<p>The (Hicks-) substitution effect is by definition the change in consumption of X induced by a change of the relative prices, holding utility fixed. Thus, the original budget line is "rolled along" its indifference curve until it has the same slope (same relative prices) as the actual budget constraint after the price change. This determines <span class="math-container">$e'$</span>. You can think of the tangent to <span class="math-container">$e'$</span> as a hypothetical budget constraint that compensates you for the lost utility resulting from the increasing price of <span class="math-container">$X$</span> by adding income until you can just afford a bundle with the same utility as the original bundle <span class="math-container">$e$</span>. In total, the change from <span class="math-container">$e$</span> to <span class="math-container">$e^*$</span> is then separated into a change from <span class="math-container">$e$</span> to <span class="math-container">$e'$</span>, (rolling the budget line along the indifference curve: relative price change, holding utility constant = substitution effect) and a change from <span class="math-container">$e'$</span> to <span class="math-container">$e^*$</span> (parallel shift of budget line: income change, holding relative prices constant = income effect).</p>

CC BY-SA 4.0
2020-05-23T21:08:07.520Z
@Giskard: Right, I edited that into my answer.
VARulle
CC BY-SA 4.0
2020-05-23T21:15:00.660Z
I have nothing to give in return, I already gave my upvote before posting the previous comment.
Giskard
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2020-05-23T21:17:07.210Z
@Giskard: That's o.k., I didn't expect something in return, I just did it to make the world a better place.
VARulle
CC BY-SA 4.0
2020-05-23T14:12:42.970Z
2020-05-23T21:07:13.163Z
2020-05-23T21:07:13.163Z
VARulle
<p>I have recently been learning about the concept of utility and the indifference curve. I am having some problems understanding the effects on consumption of two goods <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> of a change in the price of <span class="math-container">$X$</span>. I understand what the substitution effect generally means. It refers to a substitution of the good <span class="math-container">$Y$</span> for the good <span class="math-container">$X$</span> that has now become more expensive. Graphically, it is represented by a shift along the indifference curve from point <span class="math-container">$e$</span> to point <span class="math-container">$e^{'}$</span>.</p>

<p>The scenario given in the diagram is as follows: Income <span class="math-container">$M$</span> = <span class="math-container">$ {$100}$</span> while original prices are at <span class="math-container">$p_x$</span> = <span class="math-container">$ {$10}$</span> and <span class="math-container">$p_y$</span> = <span class="math-container">$ {$10}$</span> The three downward-sloping lines seen in the diagram are the so-called "budgetary constraints". The original budgetary constraint line that is tangent to point <span class="math-container">$e$</span> has equation <span class="math-container">$Y = \frac{M}{p_y} - (\frac{p_x}{p_y})X$</span>. However, I am unsure how the two other budgetary constraint lines are derived. The effect here is a doubling of the price of <span class="math-container">$X$</span> from <span class="math-container">$ {$10}$</span> to <span class="math-container">$ {$20}$</span>. Hence, I would think the straightforward effect is that the the gradient of the budgetary constraint line changes from <span class="math-container">$-1$</span> to <span class="math-container">$-2$</span> but that is not what is directly observed. In fact, there is also a change in the vertical intercept of the budgetary constraint line. </p>

<p>Can someone help to explain to me why the substitution effect leads to consumption at point <span class="math-container">$e^{'}$</span> and why the income effect leads to consumption at point <span class="math-container">$e^{*}$</span>?</p>

<p>The example is taken from the book Economics with Calculus by Michael C. Lovell (Chapter 4, p. 152).</p>

<p><a href="https://i.sstatic.net/WbtWK.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/WbtWK.png" alt="enter image description here"></a></p>

<utility><consumer-theory>
Mathematics of the income and substitution effects
Tan Yong Boon
# Mathematics of the income and substitution effects

(Asked by: username_0 on 2020-05-23)

I have recently been learning about the concept of utility and the indifference curve. I am having some problems understanding the effects on consumption of two goods $X$ and $Y$ of a change in the price of $X$. I understand what the substitution effect generally means. It refers to a substitution of the good $Y$ for the good $X$ that has now become more expensive. Graphically, it is represented by a shift along the indifference curve from point $e$ to point $e^{'}$.

The scenario given in the diagram is as follows: Income $M$ = $ {$100}$ while original prices are at $p\_x$ = $ {$10}$ and $p\_y$ = $ {$10}$ The three downward-sloping lines seen in the diagram are the so-called "budgetary constraints". The original budgetary constraint line that is tangent to point $e$ has equation $Y = \frac{M}{p\_y} - (\frac{p\_x}{p\_y})X$. However, I am unsure how the two other budgetary constraint lines are derived. The effect here is a doubling of the price of $X$ from $ {$10}$ to $ {$20}$. Hence, I would think the straightforward effect is that the the gradient of the budgetary constraint line changes from $-1$ to $-2$ but that is not what is directly observed. In fact, there is also a change in the vertical intercept of the budgetary constraint line.

Can someone help to explain to me why the substitution effect leads to consumption at point $e^{'}$ and why the income effect leads to consumption at point $e^{\*}$?

The example is taken from the book Economics with Calculus by Michael C. Lovell (Chapter 4, p. 152).

[![enter image description here](https://i.sstatic.net/WbtWK.png)](https://i.sstatic.net/WbtWK.png)

**Tags:** utility, consumer-theory

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## 1 answers

### Answer by username_1 on 2020-05-23. Score: 3 | Accepted answer

The (Hicks-) substitution effect is by definition the change in consumption of X induced by a change of the relative prices, holding utility fixed. Thus, the original budget line is "rolled along" its indifference curve until it has the same slope (same relative prices) as the actual budget constraint after the price change. This determines $e'$. You can think of the tangent to $e'$ as a hypothetical budget constraint that compensates you for the lost utility resulting from the increasing price of $X$ by adding income until you can just afford a bundle with the same utility as the original bundle $e$. In total, the change from $e$ to $e^\*$ is then separated into a change from $e$ to $e'$, (rolling the budget line along the indifference curve: relative price change, holding utility constant = substitution effect) and a change from $e'$ to $e^\*$ (parallel shift of budget line: income change, holding relative prices constant = income effect).

> **Comments:**
> **username_1**: @Giskard: Right, I edited that into my answer.  
> **username_2**: I have nothing to give in return, I already gave my upvote before posting the previous comment.  
> **username_1**: @Giskard: That's o.k., I didn't expect something in return, I just did it to make the world a better place.  

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