When Coffee Gets More Expensive — Understanding the Slutsky Decomposition
Hello everyone, I am so happy to share my first blog post!
Today, I want to discuss the Slutsky decomposition. I chose this topic because it was one of the first ideas we learned in our Applied Microeconomics class. The name sounds quite fancy, and when I first saw the equations, the idea also looked more complicated than it actually was. However, once we think about what happens when a price changes, the intuition becomes much easier to understand. So, let’s take a look at it.
Suppose the price of coffee goes up. I will probably buy less coffee, but why? One reason is that coffee is now more expensive relative to other goods, so I may replace some coffee with something else. Another reason is that the same income can no longer buy as much as before. Even though the number on my paycheck has not changed, I have less purchasing power.
The problem is that these two things happen together. If I only look at how much coffee I bought before and after the price increase, I can see the total change, but I cannot tell which part came from the change in relative prices and which part came from the loss of purchasing power. This is why we need the compensation step.
Imagine that the price of coffee has already increased, but I am temporarily given some extra income. The amount is just enough to let me reach the same level of satisfaction, or utility, that I had before the price change. This does not mean that I have to buy the same bundle as before. I can still change what I buy, and because coffee is now relatively more expensive, I will probably choose less coffee and more of other goods.
The reason this step is useful is that the extra income keeps me at my original utility level, while the new relative prices are still in place. In this situation, I am as well off as before, but coffee is still relatively more expensive. So, the movement from my original choice to this compensated choice shows how I respond to the change in relative prices. This is the substitution effect.
Next, take away the extra income but keep the new prices exactly the same. Since the relative prices have not changed between these two steps, any additional change in what I buy cannot come from another substitution effect. It comes from having less purchasing power. This is the income effect.
The basic idea is
\[ \text{Total Effect} = \text{Substitution Effect} + \text{Income Effect}. \]
The numerical example below shows how we can separate these two effects and then connect them to the general Slutsky equation.
A Numerical Example
Suppose a consumer chooses between coffee, \(x\), and all other goods, \(y\), with preferences
\[ U(x,y)=xy^2. \]
The consumer has income \(m\) and faces prices \(p_x\) and \(p_y\). The budget constraint is
\[ p_xx+p_yy=m. \]
The consumer solves
\[ \max_{x,y} xy^2 \]
subject to
\[ p_xx+p_yy=m. \]
The Lagrangian is
\[ \mathcal{L} = xy^2+\lambda(m-p_xx-p_yy). \]
The first-order conditions are
\[ \frac{\partial\mathcal{L}}{\partial x} = y^2-\lambda p_x = 0 \]
and
\[ \frac{\partial\mathcal{L}}{\partial y} = 2xy-\lambda p_y = 0. \]
Dividing the two conditions gives
\[ \frac{y}{2x} = \frac{p_x}{p_y}, \]
so
\[ y = \frac{2p_x}{p_y}x. \]
Substituting into the budget constraint gives
\[ 3p_xx=m. \]
Therefore, the Marshallian demands are
\[ x(p_x,m) = \frac{m}{3p_x} \]
and
\[ y(p_y,m) = \frac{2m}{3p_y}. \]
To keep the arithmetic clean, I use normalized prices rather than literal café prices. Suppose
\[ m=120, \qquad p_x=1, \qquad p_y=1. \]
The original bundle is
\[ x_0=40 \]
and
\[ y_0=80. \]
The original utility level is
\[ u_0 = 40(80)^2 = 256000. \]
Now suppose the normalized price of coffee rises from
\[ p_x=1 \]
to
\[ p_x'=8. \]
With income unchanged,
\[ x_1 = \frac{120}{3(8)} = 5. \]
So the total effect is
\[ TE = x_1-x_0 = 5-40 = -35. \]
We know demand fell by 35 units, but we still have not separated why.
Step 1: The Substitution Effect
To isolate the substitution effect, hold the consumer at the original utility level and ask how the consumer would respond to the new relative prices. This gives Hicksian, or compensated, demand. In the standard Slutsky equation, this Hicksian demand measures the substitution effect.
