<p>Yes, I think for question (e), there is a bit of an &quot;error&quot; in the reasoning.</p>
<p>The total demand change can always be decomposed in an income and substitution effect. If there are only two goods: the substitution effect is always determined in sign: it is negative for the good that becomes relatively more expensive and positive for the good that becomes relatively more cheap (this no longer holds if there are more than two goods).</p>
<p>For (e), my intuition is the following:</p>
<ol>
<li>The relative price of good 1 increases: this means that the substition effect for good 1 is negative (SE &lt; 0)</li>
<li>The final consumption of good 1 goes up (this differs from the solution manual) (SE + IE &gt; 0)</li>
</ol>
<p>Points 1 and 2 show that the income effect of good 1 must be positive (IE &gt; 0). However we also have that the real wealth from period 1 to period 2 goes down. Positive income effect with a real income decline means that good 1 must be inferior at some point.</p>
<p>For question (f) the intuition is the folllowing:</p>
<ol>
<li>The relative price of good 2 goes down, so the subsitution effect on good 2 is positive (SE &gt; 0)</li>
<li>The final demand for good goes down (SE + IE &lt; 0)</li>
</ol>
<p>Given points 1 and 2, we have that the income effect on good 2 must be negative (IE
&lt; 0). Combining this with the fact that the real wealth from period 1 to period 2 goes up, we see that good 2 must be inferior at some point.</p>

