Calculating a Giffen Good

Abstract

This paper provides a simple example of the utility function with two consumption goods which can be calculated by hand to produce a Giffen good. It is based on the theoretical result by Kubler et al. (Am Econ Rev 103:1034–1053, 2013). Using a model of portfolio selection with a risk-free asset and a risky asset, they showed that there always exists a parameter set which assures that the risk-free asset becomes a Giffen good if the utility function belongs to the HARA (hyperbolic absolute risk aversion) family with decreasing absolute risk aversion (DARA) and decreasing relative risk aversion (DRRA). This paper investigates their result further in a usual microeconomic setting where the risk-free asset and the risky asset are changed to the first and second consumption goods, respectively. It is organized as follows. First, a standard utility maximization problem of a consumer is directly solved to obtain the conditions for the first good to be a Giffen good. Second, the same problem is analyzed by means of decompositions of the price effect due to Slutsky and Sasakura (Italian Econ J 2:258–280, 2016). As is well known, the former decomposition consists of the substitution effect and the income effect, while the latter implies the decomposition into the ratio effect and the unit-elasticity effect. Lastly these analyses are compared and summarized. It should be added that the utility function proposed in this paper can also be used for the analysis of a normal good mutatis mutandis.

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Notes

  1. 1.

    For the recent developments, see Doi et al. (2009) and Heijman and von Mouche (2012), and Fujimoto (2018). In fact a Giffen good was first analyzed by Simon Gray in 1815. See Masuda and Newman (1981).

  2. 2.

    For a further consideration of the relationship between the HARA family and a Giffen good, see also Kannai and Selden (2014).

  3. 3.

    See, for example, Campbell and Cochrane (1995), Gollier (2001), and Meyer and Meyer (2005).

  4. 4.

    It is of the type referred to as the Klein-Rubin or Stone-Geary utility function.

  5. 5.

    As far as I know, there is no such figure as panel (a) in which the three curves for \(0<\gamma <1\), \(\gamma =1\), and \(\gamma >1\) are drawn at the same time.

  6. 6.

    Also it can easily be shown that

    $$\begin{aligned}&-p_2^2u_{11} + 2p_1p_2u_{12} - p_1^2 u_{22} \\&=\alpha _1\gamma (\beta _{21}p_1-\beta _1p_2)^2A^{-\gamma -1}+\alpha _2\gamma (\beta _1p_2-\beta _{22}p_1)^2B^{-\gamma -1}>0 \end{aligned}$$

    because of Assumptions 1 and 2. As is apparent, the above result means that the bordered Hessian determinant associated with the current consumer’s constrained maximization problem is warranted to be positive before using the first-order conditions.

  7. 7.

    Using the natural logarithm, Assumption 3 can be written as

    $$\begin{aligned} (0<)\ \gamma <\log \left( \frac{\alpha _2}{\alpha _1}\frac{\beta _1p_2-\beta _{22}p_1}{\beta _{21}p_1-\beta _1p_2}\right) \left( \log \frac{\beta _{22}}{\beta _{21}}\right) ^{-1}. \end{aligned}$$
  8. 8.

    For the derivation of \(\partial q_1^*/\partial p_1\) and \(\partial q_2^*/\partial p_1\) below, see Appendix A.

  9. 9.

    It can also be said, as one of the reviewers suggested, that good 1 is more likely to become a Giffen good when its price is high compared to the case where its price is low, ceteris paribus, because the result that \(\kappa \) decreases with \(p_1\) implies that a rise in \(p_1\) relaxes the condition \(\beta _{22}-\beta _{21}\kappa >0\) (Assumption 3) for a Giffen good.

  10. 10.

    For the derivation of various effects in this section, see Appendix C.

  11. 11.

    Kubler et al. (2013, pp. 1034, 1038, 1047–1048) repeatedly mention the Slutsky decomposition to explain the behavior of a risk-free asset as a Giffen good, but they does not use it.

  12. 12.

