<p>You can show this concerning the optimization problem with the objective function $U_0 = f(x_1) + x_2$ and the budget restriction $M - p_1 x_1 - p_2 x_2 = 0$. Using the Lagrangian, this leads you to
$$
f'(x_1) = \frac{p_1}{p_2} \quad \text{or} \quad f'^{-1}(\frac{p_1}{p_2}) = x_1^{*} = D_1(p)
$$
You can see that in this special case the optimum quantity of $x_1^{*}$ (Marshallian demand function) does not depend on the income $M$
$$
\frac{\partial D_1}{\partial M} = 0,
$$
The income effect is therefore zero, and you will not consume a different amount of $x_1^{*}$ if the income $M$ varies.</p>

<p><strong>Some further considerations:</strong> Based on the Marshallian $D_i(p, M) = x_i^{*}$ and Hicksian $H_i(p, u) = x_i^{*}$ demand function, you can show some interesting properties of this particular utility function using the Slutsky equation:
$$
\frac{\partial D_i}{\partial p_i} = \frac{\partial H_i}{\partial p_i} - x_i^{*} \frac{\partial D_i}{\partial M}
$$
This shows that the derivative of the Marshallian demand function with respect to price equals the derivative of the Hicksian demand function with respect to price minus the optimal $x_i^{*}$ times the derivative of the Marshallian demand function with respect to income. In this special case, the Marshallian demand function equals the Hicksian demand function, as $\frac{\partial D_i}{\partial M} = 0$.</p>

CC BY-SA 3.0
2016-11-03T13:23:09.923Z
Ah, thanks, that makes a lot of sense! So, since the equivalent variation (EV) and compensated variation (CV) both measure the substitution effect, they have to offset each other, don't they?
Louki
CC BY-SA 3.0
2016-11-03T13:25:54.773Z
As there is no income effect and only a single Hicksian demand function that equals the Marshallian demand function, this means that CV = EV.
Michael Pho
CC BY-SA 3.0
2016-11-03T13:17:08.623Z
2016-11-03T13:17:08.623Z
Michael Pho
<p>I have the following quasi-linear utility function given: $u_0 = f(x_1) + x_2$ (with $f'&gt;0$,$f''&lt;0$). </p>

<p>I know that the indifference curves are vertically parallel, which means that the slope is independent of the consumption of $x_2$. I suppose that there is no income effect, but how can i show this?</p>

<p>Cheers</p>

<microeconomics><utility>
Quasi-linear utility functions
Louki
# Quasi-linear utility functions

(Asked by: username_0 on 2016-11-03)

I have the following quasi-linear utility function given: $u\_0 = f(x\_1) + x\_2$ (with $f'>0$,$f''<0$).

I know that the indifference curves are vertically parallel, which means that the slope is independent of the consumption of $x\_2$. I suppose that there is no income effect, but how can i show this?

Cheers

**Tags:** microeconomics, utility

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## 1 answers

### Answer by username_1 on 2016-11-03. Score: 0 | Accepted answer

You can show this concerning the optimization problem with the objective function $U\_0 = f(x\_1) + x\_2$ and the budget restriction $M - p\_1 x\_1 - p\_2 x\_2 = 0$. Using the Lagrangian, this leads you to
$$
f'(x\_1) = \frac{p\_1}{p\_2} \quad \text{or} \quad f'^{-1}(\frac{p\_1}{p\_2}) = x\_1^{\*} = D\_1(p)
$$
You can see that in this special case the optimum quantity of $x\_1^{\*}$ (Marshallian demand function) does not depend on the income $M$
$$
\frac{\partial D\_1}{\partial M} = 0,
$$
The income effect is therefore zero, and you will not consume a different amount of $x\_1^{\*}$ if the income $M$ varies.

**Some further considerations:** Based on the Marshallian $D\_i(p, M) = x\_i^{\*}$ and Hicksian $H\_i(p, u) = x\_i^{\*}$ demand function, you can show some interesting properties of this particular utility function using the Slutsky equation:
$$
\frac{\partial D\_i}{\partial p\_i} = \frac{\partial H\_i}{\partial p\_i} - x\_i^{\*} \frac{\partial D\_i}{\partial M}
$$
This shows that the derivative of the Marshallian demand function with respect to price equals the derivative of the Hicksian demand function with respect to price minus the optimal $x\_i^{\*}$ times the derivative of the Marshallian demand function with respect to income. In this special case, the Marshallian demand function equals the Hicksian demand function, as $\frac{\partial D\_i}{\partial M} = 0$.

> **Comments:**
> **username_0**: Ah, thanks, that makes a lot of sense! So, since the equivalent variation (EV) and compensated variation (CV) both measure the substitution effect, they have to offset each other, don't they?  
> **username_1**: As there is no income effect and only a single Hicksian demand function that equals the Marshallian demand function, this means that CV = EV.  

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