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.

Appendices

Appendix 1: proofs

1.1 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.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.

1.2 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.