    When \(q_1=q_1^*\) and \(q_2=q_2^*\), the value of the utility function (1) is written as

    $$\begin{aligned} u(q_1^*, q_2^*)= & {} \alpha _1\frac{(A^*)^{1-\gamma }}{1-\gamma } +\alpha _2\frac{(B^*)^{1-\gamma }}{1-\gamma }=\frac{\alpha _1+ \alpha _2\kappa ^{1-\gamma }}{1-\gamma }\chi ^{1-\gamma } \ \text{ for }\ \gamma \ne 1, \\= & {} \alpha _1\log (A^*)+\alpha _2\log (B^*)=(\alpha _1+\alpha _2) \log \chi +\alpha _2\log \kappa \ \text{ for }\ \gamma =1. \end{aligned}$$

    Note that \(\chi \) is an increasing function of I. It follows that the maximized value of utility is an increasing function of I, though \(q_1^*\) and \(q_2^*\) move in the opposite direction to changes in I.

  13. 13.

    See Sasakura (2016) for details. Using such a decomposition of the price effect, the CES utility function as well as the Cobb-Douglas utility function are analyzed there. The CRRA utility function can also be analyzed in a similar way.

  14. 14.

    Jensen and Miller (2008) showed empirically that rice (or wheat) became a Giffen good for poor households in China. Their theoretical consideration to find a Giffen good followed the Slutsky decomposition faithfully. For that purpose they drew a plane (Fig. 1 on p. 1557 of their paper) with a staple food (bread) on the horizontal axis and a fancy good (meat) on the vertical axis. The plane is divided into three regions, a calorie-deprived zone, a subsistence zone, and a standard zone. Since their plane is similar to the \(q_1\)-\(q_2\) plane of this paper, it can be said that the calorie-deprived zone, the subsistence zone, and the standard zone are regarded respectively as the cases for \(I=I_{\text{ min }}\), for \(I_{\text{ min }}<I<I^G\), and for \(I^G<I<I^Z\) in Fig. 3.

  15. 15.

    Figures 4 and 5 are so drawn that the intercept of each budget line is above that of the \(B=0\) line (\(\beta _3/\beta _{22}\)). In fact it can be shown that if \((0<) \gamma \le 1\), the intercepts of both budget lines are below that of the \(B=0\) line. So a relatively large value of \(\gamma \) is assumed in Figs. 4 and 5.

References

  1. Arrow KJ (1971) Essays in the theory of risk-bearing. Markham, Chicago

    Google Scholar 

  2. Campbell YJ, Cochrane JH (1995) By force of habit: a consumption-based explanation of aggregate stock market behavior. J Polit Econ 107:205–251

  3. Doi J, Iwasa K, Shimomura K (2009) Giffen behavior independent of the wealth level. Econ Theor 41:247–267

    Article  Google Scholar 

  4. Fujimoto M (2018) A geometrical approach to Giffen behavior: the Epstein and Spiegel utility function revisited. Manchester Sch 86:681–694

    Article  Google Scholar 

  5. Gollier C (2001) The economics of risk and time. The MIT Press, Cambridge, Massachusetts

    Google Scholar 

  6. Heijman W, von Mouche P (eds) (2012) New insights into the theory of Giffen goods. Springer, Berlin

  7. Jensen RT, Miller NH (2008) Giffen behavior and subsistence consumption. Am Econ Rev 98:1553–1577

    Article  Google Scholar 

  8. Kannai Y, Selden L (2014) Violation of the law of demand. Econ Theor 55:47–61

    Article  Google Scholar 

  9. Kubler F, Selden L, Wei X (2013) Inferior good and Giffen behavior for investing and borrowing. Am Econ Rev 103:1034–1053

    Article  Google Scholar 

  10. Marshall A (1895) Principles of economics, 3rd edn. Macmillan, London

    Google Scholar 

  11. Masuda E, Newman P (1981) Gray and Giffen goods. Econ J 91:1011–1014

    Article  Google Scholar 

  12. Meyer DJ, Meyer J (2005) Risk preferences in multi-period consumption models, the equity premium puzzle, and habit formation utility. J Monetar Econ 52:1497–1515

    Article  Google Scholar 

  13. Sasakura K (2016) Slutsky revisited: a new decomposition of the price effect. Italian Econ J 2:258–280

    Article  Google Scholar 

Download references

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Correspondence to Kazuyuki Sasakura.

