Macro assignment

Macroeconomic Theory

1 Chapter 7: Intertemporal Fiscal Policy

We introduce a central government that collects taxes from households (and/or �rms) and makes

consumption expenditures. In doing so, we discuss the government's intertemporal budget con-

straint which involves saving or borrowing across periods when taxes are unequal to expenditures.

• Primary Budget De�cit (Surplus): exists in any period where tax revenue is smaller (larger) than government expenditures.

• Secondary Budget De�cit (Surplus): exists in any period where tax revenue plus net interest income is smaller (larger) than government expenditures.

→ We will generally refer to a secondary 'budget de�cit (surplus)

⇒ Let sgovt denote the (secondary) budget surplus, tt denote tax revenue, gt denote govern- ment expenditures and bt denote net government wealth (all in real terms). Then:

s gov t ≡ tt + rtbt−1 −gt (1)

→ if sgovt > 0 then government is running a surplus → if sgovt < 0 then government is running a de�cit

1.1 Two-Period Partial Equilibrium Fiscal Model

We will start with a dynamic two-period model of the government's intertemporal budget for

simplicity. We later generalize this to the in�nite-period framework.

Figure 1: Timeline of Events

Spring 2014 | © Sanjay K. Chugh 119

Period 1 Period 2

b0 b2

Government activities during period 1:

government spending and tax collection

b1

Government activities during period 2:

government spending and tax collection

Beginning of analysis

End of analysis

�

Figure 39. Timing of events for the government.

Unlike household or �rm, there is assume to be no utility or pro�t maximization here. Tax

and spending policies are assumed to be exogenous.

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1.1.1 Government Budget Constraint

⇒ Periods t=1,2 → Resources available to government: real tax revenue, tt; real wealth, bt−1; net real interest

income, bt−1rt

→ Possible expenditures: real expenditures, gt; real wealth for beginning of next period, bt −→ Thus the period-t budget constraint is:

gt + bt = bt−1(1 + rt) + tt (2)

⇒ Lifetime Budget Constraint Evaluating Equation (2) for t = 1,2 we have:

g1 + b1 = b0(1 + r1) + t1

g2 + b2 = b1(1 + r2) + t2

→ We can derive the lifetime budget constraint for the government by combining periods-1 and -2 budget constraints and imposing the corresponding initial and terminal conditions b0 = b2 = 0:

g2 + b2 = (t1 + b0(1 + r1)−g1) (1 + r2) + t2 g2 = (t1 −g1) (1 + r2) + t2

g2 1 + r2

= t1 −g1 + t2

1 + r2

⇒ g1 + g2

1 + r2 = t1 +

t2 1 + r2

(3)

→ Equation (3) is the Lifetime Budget Constraint (LBC) for the government, which equates the present discounted value of lifetime expenditures to the present discounted value of lifetime

resources.

⇒ Implication of the LBC for Tax Policy: For a given path of real interest rates and govern- ment expenditures, any changes to tax revenue now must be o�set by tax changes in the future.

→ Re-arranging Equation (3) for t2:

t2 = g1(1 + r2) + g2 − t1(1 + r2)

Di�erentiating with respect to t1:

dt2 dt1

= −(1 + r2)

⇒ dt2 = −(1 + r2)dt1 (4)

→ Intuition: Since the left-hand side of Equation (3) does not change when t1 changes, t2 must change in an o�setting fashion so that the LBC holds.

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1.1.2 Ricardian Equivalence

As special case in our dynamic framework, the Ricardian-Equivalence Proposition asserts

that the timing of taxes do not matter for private agent's economic behavior because they in-

ternalize the government's lifetime budget constraint into their own.

→ Requires all taxes to be lump-sum (or non-distortionary) taxes, where the amount owed by an economic agent does not depend on their choices.

EXAMPLE: Consider the representative household in the two-period consumption-savings model

with exogenous labor income. Suppose that the government collects lump-sum taxes tt from

them. Their period-t budget constraint would then be:

ct + at = (1 + rt)at−1 + yt − tt (5)

This yields a lifetime budget constraint for the household of:

c1 + c2

1 + r2 = y1 +

y2 1 + r2

− t1 − t2

1 + r2 (6)

where the initial and terminal conditions a0 = a2 = 0 have been imposed.

