Suppose the income elasticity of demand for food is 0.5 and the price elasticity of demand is \(-1.0 .\) Suppose also that Felicia spends \(\$ 10,000\) a year on food, the price of food is \(\$ 2,\) and that her income is \(\$ 25,000\) a. If a sales tax on food caused the price of food to increase to \(\$ 2.50,\) what would happen to her consumption of food? (Hint: Because a large price change is involved, you should assume that the price elasticity measures an arc elasticity, rather than a point elasticity.) b. Suppose that Felicia gets a tax rebate of \(\$ 2500\) to ease the effect of the sales tax. What would her consumption of food be now? c. Ts she better or worse off when given a rebate equal to the sales tax payments? Draw a graph and explain.

First, we apply the formula of arc price elasticity of demand: \( \text{PED} = \frac{\text{\% change in quantity demanded}}{\text{\% change in price}} \). In this case, the \text{PED} value is given as \(-1.0\). We know that the old price was \$2 and the new price is \$2.50. So, the percentage change in price = \( \frac{\text{(new price - old price)}}{\text{(Average of old and new price)}} \times 100\% = \frac{\$0.50}{\$2.25} \times 100\% \approx 22.22\%\). Then, to find the change in quantity, we rearrange the PED formula and plug in the numbers: \text{\% change in quantity demanded} = \( \text{PED} \times \text{\% change in price} = -1.0 \times 22.22\%\approx -22.22\% \). That means her food consumption will decrease by approximately 22.22%.

Felicia gets a tax rebate of \$2500, which increases her income. Using the income elasticity of demand (IED), we can calculate the effect of this income increase on her food consumption. The formula for the IED is: \(\text{IED} = \frac{\text{\% change in quantity demanded}}{\text{\% change in income}} \). The IED for food is given as 0.5. The percentage change in her income is: \( \frac{\text{(new income - old income)}}{\text{(average of old and new income)}} \times 100\% = \frac{\$2500}{\$26250} \times 100\% \approx 9.52\% \). Rearrange the IED formula and plug in the numbers: \(\text{\% change in quantity demanded} = \text{IED} \times \text{\% change in income} = 0.5 \times 9.52\%\approx 4.76\% \). This indicates her consumption will increase by approximately 4.76% due to the income increment.

From above, Felicia's food consumption decreases by 22.22% due to price increase and it increases by 4.76% due to income increase. The net effect is that her food consumption will change by -22.22% + 4.76% = -17.46%. That means her food consumption will still decrease by ~17.46%, even considering both the price increase and the income increment. So, she is worse off when given a rebate equal to the sales tax payments. You can draw a simple graph with quantity of food on the x-axis, price on the y-axis, and show that after the sales tax and rebate, Felicia's demand curve for food shifts to the left.