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I am very grateful to Professor Kazumichi Iwasa, Kobe University, and two anonymous reviewers of this journal for their valuable comments. Needless to say, all remaining errors are mine. An earlier version of this paper was presented at the Singapore Economic Review Conference (SERC) 2019.

Appendices

Appendix

A Calculation of \(\frac{\partial q_1^*}{\partial p_1}\) and \(\frac{\partial q_2^*}{\partial p_1}\)

$$\begin{aligned} \frac{\partial q_1^*}{\partial p_1}= & {} \frac{1}{D^2}\ \left\{ (-\beta _3p_2+\beta _{21}I)\kappa 'D \right. \\&\left. +[(1-\kappa )\beta _3p_2-(\beta _{22}-\beta _{21}\kappa )I] [-\kappa '(\beta _{21}p_1-\beta _1p_2)+(\beta _{22}-\beta _{21}\kappa )]\right\} \\= & {} \frac{1}{D^2}\Bigl (\beta _3p_2\{-\kappa 'D+(1-\kappa ) [-\kappa '(\beta _{21}p_1-\beta _1p_2)+(\beta _{22}-\beta _{21}\kappa )]\} \\&-\{-\beta _{21}\kappa 'D+(\beta _{22}-\beta _{21}\kappa ) [-\kappa '(\beta _{21}p_1-\beta _1p_2)+(\beta _{22}-\beta _{21}\kappa )]\}I\Bigr ) \\= & {} \frac{1}{D^2}\Bigl (\beta _3p_2\{-\kappa ' (\beta _{21}-\beta _{22})p_1+(1-\kappa )(\beta _{22}-\beta _{21}\kappa )\} \\&-\{-\kappa '(\beta _{21}-\beta _{22})\beta _1p_2+(\beta _{22} -\beta _{21}\kappa )^2\}I\Bigr ), \end{aligned}$$

and

$$\begin{aligned} \frac{\partial q_2^*}{\partial p_1}= & {} \frac{1}{D^2}\ \left\{ [-\kappa '(\beta _1I-\beta _3p_1)-(1-\kappa )\beta _3]D \right. \\&\left. +(1-\kappa )(\beta _1I-\beta _3p_1)[-\kappa '(\beta _{21}p_1-\beta _1p_2) +(\beta _{22}-\beta _{21}\kappa )]\right\} \\= & {} \frac{1}{D^2}\Bigl (-\{-\kappa '\beta _3p_1D+(1-\kappa )\beta _3D +\beta _3p_1(1-\kappa )[-\kappa '(\beta _{21}p_1-\beta _1p_2) \\&+(\beta _{22}-\beta _{21}\kappa )]\} \\&+\{-\beta _1\kappa 'D+\beta _1(1-\kappa )[-\kappa '(\beta _{21}p_1-\beta _1p_2) +(\beta _{22}-\beta _{21}\kappa )]\}I\Bigr ) \\= & {} \frac{1}{D^2}\Bigl (-\beta _3\{-\kappa '(\beta _{21}-\beta _{22})p_1^2 +(1-\kappa )^2\beta _1p_2\} \\&+\beta _1\{-\kappa '(\beta _{21}-\beta _{22})p_1+(1-\kappa )(\beta _{22} -\beta _{21}\kappa )\}I\Bigr ). \end{aligned}$$

B Proof of Lemma 1

Here are the results of subtraction between two incomes in Lemma 1.

