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

Question

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?

Answer 1

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.