## 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.

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## Notes

- 1.
This ratio is related to the diversion ratio, which is used in analysis of antitrust (Farrell and Shapiro 2010).

- 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.
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*k*th good has specific meaning such as a particular brand and size of potato chip. - 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.
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.
See Rieskamp et al. (2006) where behavior from the attraction and compromise effect could be viewed as a form of complementarity.

- 7.
- 8.
See Amir et al. (2017) for more details on demand systems arising from quadratic utilities.

- 9.
Here \(v'\) denotes the transpose of a vector

*v*.

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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

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

where \(D_k\) is the *k*th 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

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

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

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

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

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

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,

Applying Cramer’s rule for good *j* results in

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

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

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

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

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

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

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

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.

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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

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### Keywords

- Hicksian complementarity
- Demand
- Instrumental variables

### JEL Classification

- D01
- D11
- C10