Alternative to Ricardian equivalence and calculus

An alternative interpretation of Ricardian equivalence? Consider a modi�ed version of the two-period framework with government studied in the chapter. By �government� here we will mean just the ��scal authority� ; suppose that there is no �monetary authority� at all.

The government and the representative consumer each live for both periods of the economy, and suppose that there are never any credit constraints on the consumer. The government does not have access to lump- sum taxes, only proportional consumption taxes. However (this is di�erent from our baseline framework), the consumption taxes the government collects in a given period are not restricted to be levied on consumption from only in that period. To be more precise, suppose that total consumption tax revenues the government collects in period 1 are based only on period-1 consumption (e.g., because there was no period zero). However, total consumption tax revenues the government collects in period 2 are based on both period-1 consumption and period-2 consumption. That is, a portion of the revenue collected in period 2 is based on period-1 consumption, and the remaining portion of the revenue collected in period 2 is based on period-2 consumption.

Denote by τ1,1 , the tax rate on period-1 consumption that is levied in period 1; denote by τ1,2 the tax rate on period-1 consumption that is levied in period 2; and by τ2,2 the tax rate on period-2 consumption that is levied in period 2. There is no τ2,3, (which would represent the tax rate on period-2 consumption that is levied in period 3) because the economy does not exist in period 3.

With this notation, the government ' s period-1 and period-2 budget constraints in real terms are

g1 + b1 = (1 + r)b0 + t1,1c1

g2 + b2 = (1 + r)b1 + t1,2c1 + t2,2c2

The representative consumer ' s period-1 and period-2 budget constraints in real terms are

(1 + τ1,1)c1 + a1 = (1 + r)a0 + y1

τ1,2c1 + (1 + τ2,2)c2 + a2 = (1 + r)a1 + y2

For simplicity, suppose that the government and consumer each begin period 1 with zero assets. As usual, you can think of all the tax rates as being numbers between zero and one (but they need not be so restricted). The remainder of the notation is as in the chapter.

1. Construct the government's lifetime budget constraint (LBC), showing important steps. Provide brief economic interpretation. For simplicity, assume b0 = b2 = 0

g1 + b1 = t1,1c1

g2 = (1 + r)b1 + t1,2c1 + t2,2c2

g2 = (1 + r) (t1,1c1 − g1) + t2,2c2 g2

1 + r = (t1,1c1 − g1) +

t1,2c1 + t2,2c2 1 + r

g1 + g2

1 + r = t1,1c1 +

t1,2c1 + t2,2c2 1 + r

The present value of government spending must equal the present value of tax revenues.

2. Construct the consumer's lifetime budget constraint (LBC), showing important steps. Provide brief economic interpretation.

(1 + τ1,1)c1 + a1 = y1

τ1,2c1 + (1 + τ2,2)c2 = (1 + r)a1 + y2

τ1,2c1 + (1 + τ2,2)c2 = (1 + r) (y1 − (1 + τ1,1)c1) + y2( 1 + τ1,1 +

τ1,2 1 + r

) c1 +

(1 + τ2,2)c2 1 + r

= y1 + y2

1 + r

3. The essence of the way we de�ned Ricardian equivalence was that an economy exhibits Ricardian equivalence if, holding �xed its sequence of government spending � and also assuming no credit con- straints and that consumers' planning horizons are the same (in length) as the government's planning

1

horizon � a change in the timing of lump-sum taxes has no e�ect on consumption or national savings. In the analysis here, suppose that the government keeps its sequence of g1 and g1 unchanged, but decides to cut the tax rate in period 1 on period-1 consumption � that is, it lowers the tax rate τ1,1. Is it possible for this economy to exhibit Ricardian equivalence even though in the framework in this problem taxes are not lump sum (and, as stated in the problem, there are no credit constraints or any mismatches between the planning horizons of consumers or the government)? If so, carefully show how/why and provide brief economic interpretation. If not, precisely explain why not. (Hint: Base the analysis on one or both of the LBCs derived in parts a and b.)(

1 + τ1,1 + τ1,2 1 + r

) c1 +

(1 + τ2,2)c2 1 + r

= y1 + y2

1 + r

(1 + τ2,2)c2 1 + r

= y1 + y2

1 + r −

( 1 + τ1,1 +

τ1,2 1 + r

) c1

(1 + τ2,2)c2 = (1 + r)y1 + y2 − ( 1 + τ1,1 +

τ1,2 1 + r

) (1 + r)c1

c2 = (1 + r)y1 + y2

(1 + τ2,2) −

1 + τ1,1 + τ1,2 1+r

1 + τ2,2 (1 + r)c1

Previously, a change in the tax on c1 required an o�-setting tax on c2, causing the consumer to change her consumption in each period. But here, a change in τ1,1 can be o�set by a change in τ1,2/(1 + r), leaving the overall tax on c1 unchanged. So it is possible that Ricardian Equivalence holds.

2