→ Consider the intertemporal dimension for c1 and c2. The relevant price for optimal choice is given by the slope of the LBC, ∂c2/∂c1. Using Equation (6) to compute that slope:

c2 = y1(1 + r2) + y2 − c1(1 + r2)− t1(1 + r2)− t2

⇒ ∂c2 ∂c1

= −(1 + r2)

(i) From above, the relevant price for the intertemporal decision does not depend on taxes

(ii) From equation (4), changes to timing of will not change the PDV of lifetime resources

⇒ Because of (i) and (ii), Ricardian Equivalence holds.

EXAMPLE: Suppose that the government collects distortionary taxes on the representative

households' consumption so that tt = τtct for t = 1,2. Using this expression into Equation (6),

and re-arranging for c2 to compute the intertemporal price of consumption:

c1 + c2

1 + r2 = y1 +

y2 1 + r2

− τ1c2 − τ2c2 1 + r2

c1(1 + τ1) + c2(1 + τ2)

1 + r2 = y1 +

y2 1 + r2

c2 =

( 1 + r2 1 + τ2

) ( y1 +

y2 1 + r2

) − c1

(1 + τ1)

(1 + τ2) (1 + r2)

⇒ ∂c2 ∂c1

= − (1 + τ1)

(1 + τ2) (1 + r2)

⇒ Since the distortionary tax a�ects the intertemporal price of consumption, tax changes will a�ect private behavior and Ricardian Equivalence will fail.

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1.2 National Savings and Generalization to the In�nite-Period Framework

Whether using the two-period model or the in�nite period framework, we de�ne real national

savings as the sum of private savings and government savings for a given period t.

→ Recall the de�nition of real private savings:

s priv t ≡ at −at−1

→ Using the general expression for the household's period-t budget constraint with labor and interest income, in addition to lump-sum taxes, we have:1

s priv t = wtnt + rtat−1 − ct − tt (7)

→ Using our de�nition of the real budget surplus for government savings, we have:

s gov t ≡ tt + rtbt−1 −gt (8)

⇒ National Savings, snatt , is the sum of private and government savings for a given period t :

snatt = s priv t + s

gov t

snatt = (wtnt + rtat−1 − ct − tt) + (tt + rtbt−1 −gt)

snatt = wtnt + rtat−1 − ct + rtbt−1 −gt (9)

Intuitively: National savings is household income less consumption, plus government income less

expenditures.

1.2.1 Ricardian Equivalence and National Savings

⇒ Under non-distortionary taxation, Ricardian Equivalence will hold: → From inspection of Equation (7), a decrease in tt will increase private savings. → From inspection of Equation (8), a decrease in tt will decrease government savings → From inspection of Equation (9), a decrease in tt will not a�ect national savings because

the increase in private savings and government savings are exactly o�setting.

⇒ Under distortionary taxation, Ricardian Equivalence will fail, and changes to the timing of taxation will a�ect national savings through the impact on private savings.

1 Note that yt = wtnt in the previous example with exogenous labor income

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1.2.2 Financial Market Equilibrium With Government

Recall that real private savings is increasing in the future real interest rate, rt+1. Since real

government expenditures and tax rates are assumed to be exogenous, real national savings is

then also increasing in the future real interest rate:

⇒ Financial market equilibrium occurs at the real interest rate rt+1∗ that equates the quan- tity supplied of national savings snatt ∗ with the quantity demanded of investment invt∗.

EXAMPLE: Suppose that a government taxing consumption in a distortionary fashion tem-

porarily reduces the period t tax rate. Holding constant current government expenditures, there

will be a reduction in the budget surplus (or increase in the budget de�cit).

→ Since Ricardian Equivalence fails, there will be an increase in current consumption and decrease in current saving at every possible future real interest rate:

• Crowding Out: The decrease in private investment that results from an increase in the equilibrium real interest rate caused by government policy.

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