CC BY-SA 4.0
2024-02-16T14:41:28.960Z
2024-02-16T14:41:28.960Z
tdm
<p>I am doing exercises in Chapter 2 of MWG. I feel I got completely lost in exercise 2.F.3 parts (e) and (f).</p>
<h2><span class="math-container">$\textbf{Exercise}$</span></h2>
<p>Here is the question:
<a href="https://i.sstatic.net/eHfKo.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/eHfKo.png" alt="enter image description here" /></a>
I have solved parts (a) to (d). In particular,</p>
<blockquote>
<p>I denote <span class="math-container">$\mathbf{p}^1 = (p_1^1,p_2^1) = (100,100)$</span>, <span class="math-container">$\mathbf{p}^2 = (p_1^2,p_2^2) = (100,80)$</span>, <span class="math-container">$w_1 = 100 \times 100 + 100 \times 100 = 20000$</span>, <span class="math-container">$w_2 = 120 \times 100 + 80y = 12000 + 80y$</span>, <span class="math-container">$x(\mathbf{p}^1,w_1) = \{(x_1(\mathbf{p}^1,w_1),x_2(\mathbf{p}^1,w_1))\} = \{(100,100)\}$</span>, and <span class="math-container">$x(\mathbf{p}^2,w_2) = \{(x_1(\mathbf{p}^2,w_2),x_2(\mathbf{p}^2,w_2))\} = \{(120,y)\}$</span>, where <span class="math-container">$y$</span> is the quantity of consumption of good 2 in year 2.</p>
</blockquote>
<p>So this is how I proceeded part (a)</p>
<blockquote>
<p>Notice that we always have <span class="math-container">$x(\mathbf{p}^1,w_1) \neq x(\mathbf{p}^2,w_2)$</span> because <span class="math-container">$(100,100) \neq (120,y)$</span>. Suppose <span class="math-container">$\mathbf{p}^1 \cdot x(\mathbf{p}^2,w_2) \leq w_2$</span>, that is,
<span class="math-container">\begin{align*}
100 \cdot 120 + 100 \cdot y \leq 20000 \implies y \leq 80.
\end{align*}</span>
For his behavior to violate the weak axiom, we need <span class="math-container">$\mathbf{p}^2 \cdot x(\mathbf{p}^1,w_1) \leq w_2$</span>, that is,
<span class="math-container">\begin{align*}
100 \cdot 100 + 80 \cdot 100 \leq 12000 + 80 \cdot y \implies 75 \leq y.
\end{align*}</span>
So, we have <span class="math-container">$75 \leq y \leq 80$</span>.</p>
</blockquote>
<p>From part (a), we know that the consumer's behavior is consistent (i.e., satisfies the weak axiom) if and only if <span class="math-container">$y \in [0,75) \cup (80,\infty]$</span>. For parts (b) to (d), please refer to my previous post: <a href="https://economics.stackexchange.com/questions/57607/question-on-the-weak-axiom-of-revealed-preference-and-the-definition-of-revealed">Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation</a>.</p>
<h2><span class="math-container">$\textbf{My Question}$</span></h2>
<p>Now, I am confused about parts (e) and (f). Let us take (e) for an example. It asked over what range of <span class="math-container">$y$</span> would you conclude that good 1 is inferior (at some price) for this consumer. According to the textbook (page 25), a commodity is called inferior at <span class="math-container">$(\mathbf{p},w)$</span> if it's wealth effect is negative at <span class="math-container">$(\mathbf{p},w)$</span>. However, in this question, both price and wealth might have changed, from <span class="math-container">$(\mathbf{p}^1,w_1)$</span> to <span class="math-container">$(\mathbf{p}^2,w_2)$</span>. I have difficulties isolate the wealth effect from such a change in price-wealth combination. The same problem applies to part (f). Could someone please help me out? How should I proceed the analysis?</p>
<h2><span class="math-container">$\textbf{Some of My Thought}$</span></h2>
<p>Here is some of my thought using graph, though I honestly doubt it is all correct. <a href="https://i.sstatic.net/AdxOy.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/AdxOy.png" alt="enter image description here" /></a> In the graph above, the red line is the original budget line <span class="math-container">$B_{\mathbf{p}^1,w_1}$</span>, where <span class="math-container">$w_1 = 20000$</span>. Consider the compensated price change from <span class="math-container">$(\mathbf{p}^1,w_1)$</span> to <span class="math-container">$(\mathbf{p}^2,w_3)$</span>, where <span class="math-container">$w_3 = 18000$</span>, so that the consumer can just afford his original consumption bundle <span class="math-container">$x(\mathbf{p}^1,w_1) = (100,100)$</span>. The blue line is the budget line for this compensated price change. Since the weak axiom holds, we have that <span class="math-container">$x(\mathbf{p}^2,w_3)$</span> can only be on the line segment (of the blue budget line) below (and not include) the point <span class="math-container">$(100,100)$</span>, denoted set <span class="math-container">$A$</span>. This implies that <span class="math-container">$x_1(\mathbf{p}^2,w_3) &lt; 100$</span>.</p>
<p>Now we impose wealth change. <strong>First, let us focus on part (e).</strong> Notice that the blue budget line intersect with the line <span class="math-container">$q_1 = 120$</span> at the point <span class="math-container">$(75,120)$</span>. Suppose first <span class="math-container">$w_2 &gt; w_3$</span>; that is, suppose that the blue budget line were to be shifted upward to reach <span class="math-container">$B_{\mathbf{p}^2,w_2}$</span>, as is shown in the graph below (the blue dashed line being <span class="math-container">$B_{\mathbf{p}^2,w_2}$</span>). Since the question asserts that <span class="math-container">$x(\mathbf{p}^2,w_2) = (120,y)$</span>, we must have that the consumer's choice would have to move upward from <span class="math-container">$x(\mathbf{p}^2,w_3)$</span> to <span class="math-container">$x(\mathbf{p}^2,w_2)$</span>. However, this means that such an increase in wealth will have a positive wealth effect on the consumption of good 1, because the consumption of good 1 goes up as wealth goes up. Thus, this implies that good 1 is a normal good. <a href="https://i.sstatic.net/497x8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/497x8.png" alt="enter image description here" /></a> Suppose next that <span class="math-container">$w_2 &lt; w_3$</span>; that is, suppose that the blue budget line were to be shifted downward to reach <span class="math-container">$B_{\mathbf{p}^2,w_2}$</span>, as is shown in the graph below. Again, since the question asserts that <span class="math-container">$x(\mathbf{p}^2,w_2) = (120,y)$</span>, we must have that the consumer's choice would have to move upward from <span class="math-container">$x(\mathbf{p}^2,w_3)$</span> to <span class="math-container">$x(\mathbf{p}^2,w_2)$</span>. But this time, it means that such a decrease in wealth will have a negative wealth effect on the consumption of good 1, because the consumption of good 1 goes up as wealth goes down. Thus, this implies that good 1 is an inferior good. Therefore, good 1 is inferior if and only if <span class="math-container">$w_2 &lt; w_3$</span>, which implies that <span class="math-container">$y &lt; 75$</span>. <a href="https://i.sstatic.net/s0XXn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/s0XXn.png" alt="enter image description here" /></a></p>
<p><strong>Now, let us focus on part (f).</strong> When impose wealth change, suppose first that <span class="math-container">$w_2 &lt; w_3$</span>, as is shown in the graph below. Again, since the question asserts that <span class="math-container">$x(\mathbf{p}^2,w_2) = (120,y)$</span>, we must have that the consumer's choice would have to move left-ward from <span class="math-container">$x(\mathbf{p}^2,w_3)$</span> to <span class="math-container">$x(\mathbf{p}^2,w_2)$</span>. However, it means that such a decrease in wealth will have a positive wealth effect on the consumption of good 2, because the consumption of good 2 goes down as wealth goes down. Thus, in this case, good 2 is a normal good. <a href="https://i.sstatic.net/2Q5It.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2Q5It.png" alt="enter image description here" /></a> Suppose next that <span class="math-container">$w_2 &gt; w_3$</span>. Since the weak axiom holds, we only need to consider when <span class="math-container">$y &gt; 80$</span>. First, consider the case when <span class="math-container">$80 &lt; y &lt; 100$</span>, as is shown in the graph below. Again, since the question asserts that <span class="math-container">$x(\mathbf{p}^2,w_2) = (120,y)$</span>, we must have that the consumer's choice would have to move left-ward from <span class="math-container">$x(\mathbf{p}^2,w_3)$</span> to <span class="math-container">$x(\mathbf{p}^2,w_2)$</span>. But this time, it means that  such an increase in wealth will have a negative wealth effect on the consumption of good 2, because the consumption of good 2 goes down as wealth goes up. Thus, in this case, good 2 would be an inferior good. <a href="https://i.sstatic.net/bsQLD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bsQLD.png" alt="enter image description here" /></a> Next, consider the case when <span class="math-container">$y \geq 100$</span>. In particular, consider the case shown in the graph below. Notice that we do not know where the point <span class="math-container">$x(\mathbf{p}^2,w_3)$</span> would locate on <span class="math-container">$A$</span>. So, if <span class="math-container">$y &gt; 100$</span>, we might as well have that the consumer's choice would move to the right from <span class="math-container">$x(\mathbf{p}^2,w_3)$</span> to <span class="math-container">$x(\mathbf{p}^2,w_2)$</span>. In such cases, the increase in wealth will have a positive wealth effect on the consumption of good 2, because the consumption of good 2 goes up as wealth goes up. Therefore, the scenario of <span class="math-container">$y &gt; 100$</span> is inconclusive. <a href="https://i.sstatic.net/Z09Gw.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Z09Gw.png" alt="enter image description here" /></a> Hence, we conclude that good 2 is inferior if and only if <span class="math-container">$80 &lt; y &lt; 100$</span>.</p>
<p>Again, I am not sure if this makes sense or if it is correct. I would really appreciate it if someone could help me check!</p>
<h2><span class="math-container">$\textbf{Answer from The Solution Manual}$</span></h2>
<p>Here is the answer provided by the Solution Manual:
<a href="https://i.sstatic.net/6Fo7I.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6Fo7I.png" alt="enter image description here" /></a>
Honestly, I do not quite understand how it draws its conclusions. For example, in part (e), it says that &quot;the real wealth decreases&quot;, &quot;the relative price of good 1 increases&quot;, and &quot;the demand for good 2 decreases&quot; together implies that the wealth effect on good 1 must be negative. Why?</p>
<p>Similarly, for part (f), it says that &quot;the real wealth increases&quot;, &quot;the relative price of good 2 decreases&quot;, and &quot;the demand for good 2 decreases&quot; together implies that the wealth effect on good 2 must be negative. Why?</p>

