Question

Consider a two-period model in which the consumer receives income of y in the current period and y' in the future period. Rather than imposing lump-sum taxes, the government imposes a proportional tax on consumption. In the current period the tax rate is σ and in the future period the tax rate is σ'. As a result, the government collects σc in goods in the current period and σ'c' in goods in the future period. Since G and G' are given, it follows that the present value of the revenue it collects per consumer, σc + σ'c'/(1 + r), equals a constant, G + G'/(1 + r).

a. In this economy the current-period budget constraint is given by (1+σ)c+s=y and the future-period budget constraint is given by (1+σ’)c’=y’+ (1+r)s. Show that these two period constraints imply that the consumer's lifetime budget constraint is given given by (1+σ)c+ (1+σ')c'/ 1+r = y+y'/1+r

b. Suppose that the government chooses to reduce σ and raise σ' such that it continues to collect the same amount of revenue in present value from each consumer (as it must). Explain why the consumer’s lifetime wealth is unaffected by the change in tax rates.

c. In this economy, MRS_c,c' = (1+r)(1+σ)/1+σ'. Does MRS_c,c'  increase or decrease when the government reduces σ and raises σ'? What does this imply about how equilibrium current and future consumption respond to the changes in the two tax rates given that the present value of taxes is unchanged?

d. Why does Ricardian equivalence fail to hold for this economy?

Answer 1

Solution:

given

a two-period model in which the consumer receives income of y in the current period and y' in the future period.

In the current period the tax rate is σ and in the future period the tax rate is σ'.

As a result, the government collects σc in goods in the current period and σ'c' in goods in the future period.

the present value of the revenue it collects per consumer, σc + σ'c'/(1 + r), equals a constant, G + G'/(1 + r).

(a) given

the current-period budget constraint is given by (1+σ)c+s=y

future-period budget constraint is given by (1+σ’)c’=y’+ (1+r)s

Showing  that these two period constraints imply that the consumer's lifetime budget constraint is given given by (1+σ)c+ (1+σ')c'/ 1+r = y+y'/1+r:

poof:

Y = C + I + G + X - M

Budgetary Deficits and Surpluses Rule

•Spending –Goods and services (G) + Transfers to persons (Tr)

•Revenue –Taxes (Tx)

•Net Taxes -Tx – Tr = NT

•Surplus

G + Tr < Tx

G < Tx – Tr

G < NT

•Deficit

G + Tr > Tx

G > Tx – Tr

G > NT

Hence

This depicts the given value of current period budget constraint as C+S=Y,

Like wise it was given future period budget constraint and lifetime budgetary constraint.

(b) given that the government chooses to reduce σ and raise σ' such that it continues to collect the same amount of revenue in present value from each consumer, the consumer’s lifetime wealth is unaffected by the change in tax rates:

reason;

By applying the above rule of budgetary deficit and surplus, consumers lifetime wealth is unaffected by changing the tax rates.

(c) given

MRS_c,c' = (1+r)(1+σ)/1+σ'. Does MRS_c,c'  

increase or decrease when the government reduces σ and raises σ' is:

ie

Marginal rate of substitution is MRS = =1+r

(d) Ricardian equivalence fail to hold for this economy :

because of all the above conditions the Ricardian equivalence fail to hold for this economy