$$\begin{aligned} I_1^G-I_{\text{ min }}= & {} \frac{\beta _3}{\beta _1}\frac{(\beta _{22}- \beta _{21}\kappa )D}{-\kappa '(\beta _{21}-\beta _{22})\beta _1p_2+ (\beta _{22}-\beta _{21}\kappa )^2}>0, \\ I_1^Z-I_1^G= & {} \frac{\beta _3}{\beta _1}\frac{-\kappa '(\beta _{21}- \beta _{22})D^2}{-\kappa '(\beta _{21}-\beta _{22})p_1+(1-\kappa ) (\beta _{22}-\beta _{21}\kappa )} \\&\times \frac{1}{-\kappa '(\beta _{21}-\beta _{22})\beta _1p_2+ (\beta _{22}-\beta _{21}\kappa )^2}>0, \\ I_{\text{ max }}-I_1^Z= & {} \frac{\beta _3}{\beta _1}\frac{-\kappa ' (\beta _{21}-\beta _{22})p_1D}{(\beta _{22}-\beta _{21}\kappa ) [-\kappa '(\beta _{21}-\beta _{22})p_1+(1-\kappa )(\beta _{22}-\beta _{21}\kappa )]}>0, \end{aligned}$$

because of Assumptions 14. Therefore, \(I_{\text{ min }}<I_1^G<I_1^Z<I_{\text{ max }}\). \(\square \)

C Derivation of Various Effects

The substitution effect:

$$\begin{aligned}&-\frac{u_1^* (u_2^*)^2}{p_1|U^*|} \\&=-\frac{1}{p_1|U^*|}\beta _1\chi ^{-\gamma }(\alpha _1 +\alpha _2\kappa ^{-\gamma })\chi ^{-2\gamma }(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })^2 \\&=-\frac{\beta _1\chi ^{-3\gamma -1}}{p_1|U^*|}\chi (\alpha _1 +\alpha _2\kappa ^{-\gamma })(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })^2 \\&=-\frac{(\alpha _1+\alpha _2\kappa ^{-\gamma }) (\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })^2}{p_1\alpha _1\alpha _2\beta _1\gamma (\beta _{21}-\beta _{22}) (\alpha _1\kappa ^{-\gamma -1}+\alpha _2\kappa ^{-2\gamma }) D} (\beta _1I-\beta _3p_1). \end{aligned}$$

The income effect:

$$\begin{aligned}&-q_1^*\frac{u_1^* (u_2^*u_{12}^*-u_1^*u_{22}^*)}{p_1|U^*|} \\&=\frac{q_1^*}{p_1|U^*|}\beta _1\chi ^{-\gamma }(\alpha _1 +\alpha _2\kappa ^{-\gamma })\alpha _1\alpha _2\beta _1\gamma (\beta _{21}- \beta _{22})\chi ^{-2\gamma -1}(\beta _{22}-\beta _{21}\kappa )\kappa ^{-\gamma -1} \\&=\frac{\beta _1\chi ^{-3\gamma -1}}{p_1|U^*|}(\alpha _1+\alpha _2 \kappa ^{-\gamma })\alpha _1\alpha _2\beta _1\gamma (\beta _{21}- \beta _{22})q_1^*(\beta _{22}-\beta _{21}\kappa )\kappa ^{-\gamma -1} \\&=\frac{\kappa ^{-\gamma -1}(\beta _{22}-\beta _{21}\kappa ) (\alpha _1+\alpha _2\kappa ^{-\gamma })}{p_1(\beta _{21}-\beta _{22}) (\alpha _1\kappa ^{-\gamma -1}+\alpha _2\kappa ^{-2\gamma })D} [(1-\kappa )\beta _3p_2-(\beta _{22}-\beta _{21}\kappa )I]. \end{aligned}$$

The transfer effect:

$$\begin{aligned}&q_1^*\frac{u_2^*(u_1^*u_{12}^*-u_2^*u_{11}^*)}{p_1|U^*|} \\&=\frac{q_1^*}{p_1|U^*|}\chi ^{-\gamma }(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })\alpha _1\alpha _2\beta _1^2 \gamma (\beta _{21}-\beta _{22})\chi ^{-2\gamma -1}(1-\kappa )\kappa ^{-\gamma -1} \\&=\frac{\chi ^{-3\gamma -1}}{p_1|U^*|}(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })\alpha _1\alpha _2\beta _1^2 \gamma (\beta _{21}-\beta _{22})q_1^*(1-\kappa )\kappa ^{-\gamma -1} \\&=\frac{\kappa ^{-\gamma -1}(1-\kappa )(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })}{p_1(\beta _{21}-\beta _{22}) (\alpha _1\kappa ^{-\gamma -1}+\alpha _2\kappa ^{-2\gamma })D}[(1-\kappa ) \beta _3p_2-(\beta _{22}-\beta _{21}\kappa )I]. \end{aligned}$$