<microeconomics><mathematical-economics><consumer-theory><preferences><demand>
Question on Isolating The Wealth Effect in Analysis of Changes in Price-Wealth Combinations - MWG Exercise 2.F.3 Parts (e) and (f)
Shenron
# Question on Isolating The Wealth Effect in Analysis of Changes in Price-Wealth Combinations - MWG Exercise 2.F.3 Parts (e) and (f)

(Asked by: username_0 on 2024-02-14)

I am doing exercises in Chapter 2 of MWG. I feel I got completely lost in exercise 2.F.3 parts (e) and (f).

## $\textbf{Exercise}$

Here is the question:
[![enter image description here](https://i.sstatic.net/eHfKo.png)](https://i.sstatic.net/eHfKo.png)
I have solved parts (a) to (d). In particular,

> I denote $\mathbf{p}^1 = (p\_1^1,p\_2^1) = (100,100)$, $\mathbf{p}^2 = (p\_1^2,p\_2^2) = (100,80)$, $w\_1 = 100 \times 100 + 100 \times 100 = 20000$, $w\_2 = 120 \times 100 + 80y = 12000 + 80y$, $x(\mathbf{p}^1,w\_1) = \{(x\_1(\mathbf{p}^1,w\_1),x\_2(\mathbf{p}^1,w\_1))\} = \{(100,100)\}$, and $x(\mathbf{p}^2,w\_2) = \{(x\_1(\mathbf{p}^2,w\_2),x\_2(\mathbf{p}^2,w\_2))\} = \{(120,y)\}$, where $y$ is the quantity of consumption of good 2 in year 2.

