# Hick's and Slutsky's approaches lead to different income effects. Why?

Suppose a cup of coffee and a plate of beans are sold at € 1 and € 3 respectively during the winter. In summer, the government decides to remove the subsidy on coffee and its new price per cup goes up to € 2. If a customer has an income of € 10 and the utility function $$u(c,b) = cb$$, what is the income effect?

It appears that Hicks' way and Slutsky's way lead to two different income effects.

The initial demands are $$(c_0, b_0) = (\frac{0.5 \times 10}{1}, \frac{0.5 \times 10}{3}) = (5, 5/3)$$.

Hick's way: The new demands in summer are $$(c_1, b_1) = (\frac{0.5 \times 10}{2}, \frac{0.5 \times 10}{3}) = (5/2, 5/3)$$. The Hicksian demand with utility $$u(c_0, b_0)$$ is $$(c_2, b_2) = \left(\frac{5 \sqrt 2}{2}, \frac{5 \sqrt 2}{3}\right)$$. The income effect we get from this is $$c_1 - c_2 \approx 1.036$$.

Slutsky's way: The new demands in summer will be $$(c_1, b_1)$$ as we calculated above. Then the Marshallian demand on the budget line $$2x + 3y = 2c_0 + 3b_0 = 15$$ will be $$(c_2, b_2)=\left(\frac{15}{4}, \frac{15}{6}\right)$$. The income effect we get from this is $$c_1 - c_2 = 2.5-3.75 \approx 2.083$$.

If the EMP is a dual of the UMP, how are the two methods leading to different income effects?

EMP being a dual of UMP is unrelated to income effects.

If $$(c_0, b_0)$$ denotes the initial demand, and $$(c_1, b_1)$$ denotes the demand after the price of coffee has changed, then

• To find the Hicksian substitution and income effect, we solve the following problem: $$\begin{eqnarray*} \min_{(c, b)\in\mathbb{R^2_+}} & \ 2c + 3b \\ \text{s.t. } & cb \geq c_0b_0\end{eqnarray*}$$

Let $$(c_2^h, b_2^h)$$ denotes the solution to the above problem. This implies that $$2c_2^h + 3b_2^h < 2c_0 + 3b_0$$. The inequality will be strict because $$(c_0, b_0)$$ is the equilibrium bundle when price ratio is $$\frac{1}{3}$$, and since $$u$$ is increasing, differentiable and strictly quasi-concave in $$\mathbb{R}^2_{++}$$, cost will be minimised at a different bundle. [Please note that duality says that if you minimise $$c + 3b$$ subject to $$cb \geq c_0b_0$$, you'll get $$(c_0, b_0)$$ as the solution]

• To find the Slutsky substitution and income effect, we solve the following problem: $$\begin{eqnarray*} \max_{(c, b)\in\mathbb{R^2_+}} & \ cb \\ \text{s.t. } & 2c+3b \leq 2c_0+3b_0\end{eqnarray*}$$

Let $$(c_2^s, b_2^s)$$ denotes the solution to this problem. Since $$u$$ is an increasing function, the solution will satisfy $$2c_2^s+3b_2^s = 2c_0+3b_0 > 2c_2^h + 3b_2^h$$. Therefore, the two bundles $$(c_2^s, b_2^s)$$ and $$(c_2^h, b_2^h)$$ will be different, and consequently, the substitution effect and income effect will be different for the two methods.

• Certain parts are confusing. How is the dual min$c+3b$ subject to $cb = c_0b_0$? Isn't it supposed to be max $cb$ subject to $2c+3b = 2c+0 + 3b_0$ which is the Marshallian demand used in Slutsky's approach? I am referring to 1.2 and Figure 1.
– Isa
Commented Jul 21, 2022 at 10:13
• What is the income effect in my example if two methods lead to different income effects? Also, if there's inconsistency in the effects when used with different methods, is it used in practice? Thank you! ^_^
– Isa
Commented Jul 21, 2022 at 10:17
• Answer to your first question is that since $(c_0, b_0)$ is the solution to the the maximization problem $\max \ cb$ subject to $c + 3b \leq 10$, it will also solve the expenditure mimimization problem $\min \ c+3b$ subject to $cb \geq c_0b_0$. That is what I meant when I made a comment about Duality.
– Amit
Commented Jul 21, 2022 at 13:16
• To your second question, the magnitude of income effect can vary depending on the method used. So it is appropriate to specify that whether it is the Hicksian Income effect or the Slutsky income effect.
– Amit
Commented Jul 21, 2022 at 13:18
• I get it now ^_^
– Isa
Commented Jul 21, 2022 at 22:44