The ratio effect (= the substitution effect + the transfer effect):

$$\begin{aligned}&\frac{\ \frac{\partial \theta }{\partial p_1}I\ }{p_1} \\&=\frac{\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma }}{p_1\alpha _1\alpha _2\beta _1\gamma (\beta _{21}-\beta _{22}) (\alpha _1\kappa ^{-\gamma -1}+\alpha _2\kappa ^{-2\gamma }) D} \\&\ \ \ \times \Bigl \{-(\alpha _1+\alpha _2\kappa ^{-\gamma }) (\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })(\beta _1I-\beta _3p_1) \\&\ \ \ +\alpha _1\alpha _2\beta _1\gamma \kappa ^{-\gamma -1} (1-\kappa )[(1-\kappa )\beta _3p_2-(\beta _{22}-\beta _{21}\kappa )I]\Bigr \} \\&=\frac{\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma }}{p_1\alpha _1\alpha _2\beta _1\gamma (\beta _{21}-\beta _{22}) (\alpha _1\kappa ^{-\gamma -1}+\alpha _2\kappa ^{-2\gamma }) D} \\&\ \ \ \times \Big \{(\alpha _1+\alpha _2\kappa ^{-\gamma }) (\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma }) \beta _3p_1+\alpha _1\alpha _2\beta _1\gamma \kappa ^{-\gamma -1}(1-\kappa )^2\beta _3p_2 \\&\ \ \ -\left[ (\alpha _1+\alpha _2\kappa ^{-\gamma })(\alpha _1\beta _{21}+ \alpha _2\beta _{22}\kappa ^{-\gamma })\beta _1+\alpha _1\alpha _2\beta _1 \gamma \kappa ^{-\gamma -1}(1-\kappa )(\beta _{22}-\beta _{21}\kappa )\right] I\Big \}. \end{aligned}$$

The price effect (= the substitution effect + the income effect = the unit-elasticity effect + the ratio effect):

$$\begin{aligned}&\frac{\partial q_1}{\partial p_1}\bigg |_{p_2, I=\text{ const }} \\&=\frac{\alpha _1+\alpha _2\kappa ^{-\gamma }}{p_1\alpha _1\alpha _2 \beta _1\gamma (\beta _{21}-\beta _{22})(\alpha _1\kappa ^{-\gamma -1}+ \alpha _2\kappa ^{-2\gamma }) D} \\&\quad \times \Bigl \{-(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })^2 (\beta _1I-\beta _3p_1) \\&\quad +\alpha _1\alpha _2\beta _1\gamma \kappa ^{-\gamma -1}(\beta _{22}- \beta _{21}\kappa )[(1-\kappa )\beta _3p_2-(\beta _{22}-\beta _{21}\kappa )I]\Bigr \} \\&=\frac{\alpha _1+\alpha _2\kappa ^{-\gamma }}{p_1\alpha _1\alpha _2\beta _1 \gamma (\beta _{21}-\beta _{22})(\alpha _1\kappa ^{-\gamma -1}+\alpha _2\kappa ^{-2\gamma }) D} \\&\quad \quad \times \Bigl \{(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma }) ^2 \beta _3p_1+\alpha _1\alpha _2\beta _1\gamma \kappa ^{-\gamma -1}(1-\kappa ) (\beta _{22}-\beta _{21}\kappa )\beta _3p_2 \\&\quad \quad -[(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })^2 \beta _1 +\alpha _1\alpha _2\beta _1\gamma \kappa ^{-\gamma -1}(\beta _{22}-\beta _{21}\kappa )^2]I\Bigr \}. \end{aligned}$$

D Proof of Lemma 2

Here are the results of subtraction between two incomes in Lemma 2.