So this is how I proceeded part (a)

> Notice that we always have $x(\mathbf{p}^1,w\_1) \neq x(\mathbf{p}^2,w\_2)$ because $(100,100) \neq (120,y)$. Suppose $\mathbf{p}^1 \cdot x(\mathbf{p}^2,w\_2) \leq w\_2$, that is,
> \begin{align\*}
> 100 \cdot 120 + 100 \cdot y \leq 20000 \implies y \leq 80.
> \end{align\*}
> For his behavior to violate the weak axiom, we need $\mathbf{p}^2 \cdot x(\mathbf{p}^1,w\_1) \leq w\_2$, that is,
> \begin{align\*}
> 100 \cdot 100 + 80 \cdot 100 \leq 12000 + 80 \cdot y \implies 75 \leq y.
> \end{align\*}
> So, we have $75 \leq y \leq 80$.

From part (a), we know that the consumer's behavior is consistent (i.e., satisfies the weak axiom) if and only if $y \in [0,75) \cup (80,\infty]$. For parts (b) to (d), please refer to my previous post: [Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation](https://economics.stackexchange.com/questions/57607/question-on-the-weak-axiom-of-revealed-preference-and-the-definition-of-revealed).

## $\textbf{My Question}$

Now, I am confused about parts (e) and (f). Let us take (e) for an example. It asked over what range of $y$ would you conclude that good 1 is inferior (at some price) for this consumer. According to the textbook (page 25), a commodity is called inferior at $(\mathbf{p},w)$ if it's wealth effect is negative at $(\mathbf{p},w)$. However, in this question, both price and wealth might have changed, from $(\mathbf{p}^1,w\_1)$ to $(\mathbf{p}^2,w\_2)$. I have difficulties isolate the wealth effect from such a change in price-wealth combination. The same problem applies to part (f). Could someone please help me out? How should I proceed the analysis?

## $\textbf{Some of My Thought}$

Here is some of my thought using graph, though I honestly doubt it is all correct. [![enter image description here](https://i.sstatic.net/AdxOy.png)](https://i.sstatic.net/AdxOy.png) In the graph above, the red line is the original budget line $B\_{\mathbf{p}^1,w\_1}$, where $w\_1 = 20000$. Consider the compensated price change from $(\mathbf{p}^1,w\_1)$ to $(\mathbf{p}^2,w\_3)$, where $w\_3 = 18000$, so that the consumer can just afford his original consumption bundle $x(\mathbf{p}^1,w\_1) = (100,100)$. The blue line is the budget line for this compensated price change. Since the weak axiom holds, we have that $x(\mathbf{p}^2,w\_3)$ can only be on the line segment (of the blue budget line) below (and not include) the point $(100,100)$, denoted set $A$. This implies that $x\_1(\mathbf{p}^2,w\_3) < 100$.

Now we impose wealth change. **First, let us focus on part (e).** Notice that the blue budget line intersect with the line $q\_1 = 120$ at the point $(75,120)$. Suppose first $w\_2 > w\_3$; that is, suppose that the blue budget line were to be shifted upward to reach $B\_{\mathbf{p}^2,w\_2}$, as is shown in the graph below (the blue dashed line being $B\_{\mathbf{p}^2,w\_2}$). Since the question asserts that $x(\mathbf{p}^2,w\_2) = (120,y)$, we must have that the consumer's choice would have to move upward from $x(\mathbf{p}^2,w\_3)$ to $x(\mathbf{p}^2,w\_2)$. However, this means that such an increase in wealth will have a positive wealth effect on the consumption of good 1, because the consumption of good 1 goes up as wealth goes up. Thus, this implies that good 1 is a normal good. [![enter image description here](https://i.sstatic.net/497x8.png)](https://i.sstatic.net/497x8.png) Suppose next that $w\_2 < w\_3$; that is, suppose that the blue budget line were to be shifted downward to reach $B\_{\mathbf{p}^2,w\_2}$, as is shown in the graph below. Again, since the question asserts that $x(\mathbf{p}^2,w\_2) = (120,y)$, we must have that the consumer's choice would have to move upward from $x(\mathbf{p}^2,w\_3)$ to $x(\mathbf{p}^2,w\_2)$. But this time, it means that such a decrease in wealth will have a negative wealth effect on the consumption of good 1, because the consumption of good 1 goes up as wealth goes down. Thus, this implies that good 1 is an inferior good. Therefore, good 1 is inferior if and only if $w\_2 < w\_3$, which implies that $y < 75$. [![enter image description here](https://i.sstatic.net/s0XXn.png)](https://i.sstatic.net/s0XXn.png)

