# Question on Isolating The Wealth Effect in Analysis of Changes in Price-Wealth Combinations - MWG Exercise 2.F.3 Parts (e) and (f)

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: 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.

## $$\textbf{My Question}$$

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. 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. 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 $$y < 75$$.

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. 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. 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. 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: 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?

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.