# is income effect in the given quasilinear function equal to 0?

utility function given to me is as follows, $$u(x,y) = x+4 \sqrt{y}$$ which is a simple quasilinear function. There is a change in the given price vector, $$(p_x,p_y) = (1,1) \to (0.25,1)$$, and the income, M=1. the question asks about the absolute values of substitution and Income Effect using Hicks method.

approaching this in any way results in the demand $$(x^* , y^*) = (0,1)$$, consumer would have prefered to consume $$y^* = 4$$ but as it is not affordable for the given budget so consumer would choose to consume all of y and nothing of x initially.

upon further solving i am getting $$SE= 3$$ and $$ME=15$$ but the income effect in quasilinear case is always zero.

can anyone please assist me with this?

• You forgot to specify the income of the consumer. Jun 12, 2022 at 11:30
• Income effect in case of quasi-linear is not always $0$.
– Amit
Jun 12, 2022 at 11:34
• @Amit it just occurred to me that ME = 0 only for the non linear good of the function, in this case it is y. Jun 12, 2022 at 11:39
• Yes, but in the above case the demand for $Y$ is not independent of income. Demand for $Y$ is 1 when income is 1 and both prices are also 1, and demand for $Y$ is 2 when income is 2 and both prices are still 1. So demand for $Y$ increased with income.
– Amit
Jun 12, 2022 at 11:48
• I haven't solved it, but they don't look correct. Reason is that with $M=1$, and $0.25\leq p_X \leq 1$, such a high income effect is impossible because even if you spend all your income on $X$ when price of $X$ is $p_X =0.25$, the maximum possible demand for $X$ is not more than $4$ which is a strict upper bound on these effects because you'll always spend some amount on $Y$.
– Amit
Jun 12, 2022 at 12:08

First let's consider the following problem: $$\begin{eqnarray*} \max_{(x, y) \in \mathbb{R}^2_+} & x + 4\sqrt{y} \\ \text{s.t.} \ & p_Xx + p_Yy \leq M\end{eqnarray*}$$ Solution to the above problem is known as demand, and it is given by $$\begin{eqnarray*} (x^d, y^d) (p_X, p_Y, M) = \begin{cases} \left(0, \frac{M}{p_Y}\right) &\text{if } \frac{M}{p_Y} \leq \frac{4p_X^2}{p_Y^2} \\ \left(\frac{M}{p_X}-\frac{4p_X}{p_Y}, \frac{4p_X^2}{p_Y^2}\right) &\text{if } \frac{M}{p_Y} > \frac{4p_X^2}{p_Y^2}\end{cases} \end{eqnarray*}$$ To see how to find it, procedure is similar to this one: https://economics.stackexchange.com/a/16475/11824

Now let's consider the following problem: $$\begin{eqnarray*} \min_{(x, y) \in \mathbb{R}^2_+} & p_Xx + p_Yy \\ \text{s.t.} \ & x + 4\sqrt{y} \geq \mu\end{eqnarray*}$$ Solution to the above problem is known as Hicksian demand, and it is given by $$\begin{eqnarray*} (x^h, y^h) (p_X, p_Y, \mu) = \begin{cases} \left(0, \frac{\mu^2}{16}\right) &\text{if } \frac{p_X}{p_Y} \geq \frac{\mu}{8} \\ \left(\mu-\frac{8p_X}{p_Y}, \frac{4p_X^2}{p_Y^2}\right) &\text{if } \frac{p_X}{p_Y} < \frac{\mu}{8}\end{cases} \end{eqnarray*}$$

So, total price effect $$= (x^d, y^d)(0.25,1,1) - (x^d, y^d)(1,1,1) = \left(3,\frac{1}{4}\right) -(0,1) = \left(3, -\frac{3}{4}\right)$$

and the initial satisfaction level is $$\mu = u(0,1) = 4$$.

Now we can find the Income and Substitution effect using Hicksian method:

Hicksian SE $$=(x^h, y^h)(0.25, 1, 4) - (x^d, y^d)(1,1,1) = \left(2,\frac{1}{4}\right)-(0,1) = \left(2, -\frac{3}{4}\right)$$

Hicksian IE $$= (x^d, y^d)(0.25,1,1)-(x^h, y^h)(0.25, 1, 4) = \left(3,\frac{1}{4}\right)-\left(2, \frac{1}{4}\right) = (1,0)$$

• i found the hicksian demand by simply solving the new budget with new prices and i did get the same answer, i.e. (3, 1/4). But i prefer your method more, could you please tell how you got μ/8 while solving for the hicksian demand? Jun 15, 2022 at 7:09
• i think that is MRS Jun 15, 2022 at 7:09