$$\begin{aligned} I_2^G-I_{\text{ min }}= & {} \frac{\beta _3}{\beta _1}\frac{\alpha _1\alpha _2 \gamma \kappa ^{-\gamma -1}(\beta _{22}-\beta _{21}\kappa )D}{(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })^2 +\alpha _1\alpha _2\gamma \kappa ^{-\gamma -1}(\beta _{22}-\beta _{21}\kappa )^2}>0, \\ I_2^Z-I_2^G= & {} \frac{\beta _3}{\beta _1}\frac{\alpha _1\alpha _2 \gamma \kappa ^{-\gamma -1}(\beta _{21}-\beta _{22})(\alpha _1+\alpha _2 \kappa ^{-\gamma +1})(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^ {-\gamma })D}{(\alpha _1+\alpha _2\kappa ^{-\gamma })(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })+\alpha _1\alpha _2\gamma \kappa ^{-\gamma -1} (1-\kappa )(\beta _{22}-\beta _{21}\kappa )} \\&\times \frac{1}{(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })^2 +\alpha _1\alpha _2\beta _1\gamma \kappa ^{-\gamma -1}(\beta _{22}-\beta _{21}\kappa )^2}>0, \\ I_{\text{ max }}-I_2^Z= & {} \frac{\beta _3}{\beta _1}\frac{(\alpha _1 +\alpha _2\kappa ^{-\gamma })(\alpha _1\beta _{21}+\alpha _2\beta _{22} \kappa ^{-\gamma })D}{(\alpha _1+\alpha _2\kappa ^{-\gamma })(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })+\alpha _1\alpha _2\gamma \kappa ^{-\gamma -1}(1-\kappa )(\beta _{22}-\beta _{21}\kappa )} \\&\times \frac{1}{\beta _{22}-\beta _{21}\kappa }>0, \end{aligned}$$

because of Assumptions 1-4. Therefore, \(I_{\text{ min }}<I_2^G<I_2^Z<I_{\text{ max }}\). \(\square \)

E Proof of Lemma 3

Using (13), it can be shown that

$$\begin{aligned} \frac{[(\beta _{21}-\beta _{22})\beta _1p_2]^2}{(\beta _{21}p_1- \beta _1p_2)(\beta _1p_2-\beta _{22}p_1)}=\frac{(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })^2}{\alpha _1\alpha _2\kappa ^{-\gamma }}. \end{aligned}$$
(14)

Substituting (12) and (14) into (9) yields

$$\begin{aligned} I_1^Z= & {} \frac{\beta _3}{\beta _1}\frac{-\kappa '(\beta _{21}- \beta _{22})p_1^2+(1-\kappa )^2\beta _1p_2}{-\kappa '(\beta _{21}-\beta _{22})p_1 +(1-\kappa )(\beta _{22}-\beta _{21}\kappa )} \\= & {} \frac{\beta _3}{\beta _1}\bigg \{\frac{\kappa }{\gamma }\frac{[(\beta _{21}- \beta _{22})\beta _1p_2]^2}{(\beta _{21}p_1-\beta _1p_2)(\beta _1p_2-\beta _{22}p_1)} \frac{p_1}{\beta _1p_2}p_1+(1-\kappa )^2\beta _1p_2\bigg \} \\&\times \bigg \{\frac{\kappa }{\gamma }\frac{[(\beta _{21}- \beta _{22})\beta _1p_2]^2}{(\beta _{21}p_1-\beta _1p_2)(\beta _1p_2- \beta _{22}p_1)}\frac{p_1}{\beta _1p_2}+(1-\kappa )(\beta _{22}-\beta _{21}\kappa )\bigg \}^{-1} \\= & {} \frac{\beta _3}{\beta _1}\bigg \{\frac{\kappa }{\gamma }\frac{ (\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })^2}{\alpha _1\alpha _2\kappa ^{-\gamma }}\frac{\alpha _1+\alpha _2\kappa ^{-\gamma }}{\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma }}p_1+ (1-\kappa )^2\beta _1p_2\bigg \} \\&\times \bigg \{\frac{\kappa }{\gamma }\frac{(\alpha _1\beta _{21}+\alpha _2 \beta _{22}\kappa ^{-\gamma })^2}{\alpha _1\alpha _2\kappa ^{-\gamma }}\frac{\alpha _1 +\alpha _2\kappa ^{-\gamma }}{\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma }} +(1-\kappa )(\beta _{22}-\beta _{21}\kappa )\bigg \}^{-1} \\= & {} \frac{\beta _3}{\beta _1}\frac{(\alpha _1+\alpha _2\kappa ^{-\gamma }) (\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma })p_1+\alpha _1\alpha _2 \beta _1\gamma \kappa ^{-\gamma -1}(1-\kappa )^2p_2}{(\alpha _1+\alpha _2 \kappa ^{-\gamma })(\alpha _1\beta _{21}+\alpha _2\beta _{22}\kappa ^{-\gamma }) +\alpha _1\alpha _2\gamma \kappa ^{-\gamma -1}(1-\kappa )(\beta _{22}-\beta _{21}\kappa )} \\= & {} I_2^Z, \end{aligned}$$

