 4)′,

P2=(2 1 2)′,

P3=(2 2 1)′.

The observed demands are given by Xt⁢(ε)=qt⁢(1). The matrix P⁢X⁢(ε) with elements p⁢xτ⁢t⁢(ε)=〈Pτ,Xt⁢(ε)〉 is equal to

P⁢X⁢(ε)=(2+5⁢ε1+6⁢ε4+3⁢ε2+3⁢ε1+4⁢ε2+3⁢ε2+3⁢ε2+3⁢ε1+4⁢ε).

If ε=0, then the direct revealed preference relation is given by

R⁢(0)=(110010001),

with transitive closure R*⁢(0)=R⁢(0). The trade statistics {(Pt,Xt⁢(0))}t=1T satisfies GARP and HARP. In order to simplify the calculations we set ε=0. Having zero demand may seem unnatural, however the forecasting sets for small enough positive ε should differ only a little unless setting ε>0 leads to failure of GARP.^[12] It may be shown that R⁢(ε)=R⁢(0) for ε<1. Therefore, choosing ε from (0,1) does not lead to failure of GARP.

Let P4=(1 1 1)′ and x4=2. We start with G⁢(P4,x4,{x~t}t=13). The intersection demands are given by

〈P4,qt⁢(x~t)〉=x~t=2.

and the set {X∈ℝ+m:X∈G(P4,x4,{x~t}t=13) is given by

X1⩽2-32⁢X2,

X3=2-X1-X2,

X1⩾0,X2⩾0.

From this we see that the closure of S⁢(P4,x4,{x~t}t=13) is G⁢(P4,x4,{x~t}t=13). Therefore, (2.20) implies that

K~G1⁢(P4,x4)=G⁢(P4,x4,{x~t}t=13).

Since Engel curves are rays,

KH1⁢(P4,x4)=K~G1⁢(P4,x4).

Therefore,

G⁢(P4,x4,{x~t}t=13)=KH1⁢(P4,x4).

The set {X∈ℝ+m:X∈KH1(P4,x4) is given by

X3=2-X1,

X2=0,

X1⩾0,X1⩽2.

We see that the set KH1⁢(P4,x4) is a proper subset of G⁢(P4,x4,{x~t}t=13). This implies that (2.20) is not true. If we set ε=0.5, then G⁢(P4,x4,{x~t}t=13) is been given by

X1⩽1.75-1.5⁢X2,

X2⩽1,

1⩽X1+X2,

X3=2-X1-X2,

X1⩾0,X2⩾0,

while the set KH1⁢(P4,x4) is given by

X1⩽1.75-1.5⁢X2,

X2⩽0.625,

1⩽X1+X2,

X3=2-X1-X2,

X1⩾0,X2⩾0.

The set KH1⁢(P4,x4) is still a proper subset of G⁢(P4,x4,{x~t}t=13).

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Received: 2015-2-2

Accepted: 2015-4-3

Published Online: 2015-5-21

Published in Print: 2016-8-1

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https://doi.org/10.1515/jiip-2015-0015

Keywords for this article

Inverse problem; mathematical programming; revealed preference; economic indices; forecasting