Using the Slutsky equation

Question

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.

Answer 1

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.