Hicksian complementarity and perturbed utility models

Abstract

This paper studies aggregate complementarity without price or income variation. We show that for a class of utility functions, variation in non-price observables allows one to recover a measure of complementarity similar to Hicksian complementarity. In addition, the entire Slutsky matrix can be recovered up to scale without price variation. We then examine aggregate complementarity in latent utility models used in discrete choice, bundles, and matching. We show that classical linear instrumental variables can recover Hicksian complementarity for the special case of quadratic utility.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    This ratio is related to the diversion ratio, which is used in analysis of antitrust (Farrell and Shapiro 2010).

  2. 2.

    McFadden and Fosgerau (2012) consider the model with \(u_k(x_k)\) replaced by a perturbation parameter \(u_k\). We differ in interpretation since it is an index of observable characteristics. Here the \(D(\cdot )\) function can be viewed as a disturbance or utility that does not depend on other observables.

  3. 3.

    The vector \(x_k\) may also include other variables that shift across decision problems, such as sugar, calories, etc. Importantly, a perturbed utility model is not a pure characteristics model in the spirit of Lancaster (1966). This is because the term D(y) encodes certain features of preferences that do not depend on non-price observables. Moreover, the function D may not be symmetric in its arguments, reflecting that the kth good has specific meaning such as a particular brand and size of potato chip.

  4. 4.

    This is a standard regularity condition; see Varian (1992), p. 498. This essentially ensures the second-order conditions are satisfied with a strict inequality.

  5. 5.

    If non-price observables are treated as non-random, then this relationship holds with \(y^*(x)\) defined by integrating over the marginal distribution of \(\varepsilon \). If non-price observables are random, we require independence between non-price observables and taste shocks.

  6. 6.

    See Rieskamp et al. (2006) where behavior from the attraction and compromise effect could be viewed as a form of complementarity.

  7. 7.

    See Choo and Siow (2006) and Galichon and Salanié (2015), who consider a model of shares of types matched.

  8. 8.

    See Amir et al. (2017) for more details on demand systems arising from quadratic utilities.

  9. 9.

    Here \(v'\) denotes the transpose of a vector v.

References

  1. Allen, R., Rehbeck, J.: Latent complementarity in bundles models. Available at SSRN 3257028, (2019a)

  2. Allen, R., Rehbeck, J.: Identification with additively separable heterogeneity. Econometrica 87(3), 1021–1054 (2019b)

    Google Scholar 

  3. Allen, R., Rehbeck, J.: Revealed stochastic choice with attributes. Available at SSRN 2818041 (2019c)

  4. Amir, Rabah, Erickson, Philip, Jin, Jim: On the microeconomic foundations of linear demand for differentiated products. J. Econ. Theory 169, 641–665 (2017)

    Google Scholar 

  5. Arora, A.: Testing for complementarities in reduced-form regressions: a note. Econ. Lett. 50(1), 51–55 (1996)

    Google Scholar 

  6. Athey, S., Stern, S.: An empirical framework for testing theories about complimentarity in organizational design. Technical report, National Bureau of Economic Research (1998)

  7. Blackburn, D.: Online piracy and recorded music sales. Working paper, (2004)

  8. Chambers, C.P., Echenique, F.: Supermodularity and preferences. J. Econ. Theory 144(3), 1004–1014 (2009)

    Google Scholar 

  9. Chiappori, P.-A., Salanié, B.: The econometrics of matching models. J. Econ. Lit. 54(3), 832–61 (2016)

    Google Scholar 

  10. Choo, E., Siow, A.: Who marries whom and why. J. Polit. Econ. 114(1), 175–201 (2006)

    Google Scholar 

  11. Farrell, J., Shapiro, C.: Antitrust evaluation of horizontal mergers: an economic alternative to market definition. BE J. Theoret. Econ. 10(1), 1–41 (2010)

  12. Feng, G., Li, X., Wang, Z.: On substitutability and complementarity in discrete choice models. Oper. Res. Lett. 46(1), 141–146 (2018)

    Google Scholar 

  13. Fosgerau, M., de Palma, A. Monardo, J.: Demand models for differentiated products with complementarity and substitutability. Available at SSRN 3141041, (2018)

  14. Fox, J.T., Yang, C.H., David, H.: Unobserved heterogeneity in matching games. J. Polit. Econ. 126(4), 1339–1373 (2018)

    Google Scholar 

  15. Fudenberg, D., Iijima, R., Strzalecki, T.: Stochastic choice and revealed perturbed utility. Econometrica 83(6), 2371–2409 (2015)

