# Slutsky matrix computed from budget shares

In Haag et al. 2009 Testing and imposing Slutsky symmetry in nonparametric demand systems, the authors claim that the Slutsky matrix can be computed from share functions expressed in terms of logged prices and logged wealth. In particular they claim that, let $p$ and $w$ be logged prices and logged wealth. $s_{ij}=\frac{\partial{b_i(p,w)}}{\partial{p_j}}+\frac{\partial(b(p,w))}{\partial w}b_j(p,w)+b_j(p,w)b_i(p,w)-\delta_{i,j}b_i(p,w)$ where $\delta_{i,j}$ is the kronecker delta function that says that in the diagonal terms we have to substract $b_i(p,w)$. He claims that this is the Slutsky matrix in unlogged prices terms, but how can this beif everything is in terms of shares. I have tried to derive it myself but I cannot reproduce what they have, but maybe I am making some mistake. The authors cite Mas-Colell et al. do you know the exact reference in the book they only provide the general reference. Reference:Haag, B. R., Hoderlein, S., & Pendakur, K. (2009). Testing and imposing Slutsky symmetry in nonparametric demand systems. Journal of Econometrics, 153(1), 33–50.

• Could you please give the full reference ? – optimal control Aug 30 '15 at 16:04

In all indexed references to the Slutsky matrix in Mas-Collel et al. no such relation can be found. It is my impression that the authors meant that the expression they write conveys the same information and restrictions with the original, being a scaled version of it.

I will use indices $i$, $k$, and write the logarithm explicitly in order to avoid confusion.

$$\frac {\partial b_i}{\partial \ln p_k} = \frac {\partial (x_ip_i/w)}{\partial \ln p_k}$$

Get liberal with partial differentiation, think of it as a differential, which gives

$$\partial \ln p_k = \partial p_k/p_k$$

Insert above to get $$\frac {\partial b_i}{\partial \ln p_k} = p_k\frac {\partial (x_ip_i/w)}{\partial p_k} = \frac{p_ip_k}{w}\cdot \frac {\partial x_i}{\partial p_k}$$ and rearranging

$$\implies \frac {\partial x_i}{\partial p_k} = \frac{w}{p_ip_k} \cdot \frac {\partial b_i}{\partial \ln p_k} \tag{1}$$

So we have expressed the first term of the usual Slutsky element in terms of logarithmic derivatives of budget shares. For the second component we have

$$\frac {\partial b_i}{\partial \ln w} = w\frac {\partial (x_ip_i/w)}{\partial w} = w \frac {(\partial x_i/\partial w)p_iw - x_ip_i}{w^2} = p_i\frac {\partial x_i}{\partial w} - b_i$$

rearranging and also multiplying by $x_k$ we get

$$\frac {\partial x_i}{\partial w} x_k = \frac {x_k}{p_i}\frac {\partial b_i}{\partial \ln w} + \frac {x_k}{p_i} b_i$$

Multiply and divide each element by $p_kw$:

$$\frac {\partial x_i}{\partial w} x_k = \frac {x_kp_kw}{p_ip_kw}\frac {\partial b_i}{\partial \ln w} + \frac {x_kp_kw}{p_ip_kw} b_i$$

$$\implies \frac {\partial x_i}{\partial w} x_k = \frac {w}{p_ip_k}\frac {\partial b_i}{\partial \ln w} b_k + \frac {w}{p_ip_k} b_ib_k \tag{2}$$

Combining,

$$(1),(2) \implies s_{ik} = \frac {\partial x_i}{\partial p_k} + \frac {\partial x_i}{\partial w} x_k \\ =\frac{w}{p_ip_k} \cdot \frac {\partial b_i}{\partial \ln p_k} + \frac {w}{p_ip_k}\cdot \frac {\partial b_i}{\partial \ln w} b_k + \frac {w}{p_ip_k}\cdot b_ib_k$$

$$\implies s_{ik} = \frac{w}{p_ip_k} \Big(\frac {\partial b_i}{\partial \ln p_k} + \frac {\partial b_i}{\partial \ln w} b_k + b_ib_k\Big) \tag{3}$$

The term in the big parenthesis is what the authors give as the off-diagonal element of the Slutsky matrix. The scaling factor $\frac{w}{p_ip_k}$ is symmetrical, so the expression in the big parenthesis nevertheless reflects the same symmetry conditions as the usual $s_{ik}$ terms.

If you apply the same method for the diagonal elements you will get the $-b_i$ term.

This form of the Slutsky matrix I believe is old story, and it has come about for purposes of econometric estimation of demand systems. For example, in Theil, H., & Clements, K. W. (1980). Recent Methodological Advances in Economic Equation Systems. American Behavioral Scientist, 23(6), 789-809. we find this form of the Slutsky elements, which the authors call "elements of the Slutsky matrix" without any "explanation" as to why the textbook expressions are scaled.