Suppose the income elasticity of demand for food is0.5 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 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. Is she better or worse off when given a rebate equal to the sales tax payments? Draw a graph and explain.

The formula for arc elasticity is given as:

$$\begin{gathered} Ep=\left(\frac{△Q}{△P}\right)\left(\frac{{\displaystyle \frac{P1+P2}{2}}}{{\displaystyle \frac{Q1+Q2}{2}}}\right) \\ Now, \\ Ep=1 \\ P1=2 \\ P2=2.5 \\ △P=P1-P2 \\ =2-2.5=0.5 \end{gathered}$$

And, since Felicia spends $10000 a year on food and the price of 1 unit of food is $2, so

$Q1=\frac{10000}{2}=5000$

Now find for Q:

$$\begin{gathered} 1=\left(\frac{△Q}{0.5}\right)\left(\frac{{\displaystyle \frac{2+2.5}{2}}}{{\displaystyle \frac{5000+\left(5000+△Q\right)}{2}}}\right) \\ =-1000 \end{gathered}$$

The change in Q is = -1000 units. Thus, her consumption has decreased from 5000 units to 4000 units (5000-1000) due to increased prices. Now, as the elasticity is unitary or equal to 1, the total spending must remain the same after a change in prices; this could also be checked in the next step.

The tax rebate will increase consumption for her. To know the amount needed to use the income elasticity, the formula is given by:

$$\begin{gathered} Ep=\left(\frac{△Q}{△I}\right)\left(\frac{{\displaystyle \frac{I1+I2}{2}}}{{\displaystyle \frac{Q1+Q2}{2}}}\right) \\ Nowitisknownthat: \\ EI=0.5 \\ I1=25000 \\ △I=2500 \\ △I=I1+I2 \\ =27500 \end{gathered}$$

And Q = 4000

Therefore, solving for Q:

$$\begin{gathered} 0.5=\left(\frac{△Q}{2500}\right)\left(\frac{{\displaystyle \frac{25,000+27,500}{2}}}{{\displaystyle \frac{4000+\left(4000+△Q\right)}{2}}}\right) \\ =195 \end{gathered}$$

This means her consumption has increased from 4000 units to 4195 units.

Below is Felicia's indifference curve- budget line graph, where other goods are taken along the y-axis and food on the x-axis. B1and B2are budget lines, and U1 and U2 are indifference curves.

Felicia has been better off after the rebate. The tax rebate’s effect is equal to the sales tax effect. The budget line shifts from B1to B2, due to which Felicia can consume her initial quantity that is 5000 units.Due to tax, her quantity got reduced to 4000, and to consume her initial quantity, she needed $2500 extra, since 5000 x .5 = 2500, which is the rebate amount.