    Google Scholar 

  16. Galichon, A., Salanié, B.: Cupid’s invisible hand: Social surplus and identification in matching models. Available at SSRN 1804623, (2015)

  17. Gentzkow, M.: Valuing new goods in a model with complementarity: online newspapers. Am. Econ. Rev. 97(3), 713–744 (2007)

    Google Scholar 

  18. Gibson, M., Shrader, J.: Time use and labor productivity: the returns to sleep. Rev. Econ. Stat. 100(5), 783–798 (2018)

    Google Scholar 

  19. Heckman, J.J., Vytlacil, E.: Structural equations, treatment effects, and econometric policy evaluation 1. Econometrica 73(3), 669–738 (2005)

    Google Scholar 

  20. Hicks, J.R., Allen, R.G.D.: A reconsideration of the theory of value. Economica 1(1934), 52–176 (1934)

    Google Scholar 

  21. Hofbauer, J., Sandholm, W.H.: On the global convergence of stochastic fictitious play. Econometrica 70(6), 2265–2294 (2002)

    Google Scholar 

  22. Howard, D.H.: Rationing, quantity constraints, and consumption theory. Econometrica 45(2), 399–412 (1977)

    Google Scholar 

  23. Lancaster, K.J.: A new approach to consumer theory. J. Polit. Econ. 74(2), 132–157 (1966)

    Google Scholar 

  24. Manzini, P., Mariotti, M., Ülkü, L.: Stochastic complementarity. Econ. J. 129(619), 1343–1363 (2018)

    Google Scholar 

  25. McFadden, D.L., Fosgerau, M.: A theory of the perturbed consumer with general budgets. Technical report, National Bureau of Economic Research, (2012)

  26. McFadden, D.: Econometric models of probabilistic choice. In: Manski, C.F., McFadden, D. (eds.) Structural Analysis of Discrete data with Econometric Applications. MIT Press, Cambridge (1981)

    Google Scholar 

  27. Milgrom, P., Shannon, C., et al.: Monotone comparative statics. Econometrica 62(1), 157–180 (1994)

    Google Scholar 

  28. Novak, S., Stern, S.: Complementarity among vertical integration decisions: evidence from automobile product development. Manag. Sci. 55(2), 311–332 (2009)

    Google Scholar 

  29. Oberholzer-Gee, F., Strumpf, K.: The effect of file sharing on record sales: an empirical analysis. J. Polit. Econ. 115(1), 1–42 (2007)

    Google Scholar 

  30. Rieskamp, J., Busemeyer, J.R., Mellers, B.A.: Extending the bounds of rationality: evidence and theories of preferential choice. J. Econ. Lit. 44, 631–661 (2006)

    Google Scholar 

  31. Rob, R., Waldfogel, J.: Piracy on the high c’s: music downloading, sales displacement, and social welfare in a sample of college students. J. Law Econ. 49(1), 29–62 (2006)

    Google Scholar 

  32. Samuelson, P.A.: Complementarity: an essay on the 40th anniversary of the Hicks-Allen revolution in demand theory. J. Econ. Lit. 12, 1255–1289 (1974)

    Google Scholar 

  33. Schennach, S., White, H., Chalak, K.: Local indirect least squares and average marginal effects in nonseparable structural systems. J. Econ. 166(2), 282–302 (2012)

    Google Scholar 

  34. Topkis, D.M.: Minimizing a submodular function on a lattice. Oper. Res. 26(2), 305–321 (1978)

    Google Scholar 

  35. Varian, H.R.: Microeconomic Analysis. WW Norton, New York (1992)

    Google Scholar 

  36. Vives, X.: Nash equilibrium with strategic complementarities. J. Math. Econ. 19(3), 305–321 (1990)

    Google Scholar 

  37. Zentner, A.: Measuring the effect of file sharing on music purchases. J. Law Econ. 49(1), 63–90 (2006)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to John Rehbeck.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Some results in this paper have appeared in the deprecated working paper “Complementarity in Perturbed Utility Models”. We thank Christopher P. Chambers, Tim Conley, Ivana Komunjer, Andres Santos, Jeff Shrader, and Yixiao Sun for helpful comments.

Appendices

Appendix 1: proofs

Proof of Proposition 1

In this section, we examine comparative statics with respect to characteristic variation and prove Proposition 1. This analysis is similar to that of Howard (1977). We assume that we are examining demand functions that are positive and that derivatives \(\partial h^*_{\ell } / \partial p_{\ell } \ne 0\) and \(\partial y^*_{\ell } / \partial x_{\ell , q} \ne 0\). These conditions are enough to ensure that all commodities are goods and the budget constraint binds.

