Question

Country N produces only goods A and B. Quantities and prices of these goods for years 2017 and 2018 are shown below. The base year is 2017. Year 2017 2018 Pa 4 4 In addition, we know that aggregate private consumption c was 1 in 2017 and 2 in 2018 Finally, the country's government imposes taxes T on income (real GDP) Y. Thus, the consumption function is: There are two types of taxes and the government can use only one type: 1. Lump-sum (Fixed): T=T, where τ= 1 2. Proportional to income: T= t, Y, where t=0.1 Find the amount of γ such that the consumption c is the same if the government imposes lump-sum or proportional taxes.

Answer 1

Solution:

We are given base year as 2017, but notice that prices in base year, i.e, 2017, and another year, i.e., 2018, are same, thus to calculate real GDP, given base year won't create any difference. We are given value of C for 2017 and 2018, using which we first find C(bar) and mpc.

Year	A	B	Pa	Pb	Real GDP, Y (A*Pa + B*Pb)
2017	1	1	1	1	1*1 + 1*1 = 2
2018	4	4	1	1	4*1 + 4*1 = 8

Also C = 1 in 2017 and C = 2 in 2018; and C(bar) and mpc remain unchanged overtime.

Then using the consumption function: C = C(bar) + mpc*(Y - T)

So, in 2017, 1 = C(bar) + mpc*(2 - 1)

So, C(bar) + mpc = 1 ... (1)

Similarly, in 2018, 2 = C(bar) + mpc*(8 - 1)

So, C(bar) + 7*mpc = 2 ... (2)

Solving (1) and (2) simultaneously, we get, C(bar) = 5/6 and mpc = 1/6

So, consumption function is C = (5/6) + (1/6)*(Y - T)

Now, with lump-sum tax, this function becomes: C = (5/6) + (1/6)*(Y - 1)

And with proportional tax, the function becomes: C = (5/6) + (1/6)*(Y - 0.1*Y)

Since, we want C to be same from both, equating the two functions we get:

(5/6) + (1/6)*(Y - 1) = (5/6) + (1/6)*(Y - 0.1*Y)

(4/6) + (1/6)*Y = (5/6) + (0.9/6)*Y

(1 - 0.9)*Y = 5 - 4

0.1*Y = 1

Y = 1/0.1 = $10

So, for Y = $10, consumption is the same if the government imposes lump-tax or proportional tax.

(You may try finding C(bar) and mpc using the proportional tax of 0.1*Y, final answer would still be the same)