due to (10). \(\square \)

F Proof of Lemma 4

Substituting (12) and (14) obtained in the proof of Lemma 3 above into (8) yields

$$\begin{aligned} I_1^G= & {} \frac{\beta _3p_2[-\kappa '(\beta _{21}-\beta _{22})p_1+ (1-\kappa )(\beta _{22}-\beta _{21}\kappa )]}{-\kappa '(\beta _{21}-\beta _{22}) \beta _1p_2+(\beta _{22}-\beta _{21}\kappa )^2} \\= & {} \bigg \{\frac{\kappa }{\gamma }\frac{[(\beta _{21}-\beta _{22})\beta _1p_2]^2}{(\beta _{21}p_1-\beta _1p_2)(\beta _1p_2-\beta _{22}p_1)}\frac{\beta _3p_1}{\beta _1}+(1-\kappa )(\beta _{22}-\beta _{21}\kappa )\beta _3p_2\bigg \} \\&\times \bigg \{\frac{\kappa }{\gamma }\frac{[(\beta _{21}-\beta _{22}) \beta _1p_2]^2}{(\beta _{21}p_1-\beta _1p_2)(\beta _1p_2-\beta _{22}p_1)} +(\beta _{22}-\beta _{21}\kappa )^2\bigg \}^{-1} \\= & {} \bigg \{\frac{\kappa }{\gamma }\frac{(\alpha _1\beta _{21}+\alpha _2 \beta _{22}\kappa ^{-\gamma })^2}{\alpha _1\alpha _2\kappa ^{-\gamma }} \frac{\beta _3p_1}{\beta _1}+(1-\kappa )(\beta _{22}-\beta _{21}\kappa )\beta _3p_2\bigg \} \\&\times \bigg \{\frac{\kappa }{\gamma }\frac{(\alpha _1\beta _{21} +\alpha _2\beta _{22}\kappa ^{-\gamma })^2}{\alpha _1\alpha _2 \kappa ^{-\gamma }}+(\beta _{22}-\beta _{21}\kappa )^2\bigg \}^{-1} \\= & {} \frac{\beta _3}{\beta _1}\frac{(\alpha _1\beta _{21}+\alpha _2\beta _{22} \kappa ^{-\gamma })^2 p_1+\alpha _1\alpha _2\beta _1\gamma \kappa ^{-\gamma -1} (1-\kappa )(\beta _{22}-\beta _{21}\kappa )p_2}{(\alpha _1\beta _{21}+ \alpha _2\beta _{22}\kappa ^{-\gamma })^2 +\alpha _1\alpha _2\gamma \kappa ^{-\gamma -1}(\beta _{22}-\beta _{21}\kappa )^2} \\= & {} I_2^G, \end{aligned}$$

due to (11). \(\square \)

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Sasakura, K. Calculating a Giffen Good. Ital Econ J (2021). https://doi.org/10.1007/s40797-020-00140-1

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Keywords

  • HARA family
  • Decreasing relative risk aversion
  • Giffen good
  • Slutsky equation
  • Ratio effect

JEL Classification

  • D11
  • D01
  • G11