# Using the Slutsky equation

Suppose we have utility:

$$U(x,y)=x^{0.5}y^{0.5}$$

Then Marshallian Demand for good $x$ is:

$$x(p_{x},p_{y},I)=\frac{0.5I}{p_{x}}$$

And Hicksian Demand for good $x$ is:

$$x^{c}(p_{x},p_{y},U)=p_{x}^{-0.5}p_{y}^{0.5}U$$

If $p_{x}=1$, $p_{y}=4$, $I=8$ and $U=2$:

Marshallian Demand for $x$ is 4 and Hicksian Demand for $x$ is 4.

Supposing that the price of good $x$ rises to $p_{x}=4$, how can the Slutsky equation show us the change due to substitution effect and income effect?

Substitution effect: $\frac{\partial x^c}{\partial p_{x}}=-0.25p_{x}^{-2}I$ according to the Slutsky equation. If we plug in $p_{x}=4$, the result is -0.125. How is that meaningful in any way? The Hicksian Demand should drop from 4 to 2 after this price increase but I don't see how the Slutsky equation shows us that.

In your context, Slutsky Equation says, after $p_x$ increases from 1 to 4, the following is true:

\begin{align*} \text{total demand change in $x$} & \\ = \text{demand change in $x$, keeping $U$ fixed at 2} & \\ + \text{demand change in $x$, accounting for the income change if we do not fix $U$ at 2.} & \end{align*}

In the example, the substitution and income effects are of the same sign, so I use absolute values for the demand changes in the equation above. Thus the plus sign rather than the minus sign in Slutsky Equation.

The total demand decrease in $x$ is $3$, that is, $$x_0 - x_1 = 4 - 1 = 3.$$

The demand change in $x$, keeping $U$ fixed at 2 is: $$x^{c}_0 - x^{c}_1 = 4 - 2 = 2.$$

To fix $U$ at 2 under the new price $p_x = 4$, the consumer needs the new income to be $I_1 = 16$ instead of 8. With \$16, the consumer buys 2 units for both$x$and$y$. If we do not fix$U$at 2 when$p_x$increases, which means we allow income to decrease from 16 to 8. The demand reduction in$x$induced by this \$8 income decrease is: $$0.5 \times 16 \div 4 - 0.5 \times 8 \div 4 = 2 - 1 = 1.$$

Therefore we have $3 = 2 + 1$, as Slutsky Equation tells us.

• Thank you for your answer. I know how to derive income effects and substitution effects using the changes in Marshallian/Hicksian demands. But when using the formula $$\frac{\partial x}{\partial p_{x}}=\frac{\partial x^c}{\partial p_{x}} -x^{*}\frac{\partial x^{*}}{\partial I}$$, how do the different terms involved show us the substitution and income effects? – Omrane Nov 1 '16 at 0:46
• @Sadem, in your example the changes are discrete, it's hard to map to the continuous formula of Slutsky Equation, where only infinitesimally small change in price can be plugged into. So it does not make sense to plug in $p_x = 4$ into your expression of the substitution effect. – Paul Nov 1 '16 at 0:57
• @Sadem, if you do plug price $p_x = 4$ into the $\frac{\partial x^c}{\partial p_x}$, the quantity you get means: how much the quantity demanded of $x$ will change when $p_x$ deviates slightly from 4, holding the utility $U$ constant. – Paul Nov 1 '16 at 1:02
• I see! Thank you. Basically I have to see the formula in terms of ratios? Comparing the substitution effect term and the income effect term can tell us their percentage effect on total change, when taking the sequence of substitution THEN income effect. Am I right? – Omrane Nov 1 '16 at 1:09
• No. In your example, 300% price change is just too big to use this formula. – Paul Nov 1 '16 at 1:37