Consider the Lagrangian for the standard consumer problem with prices \(p = (p_1, \ldots , p_K)' \in {\mathbb {R}}_+^K\) and characteristics \(x=(x_1,\ldots ,x_K) \in \prod _{k=1}^K {\mathbb {R}}^{d_k}\) given by

$$\begin{aligned} {\mathcal {L}}(y, \lambda , \mu ) = \sum _{k=1}^K y_k u_k(x_k) + D(y) +\lambda \left( m-\sum _{k=1}^K p_k y_k \right) + \sum _{k=1}^K \mu _k y_K, \end{aligned}$$

with \(\lambda \ge 0\) is the Lagrange multiplier for the budget constraint and \(\mu _k \ge 0\) are Lagrange multipliers for \(y_k \ge 0\). Looking at the first-order conditions for an interior maximizer and assuming the budget constraint binds, we obtain

$$\begin{aligned} u_k(x_k) + D_k(y) - \lambda p_k&= 0 \quad \text {for } k=1,\ldots ,K \end{aligned}$$
(7)
$$\begin{aligned} \sum _{k=1}^K p_k y_k&= m, \end{aligned}$$
(8)

where \(D_k\) is the kth partial derivative of D. Let the function for demand of good k be denoted \(y_k^*=y_k^*(p,x,m)\) and the Lagrange multiplier at the maximum be denoted \(\lambda ^*=\lambda ^*(p,x,m)\). Assume \(y^*\) is a regular maximizer. We suppress dependence on parameters throughout the analysis for convenience. The bordered Hessian for the problem is given by

$$\begin{aligned} H= \begin{bmatrix} \nabla ^2 D(y^*)&-p \\ -p'&0 \end{bmatrix}, \end{aligned}$$

where \(\nabla ^2D\) is the matrix of second partial derivatives of D.Footnote 9 Examining the total derivative of Eqs. (7) and (8) with respect to \(p_\ell \), one obtains the system

$$\begin{aligned} H \begin{bmatrix} \frac{\partial y^*}{\partial p_\ell } \\ \frac{\partial \lambda ^*}{\partial p_\ell } \end{bmatrix} = e_{\ell } \lambda ^* + e_{K+1} y^*_\ell , \end{aligned}$$

where \(\frac{\partial y^*}{\partial p_\ell } = \left( \frac{\partial y_1^*}{\partial p_\ell } ,\ldots , \frac{\partial y_K^*}{\partial p_\ell } \right) '\) and \(e_{s}\) is standard basis vector for the sth dimension. Applying Cramer’s rule to the above system for good j, we obtain that

$$\begin{aligned} \frac{\partial y_j^*}{\partial p_{\ell }}&= \lambda ^* \frac{ (-1)^{\ell +j} \det (H_{\ell ,j}) }{ \det (H) } + \frac{ (-1)^{K+1+j} \det ( H_{K+1,j} ) }{\det { (H) }} y_{\ell }^*, \end{aligned}$$

where \(H_{r,c}\) is the submatrix of H which removes row r and column c. Moreover, let \(h^*_j=h_j^*(p,x,u)\) be the Hicksian demand function for good j. By the standard Slutsky equation, it follows that

$$\begin{aligned} \frac{\partial y_j^*}{\partial p_{\ell }}= \frac{\partial h^*_j}{\partial p_{\ell }} - \frac{\partial y_j^*}{\partial m} y_{\ell }^*. \end{aligned}$$

We desire to show that \(\frac{\partial h^*_j}{\partial p_{\ell }} = \lambda ^* \frac{ (-1)^{\ell +j} \det (H_{\ell ,j}) }{ \det (H) }\). This fact follows when \(\frac{\partial y_j^*}{\partial m} = -\frac{ (-1)^{K+1+j} \det ( H_{K+1,j} ) }{\det { (H) }},\) which we show is true below.

Examining the total derivative of Eqs. (7) and (8) with respect to m, one obtains the system

$$\begin{aligned} H \begin{bmatrix} \frac{\partial y^*}{\partial m} \\ \frac{\partial \lambda ^*}{\partial m} \end{bmatrix} = -e_{K+1}. \end{aligned}$$

Applying Cramer’s rule to the above system for good j, we obtain that

$$\begin{aligned} \frac{\partial y_j^*}{\partial m} = -\frac{ (-1)^{K+1+j} \det ( H_{K+1,j} ) }{\det { (H) }}. \end{aligned}$$

Thus, \(\frac{\partial h^*_j}{\partial p_{\ell }} = \lambda ^* \frac{ (-1)^{\ell +j} \det (H_{\ell ,j}) }{ \det (H) }.\)

