Marge also buys hamburgers for $$ \$ 1 heta $$ and root beer for $$ \$ 2 $$. She has income of $$ \$ 2 heta heta $$. Her budget constraint will experience a parallel outward shift if __ O A) the price of hamburgers falls to $$ \$ 5 $$, the price of root beer falls to $$ \$ 1 $$, and her income falls to $$ \$ 10 heta $$. O B) the price of hamburgers rises to $$ \$ 2 heta $$, the price of root beer rises to $$ \$ 4 $$, and her income remains the same. O C) the price of hamburgers falls to $$ \$ 8 $$, the price of root beer falls to $$ \$ 1 $$, and her income rises to $$ \$ 24 heta $$. O D) the price of hamburgers rises to $$ \$ 2 heta $$, the price of root beer rises to $$ \$ 4 $$, and her income rises to $$ \$ 50 heta $$.

Question

Marge also buys hamburgers for $$ \$ 1 	heta $$ and root beer for $$ \$ 2 $$. She has income of $$ \$ 2 	heta 	heta $$. Her budget constraint will experience a parallel outward shift if __ O A) the price of hamburgers falls to $$ \$ 5 $$, the price of root beer falls to $$ \$ 1 $$, and her income falls to $$ \$ 10 	heta $$. O B) the price of hamburgers rises to $$ \$ 2 	heta $$, the price of root beer rises to $$ \$ 4 $$, and her income remains the same. O C) the price of hamburgers falls to $$ \$ 8 $$, the price of root beer falls to $$ \$ 1 $$, and her income rises to $$ \$ 24 	heta $$. O D) the price of hamburgers rises to $$ \$ 2 	heta $$, the price of root beer rises to $$ \$ 4 $$, and her income rises to $$ \$ 50 	heta $$.

UpStudy AI · Accepted Answer

Let the original situation be described as follows. Marge’s consumption bundle satisfies

$$
\theta H + 2R = 2\theta^2,
$$

where $$H$$ is the number of hamburgers (priced at $$\theta$$ dollars each) and $$R$$ is the number of root beers (priced at $$2$$ dollars each).

A parallel outward shift of the budget line means that the slope of the budget line remains unchanged while the intercepts increase proportionally. In other words, after the change:

1. The new prices must satisfy the same ratio as the original prices.
2. The new income must allow the same linear combination of the goods but at a higher intercept (i.e. more purchasing power).

The original slope is determined by the price ratio. If we solve the original constraint for $$R$$:

$$
2R = 2\theta^2 - \theta H \quad \Longrightarrow \quad R = \theta^2 - \frac{\theta}{2}H,
$$

so the slope is

$$
-\frac{\theta}{2}.
$$

A change in prices that leaves the ratio $$\frac{\text{Price of hamburger}}{\text{Price of root beer}}=\frac{\theta}{2}$$ unchanged means that if the hamburger price becomes $$p_H'$$ and the root beer price becomes $$p_R'$$, then

$$
\frac{p_H'}{p_R'}=\frac{\theta}{2}.
$$

Now, inspect the options:

**Option A:**  
- New prices: $$p_H' = 5$$ and $$p_R' = 1$$.  
- Ratio: $$\frac{5}{1}=5$$.  
- To have $$5=\frac{\theta}{2}$$ we would need $$\theta=10$$.  
- However, the income falls to $$10\theta$$, and a lower income would shift the budget line inward rather than outward.

**Option B:**  
- New prices: $$p_H' = 2\theta$$ and $$p_R' = 4$$.  
- Ratio: $$\frac{2\theta}{4}=\frac{\theta}{2}$$, which is the same as before.  
- But the income remains unchanged at $$2\theta^2$$, so although the slope is the same, the intercepts do not move outward.

**Option C:**  
- New prices: $$p_H' = 8$$ and $$p_R' = 1$$.  
- Ratio: $$\frac{8}{1}=8$$.  
- For parallelism we would require $$8=\frac{\theta}{2}$$, or $$\theta=16$$.  
- Also, the income rises to $$24\theta$$.  
- Although the income increases, the price ratio (and hence the slope) is not the same as the original (unless $$\theta$$ specifically equals 16), so in general it is not a parallel shift.

**Option D:**  
- New prices: $$p_H' = 2\theta$$ and $$p_R' = 4$$.  
- Ratio: $$\frac{2\theta}{4}=\frac{\theta}{2}$$, which is exactly the original ratio.  
- The income rises to $$50\theta$$.  
- With the price ratio unchanged and the income increased, the entire budget constraint shifts outward in a parallel manner.

Thus, option **D** is the correct answer.
Option D is the correct answer because it maintains the original price ratio of hamburgers to root beer and increases Marge's income, resulting in a parallel outward shift of her budget constraint.

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