**Now, let us focus on part (f).** When impose wealth change, suppose first that $w\_2 < w\_3$, as is shown in the graph below. Again, since the question asserts that $x(\mathbf{p}^2,w\_2) = (120,y)$, we must have that the consumer's choice would have to move left-ward from $x(\mathbf{p}^2,w\_3)$ to $x(\mathbf{p}^2,w\_2)$. However, it means that such a decrease in wealth will have a positive wealth effect on the consumption of good 2, because the consumption of good 2 goes down as wealth goes down. Thus, in this case, good 2 is a normal good. [![enter image description here](https://i.sstatic.net/2Q5It.png)](https://i.sstatic.net/2Q5It.png) Suppose next that $w\_2 > w\_3$. Since the weak axiom holds, we only need to consider when $y > 80$. First, consider the case when $80 < y < 100$, as is shown in the graph below. Again, since the question asserts that $x(\mathbf{p}^2,w\_2) = (120,y)$, we must have that the consumer's choice would have to move left-ward from $x(\mathbf{p}^2,w\_3)$ to $x(\mathbf{p}^2,w\_2)$. But this time, it means that such an increase in wealth will have a negative wealth effect on the consumption of good 2, because the consumption of good 2 goes down as wealth goes up. Thus, in this case, good 2 would be an inferior good. [![enter image description here](https://i.sstatic.net/bsQLD.png)](https://i.sstatic.net/bsQLD.png) Next, consider the case when $y \geq 100$. In particular, consider the case shown in the graph below. Notice that we do not know where the point $x(\mathbf{p}^2,w\_3)$ would locate on $A$. So, if $y > 100$, we might as well have that the consumer's choice would move to the right from $x(\mathbf{p}^2,w\_3)$ to $x(\mathbf{p}^2,w\_2)$. In such cases, the increase in wealth will have a positive wealth effect on the consumption of good 2, because the consumption of good 2 goes up as wealth goes up. Therefore, the scenario of $y > 100$ is inconclusive. [![enter image description here](https://i.sstatic.net/Z09Gw.png)](https://i.sstatic.net/Z09Gw.png) Hence, we conclude that good 2 is inferior if and only if $80 < y < 100$.

Again, I am not sure if this makes sense or if it is correct. I would really appreciate it if someone could help me check!

## $\textbf{Answer from The Solution Manual}$

Here is the answer provided by the Solution Manual:
[![enter image description here](https://i.sstatic.net/6Fo7I.png)](https://i.sstatic.net/6Fo7I.png)
Honestly, I do not quite understand how it draws its conclusions. For example, in part (e), it says that "the real wealth decreases", "the relative price of good 1 increases", and "the demand for good 2 decreases" together implies that the wealth effect on good 1 must be negative. Why?

Similarly, for part (f), it says that "the real wealth increases", "the relative price of good 2 decreases", and "the demand for good 2 decreases" together implies that the wealth effect on good 2 must be negative. Why?

**Tags:** microeconomics, mathematical-economics, consumer-theory, preferences, demand

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## 1 answers

### Answer by username_1 on 2024-02-16. Score: 1 | Accepted answer

Yes, I think for question (e), there is a bit of an "error" in the reasoning.

The total demand change can always be decomposed in an income and substitution effect. If there are only two goods: the substitution effect is always determined in sign: it is negative for the good that becomes relatively more expensive and positive for the good that becomes relatively more cheap (this no longer holds if there are more than two goods).

For (e), my intuition is the following:

1. The relative price of good 1 increases: this means that the substition effect for good 1 is negative (SE < 0)
2. The final consumption of good 1 goes up (this differs from the solution manual) (SE + IE > 0)

Points 1 and 2 show that the income effect of good 1 must be positive (IE > 0). However we also have that the real wealth from period 1 to period 2 goes down. Positive income effect with a real income decline means that good 1 must be inferior at some point.

For question (f) the intuition is the folllowing:

1. The relative price of good 2 goes down, so the subsitution effect on good 2 is positive (SE > 0)
2. The final demand for good goes down (SE + IE < 0)

Given points 1 and 2, we have that the income effect on good 2 must be negative (IE
< 0). Combining this with the fact that the real wealth from period 1 to period 2 goes up, we see that good 2 must be inferior at some point.


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