Why does Slutsky compensation "overcompensate" the consumer?

Question

Suppose I have a Marshallian demand function $x_M(p_x^0,p_y,m^0)$. As I understand it, Slutsky compensation is defined as

$$T_S = \Delta p_x \cdot x_M(p_x^0,p_y,m^0)$$

Can someone explain why this compensation overcompensates the consumer?

Answer 1

Here's a figure to explain:

Starting from the old price line, where the optimal consumption bundle is point $A$, we increase the price of $y$ to get the new price line.

The Slutsky compensation says that we have to give the consumer enough extra income so that he can afford to old bundle ($A$) at the new price. Thus, we shift the new budget constraint out to the dashed line.

The reason this "overcompensates" is that, taking the dashed line as the budget constraint, the consumer could afford bundle $B$, which gives him higher utility than he started with.

Answer 2

Slutsky compensation makes the original consumption bundle again exactly affordable after the price change. This implies that the original utility level is reachable. But higher utility may also be reachable by changing the consumption bundle. So there may be overcompensation.

Hicksian compensation makes the original utility level again exactly reachable after the price change. The consumption bundle provided may differ from the original one.

The overcompensation is explained in many textbooks and online sources, e.g. here.

Answer 3

For both Slutsky transfer and Hicksian transfer, the only parameter changing in the problem is income $m$. You are adding money in the case of a price increase and removing money in the case of a price decrease. For Hicksian, this means

$$T_H = e(p_x^f,p_y,v^o) - m$$

For Slutsky

$$T_S = \Delta p x_M$$

To understand why Slutsky overcompensates but Hicksian does not, we need to think more carefully about them.

Consider Hicksian compensation. The Hicksian transfer provides the consumer with just enough money to return to his original indifference curve. But he doesn't have enough to buy the original bundle. He only has enough to buy a bundle with the same utility level. This is the same point we would get if we solved

$$\min p_x^f x + p_y y$$ subject to $$v^o = U(x,y)$$ In other words, the Hicksian transfer moves us to the bundle that solves that problem. So for this reason Hicks doesn't overcompensate. It is exactly the amount required given the final price and desired utility level. Hence, it is tangent.

Now consider the Slutsky transfer. Algebraically, we can see it allows us to purchase the original bundle

\begin{align*} y &= -\frac{p_x^f}{p_y} x + \frac{m + T_s}{p_y}\\ y &= -\frac{p_x^f}{p_y} x + \frac{m + p_x^f x - p_x^o x}{p_y}\\ p_y y &= -p_x^fx + m + p_x^f x - p_x^o x\\ p_y y &= m + - p_x^o x\\ p_y y + p_x^o x &= m \end{align*}

So it restores the ability to purchase the original bundle. But this line doesn't have the same slope as our original budget constraint before the price change. This one is parallel to the Hicksian transfer. Yet we already established that the Hicksian transfer was the solution to the expenditure minimization problem and hence tangent. Given that we usually assume strictly convex curves, it must form a chord and thus have bundles that offer greater utility than the Hicksian transfer. Thus, the consumer can move to a higher indifference curve. And so Slutsky transfer overcompensates the consumer because we can now buy bundles on higher indifference curves than prior to the price increase.