(For students who were assigned Chapter 11) Assume that, without taxes, the consumption schedule for an economy is as shown below:

GDP, Billions	Consumption, Billions
\(100	120
200	200
300	280
400	360
500	440
600	520
700	600

1. Graph this consumption schedule. What is the size of the MPC?

2. Assume that a lump-sum (regressive) tax of \)10 billion is imposed at all levels of GDP. Calculate the tax rate at each level of GDP. Graph the resulting consumption schedule and compare the MPC and the multiplier with those of the pretax consumption schedule.

3. Now suppose a proportional tax with a 10 percent tax rate is imposed instead of the regressive tax. Calculate and graph the new consumption schedule, and calculate the MPC and the multiplier.

4. Finally, impose a progressive tax such that the tax rate is 0 percent when GDP is \(100, 5 percent at \)200, 10 percent at \(300, 15 percent at \)400, and so forth. Determine and graph the new consumption schedule, noting the effect of this tax system on the MPC and the multiplier.

5. Use a graph similar to Figure 13.3 to show why proportional and progressive taxes contribute to greater economic stability, while a regressive tax does not.

As consumption depends on disposable income (or GDP without taxes), the consumption on Y-axis changes according to GDP on X-axis. The consumption curve for the given schedule is as follows:

As income increases, consumption also increases in some proportion. Thus, the consumption curve is positively sloped.

The income changes by $100 billion, the corresponding consumption increases by $80 billion. Thus the MPC is as follows:

$$\begin{gathered} MPC=\frac{△Consumption}{△Income} \\ =\frac{\$80billion}{\$100billion} \\ =0.8 \end{gathered}$$

Hence, MPC is 0.8.

After a regressive tax of $10 billion, the disposable income falls by $10 billion at each level of GDP. Also, the tax reduces the consumption by multiples times of the MPC.

△Consumption = MPC x Tax

= 0.8 x $10 billion

= $8 billion

The consumption will decline by $8 billion at each level of GDP.

The tax rate, percent at each level of income is calculated as:

$\mathrm{Tax}\mathrm{Rate}=\frac{\mathrm{Tax}}{\mathrm{GDP}}\times 100$

The new consumption schedule and tax rate at each level of income are as shown below:

Therefore, the new consumption curve is:

The consumption curve shifts leftwards due to a decline in income and consumption. However, the change in consecutive GDP levels (200 – 100 = 100) and consumptions (192 – 112 = 80) remain the same. Thus, MPC also remains the same, that is, 0.8.

The multiplier is calculated as:

$$\begin{gathered} k=\frac{1}{1-MPC} \\ k=\frac{1}{1-0.8} \\ k=5 \end{gathered}$$

Hence, the multiplier remains 5. The lump-sum tax does not affect MPC and multiplier.

A proportional tax of 10% will reduce the disposable income by 10% of the GDP each time.

$Tax=\frac{10\times GDP(billiondollars)}{100}$

At the same time, the consumption will reduce equally to the product of MPC and tax at that level of income.

△Consumption = MPC x Tax, where MPC IS 0.8.

The new consumption schedule and taxes are shown below:

Since consumption constantly changes by $72 billion (=184 – 112) and income changes by $90 billion (=180 – 90). The MPC (tax inclusive) is:

$$\begin{gathered} MPC=\frac{△Consumption}{△Income} \\ =\frac{72}{100} \\ =0.72 \end{gathered}$$

The multiplier value (tax inclusive) is:

$$\begin{gathered} k=\frac{1}{1-MPC} \\ k=\frac{1}{1-0.72} \\ k=\frac{1}{0.28} \\ k=3.571 \end{gathered}$$

Hence, MPC is 0.72, and the multiplier is 3.571. The proportional tax reduces the MPC and multiplier value.

With each successive level of GDP, the progressive tax rate increases by 5%. The amount of tax at each level is calculated as:

Also, the progressive tax reduces consumption by the following formula:

△Consumption = MPC x Tax

= 0.8 x Tax

Since the tax is different at each level, the amount of change in consumption will also be different, as shown in the table.

The change in consumption can be calculated as the difference between two consecutive levels of consumption (after-tax).

Since consumption is changing at a varying rate, the income MPC and multiplier have to be calculated separately, as shown below:

Hence, the MPC and multiplier values differ at each level of GDP under the progressive tax system.

The following graphs show the regressive tax, proportional tax, and progressive tax against the GDP

Since proportional tax (in respect to increasing GDP) is constant at 10%, it is a straight horizontal line. The regressive tax is a curve with a declining proportion of GDP. At the same time, the progressive tax is a positive sloped straight line because the ratio of tax constantly increases with an increase in GDP.

The regressive tax declines at a variable rate while proportional and progressive tax maintains the pace of change in tax and GDP. Therefore, the progressive and proportional tax maintains greater economic stability as compared to the regressive tax.