Similarly, we can examine the change in demand when there is a change in characteristic q of alternative \(\ell \) (\(x_{\ell ,q}\)). Examining the total derivative of Eqs. (7) and (8) with respect to \(x_{\ell ,q}\) yields,

$$\begin{aligned} H \begin{bmatrix} \frac{\partial y^*}{\partial x_{\ell ,q}} \\ \frac{\partial \lambda ^*}{\partial x_{\ell ,q}} \end{bmatrix} = -\frac{\partial u_\ell }{\partial x_{\ell ,q} } e_{\ell }. \end{aligned}$$

Applying Cramer’s rule for good j results in

$$\begin{aligned} \frac{\partial y_j^*}{\partial x_{\ell ,q}}&= -\frac{\partial u_{\ell }}{ \partial x_{\ell ,q}} \frac{ (-1)^{\ell +j} \det (H_{\ell ,j}) }{ \det (H) } \end{aligned}$$
(9)
$$\begin{aligned}&= -\frac{\partial u_{\ell } / \partial x_{\ell , q}}{\lambda ^{*}} \frac{\partial h^*_j}{\partial p_{\ell }}, \end{aligned}$$
(10)

where the second equation follows from the determinant definition of \(\frac{\partial h^*_j}{\partial p_{\ell }}\). Thus,

$$\begin{aligned} \frac{\partial y^{*}_j}{\partial x_{\ell , q}} \bigg / \frac{\partial y_{\ell }^{*}}{\partial x_{\ell , q}} = \frac{\partial h^*_j}{\partial p_{\ell }} \bigg / \frac{\partial h^*_{\ell }}{\partial p_{\ell }}. \end{aligned}$$

The law of compensated demand implies \(\frac{\partial h^*_{\ell }}{\partial p_{\ell }} \ge 0\). Since this derivative is nonzero by assumption, we complete the proof.

To obtain an additional result, one can substitute Eq. 10 into the Slutsky equation and rearrange to obtain

$$\begin{aligned} \frac{\partial {y_j^*}}{\partial x_{\ell ,q}} = \left( -\text {BS}_\ell \delta _{j,m} - \delta _{j,p_{\ell }}\right) \left( \frac{{y_j^*}}{{y_{\ell }^*} p_{\ell }} \right) \frac{ \left( \partial u_{\ell } / \partial x_{\ell ,q} \right) {y_{\ell }^*}}{ \lambda ^{*} }. \end{aligned}$$

The term \(\delta _{j,p_{\ell }} = \frac{\partial y_j^*}{\partial p_{\ell }} \frac{p_{\ell }}{y_j^*}\) is the cross-price elasticity of demand for alternative j with respect to \(p_{\ell }\), \(\delta _{j,m}= \frac{\partial y_j^*}{\partial m} \frac{m}{y_j^*}\) is the income elasticity of demand for alternative j, and \(\text {BS}_{\ell } = \frac{p_{\ell } y_{\ell }^*}{m}\) is the budget share of alternative \({\ell }\). We examine the units of the last term in the above equation and find that \(\left( \frac{\partial u_{\ell }}{\partial x_{\ell ,q} } y_{\ell }^* / \lambda ^* \right) \) is in units \(\left( \frac{ \Delta \$ }{\Delta \text {characteristic }q \text { of good }l} \text { quantity }\ell \right) \). The term of \(\frac{ \Delta \$ }{\Delta \text {characteristic }q\text { of good }l}\) could be interpreted as the willingness to pay for a marginal increase in a characteristic \(x_{\ell ,q}\) at current demand. In principle, one could attempt to elicit this information using using surveys and then incorporate this information when trying to predict how demand would change with the values of \(x_{\ell ,q}\) even when this value does not change.

Proof of Corollary 1

Recall that Proposition 1 recovers \(\frac{\partial h^*_j}{\partial p_{\ell }} \bigg / \frac{\partial h^{*}_{\ell }}{\partial p_{\ell }}\) and \(\frac{\partial h^*_{\ell }}{\partial p_{j}} \bigg / \frac{\partial h^{*}_{j}}{\partial p_{j}}\). Because \(h^*\) is continuously differentiable, Slutsky symmetry yields \(\frac{\partial h^*_j}{\partial p_{\ell }} = \frac{\partial h^*_{\ell }}{\partial p_{j}}\). Thus, we recover

$$\begin{aligned} \left( \frac{\partial h^*_j}{\partial p_{\ell }} \bigg / \frac{\partial h^{*}_{\ell }}{\partial p_{\ell }} \right) \bigg / \left( \frac{\partial h^*_{\ell }}{\partial p_{j}} \bigg / \frac{\partial h^{*}_{j}}{\partial p_{j}} \right) = \frac{\partial h^{*}_{j}}{\partial p_{j}} \bigg / \frac{\partial h^{*}_{\ell }}{\partial p_{\ell }}. \end{aligned}$$

