Slutsky equation with marshallian demand

Question

We have marshallian demands for goods 1 and 2:

$x_1^* = \frac{I}{2p_1}$ and $x_2^* = \frac{I}{2p_2}$ where $I$ is income and $p_i$ is price.

We need to solve the slutsky equation for income effect and substitution effect as such:

$\frac{D (x_i)}{D (p_j)} = \frac{D(H^i(p1, p2, i))}{ D(p_i)} - x_j*\frac{D(x_i^*)}{D(I)}$

So in balance marshallian demand is the same as the compensated demand.

I solved the left hand side of equation and got a result of $0$.

Then I moved on to the income effect and got the following result:

$-\frac{I}{2p_j}*\frac{1}{2p_i} = -\frac{I}{4p_jp_i}$

Substituting the results from above in to the slutsky equation gives a positive result for substitution effect. I thought that that the substitution effect is always negative. Can anyone help me out?

Answer 1

The first term on the RHS of the Slutsky equation (as you write it) should be $\frac{\partial h_i(p_i,p_j,I)}{\partial p_j}$, not $\frac{\partial h_i(p_i,p_j,I)}{\partial p_i}$.

So your subsequent algebra is fine but you're conflating cross-price substitution with the own-price substitution: when you let $i \neq j$ in the Slutsky equation, the first term on the RHS $\left(\text{again, }\frac{\partial h_i(p_i,p_j,I)}{\partial p_j}\right)$ measures the cross-price substitution effect of an increase in the price of $j$ on the (compensated) demand of $i$. You found it to be positive, so goods $i$ and $j$ are net substitutes (the cross-price partials of $h_i$ and $h_j$ cannot be oppositely signed -- this follows from Shephard's Lemma).

If you let $i = j$, however, you will then find that the Slutsky equation implies that the own-price substitution effect is negative for both goods, which you correctly note should be negative:

$$ \begin{align} \frac{\partial h_i(p_i,I)}{\partial p_i} &= \frac{\partial x_i}{\partial p_i} + x_i \frac{\partial x_i}{\partial I} \\ &= -\frac{I}{2p_i^2} + \frac{I}{2p_i} \left(\frac{1}{2p_i}\right) \\ &= -\frac{I}{4p_i^2} < 0. \end{align} $$