We solve
\[ \min_{x,y} \; p_x'x+p_yy \]
subject to
\[ xy^2=u_0. \]
The Lagrangian is
\[ \mathcal{L} = p_x'x+p_yy + \mu(u_0-xy^2). \]
The first-order conditions are
\[ p_x' = \mu y^2 \]
and
\[ p_y = 2\mu xy. \]
Dividing them gives
\[ \frac{p_x'}{p_y} = \frac{y}{2x}, \]
so
\[ y = \frac{2p_x'}{p_y}x. \]
Substituting into the utility constraint,
\[ x \left( \frac{2p_x'}{p_y}x \right)^2 = u_0, \]
which gives
\[ 4 \frac{(p_x')^2}{p_y^2} x^3 = u_0. \]
Therefore, the Hicksian demand for coffee is
\[ h_x(p_x',p_y,u_0) = \left( \frac{u_0p_y^2} {4(p_x')^2} \right)^{1/3}. \]
Using
\[ u_0=256000, \qquad p_x'=8, \qquad p_y=1, \]
we obtain
\[ x^H=10. \]
The substitution effect is therefore
\[ SE = x^H-x_0 = 10-40 = -30. \]
Even with utility held constant, coffee is relatively more expensive, so the consumer substitutes away from it.
Step 2: The Income Effect
Next, remove the hypothetical compensation and return the consumer to the original income.
Coffee demand falls from 10 to 5, so
\[ IE = x_1-x^H = 5-10 = -5. \]
Nominal income has not changed, but the higher coffee price has reduced real purchasing power.
Therefore,
\[ \text{Total Effect} = \text{Substitution Effect} + \text{Income Effect}, \]
and
\[ -35 = -30+(-5). \]
The example can be summarized as
\[ \boxed{ 40 \rightarrow 10 \rightarrow 5 } \]
where the first movement is the substitution effect and the second is the income effect.
Visualizing the Decomposition
In the figure, \(x_1\) represents coffee and \(x_2\) represents all other goods. Point A is the original choice. Point B is the compensated choice, so A to B is the substitution effect. Removing the compensation moves the consumer from B to C, which is the income effect. Together, they give the total movement from A to C.
From the Example to the Slutsky Equation
The same idea can be written more generally using the duality relationship
\[ x_i(p,e(p,\bar{u})) = h_i(p,\bar{u}). \]
Differentiate both sides with respect to \(p_j\):
\[ \frac{\partial x_i}{\partial p_j} + \frac{\partial x_i}{\partial m} \frac{\partial e}{\partial p_j} = \frac{\partial h_i}{\partial p_j}. \]
By Shephard’s Lemma,
\[ \frac{\partial e}{\partial p_j} = h_j(p,\bar{u}). \]
At the corresponding income level,
\[ h_j(p,\bar{u}) = x_j(p,m). \]
Therefore,
\[ \frac{\partial x_i}{\partial p_j} + x_j \frac{\partial x_i}{\partial m} = \frac{\partial h_i}{\partial p_j}. \]
Rearranging gives the Slutsky equation:
\[ \boxed{ \frac{\partial x_i}{\partial p_j} = \frac{\partial h_i}{\partial p_j} - x_j \frac{\partial x_i}{\partial m} } \]
The left side is the total price effect. The first term on the right is the compensated substitution effect, while the second captures the income effect.
Interpreting the Decomposition
For an own-price change,
\[ \frac{\partial h_i}{\partial p_i} \leq 0. \]
So the substitution effect is always non-positive: holding utility constant, a consumer does not substitute toward a good simply because it became relatively more expensive.
The income effect depends on the type of good. For a normal good, it reinforces the substitution effect after a price increase. For an inferior good, it works in the opposite direction. In the rare case of a Giffen good, the income effect can be strong enough to outweigh the substitution effect.
Takeaway
The main idea of the Slutsky decomposition is that a price change affects demand through two separate channels: relative prices and real purchasing power. In the numerical example, coffee demand falls from 40 to 5 units. Of that total decline, 30 units come from the substitution effect and 5 units come from the income effect. The compensated choice allows us to separate these two forces, even though the compensation itself is only hypothetical. More generally, the Slutsky equation shows how the total effect of a price change can be decomposed into substitution and income effects. This helps us understand not only that demand changes when prices change, but more importantly, why it changes.
Data Source
No external data are used in the numerical example. All numerical values are hypothetical and are used only to illustrate the Slutsky decomposition. The Slutsky decomposition figure is from Dr. Lipsitz’s Class_03_EMP_and_Slutsky.pptx, slide 17. The idea for this blog post was also inspired by what we recently discussed in Dr. Lipsitz’s Applied Microeconomics class.