The ratio of any diagonal to off-diagonal element of the Slutsky matrix, \(\frac{\partial h^*_j}{\partial p_{\ell }} \bigg / \frac{\partial h^{*}_{\ell }}{\partial p_{\ell }}\), is already recoverable, and so we recover the ratio of all price-derivatives of \(h^*\). Thus, the matrix of price-derivatives of \(h^*\) is recoverable up to scale.

Appendix 2: structural foundation of instrumental variables

For the quasilinear model, one can substitute out the numeraire so that the problem is

$$\begin{aligned} \max _{y\in {\mathbb {R}}_+^K}&\quad m + \sum _{k=1}^K y_k (u_k(x_k)-p_k+{\tilde{\eta }}_k) - \frac{1}{2}\sum _{k=1}^K y_k^2 +h_{1,2}y_1 y_2. \end{aligned}$$
(11)

Examining the first-order condition of the problem given in Eq. 11 and \(\eta \) is such that all goods have positive demand, we see that

$$\begin{aligned} u_1(x_1) + {\tilde{\eta }}_1 - p_1 - y_1 +h_{1,2} y_2&=0, \\ u_2(x_2) + {\tilde{\eta }}_2 - p_2 - y_2 +h_{1,2} y_1&=0, \\ u_j(x_j) + {\tilde{\eta }}_j - p_j - y_j&=0 \quad \text {for }\; j\in \{3,\ldots ,K\}. \end{aligned}$$

For this model, one may desire to estimate the parameter \(h_{1,2}\). We show that instrumental variables can be used to estimate \(h_{1,2}\).

To connect our analysis with linear instrumental variables, we assume that \(x_1 \in {\mathbb {R}}\), that \(u_1(x_1)= \alpha x_1\) where \(\alpha \in {\mathbb {R}}\), and that for all \(j\ne 1\), \(u_j(x_j)=0\). Simplifying the first-order conditions for good one and two now yields that

$$\begin{aligned} y_1&= \frac{-( h_{1,2}p_2+p_1 )}{1-h_{1,2}^2} + \frac{\alpha }{1-h_{1,2}^2} x_1 + \frac{1}{1-h_{1,2}^2}{\tilde{\eta }}_1+\frac{h_{1,2}}{1-h_{1,2}^2}{\tilde{\eta }}_2, \end{aligned}$$
(12)
$$\begin{aligned} y_2&= \frac{-(h_{1,2}p_1 + p_2)}{1-h_{1,2}^2} + \frac{\alpha h_{1,2}}{1-h_{1,2}^2} x_1 + \frac{h_{1,2}}{1-h_{1,2}^2}{\tilde{\eta }}_1+\frac{1}{1-h_{1,2}^2}{\tilde{\eta }}_2. \end{aligned}$$
(13)

These equations describe a structural relationship between shifters, prices, and unobservable heterogeneity. Treating \(Y_1, Y_2\), and \(X_1\) as random and observed, \(p_1\) and \(p_2\) as non-random, and \({\tilde{\eta }}\) as random, we may write this in a standard regression format so that

$$\begin{aligned} Y_1&= \gamma _1 + \beta _1 X_1 + \eta _1,\\ Y_2&= \gamma _2 + \beta _2 X_1 + \eta _2, \end{aligned}$$

where the coefficients and \(\eta \) terms follow Eqs. 12 and 13. In particular, \(\beta _1= \frac{\alpha }{1-h_{1,2}^2}\) and \(\beta _2= h_{1,2} \beta _1\). From this, the ratio of the coefficients \(\beta _2\) and \(\beta _1\) on \(X_1\) give \(h_{1,2}\). Combining this with Proposition 1, we see that \(h_{1,2}\) tells us both a structural utility parameter and the derivative ratio \(DR_{2,1}(x,p,m)\) of compensated demand derivatives, i.e., Hicksian complementarity.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Allen, R., Rehbeck, J. Hicksian complementarity and perturbed utility models. Econ Theory Bull 8, 245–261 (2020). https://doi.org/10.1007/s40505-019-00180-6

Download citation

Keywords

  • Hicksian complementarity
  • Demand
  • Instrumental variables

JEL Classification

  • D01
  • D11
  • C10