Government in the Two Period Model.
We will now place a government in our model.
There are a few preliminary points that we need to make.
1) All government decisions to use some of the production of our real economy (Y) can
be thought of as government expenditure (G).
2) This decision is made exogenously –therefore G is an exogenous variable.
3) The “spending” decision can be financed by:
a) lump-sum taxes (T) or
b) Using the credit market and selling bonds.
The interest rate that the government pays is the real interest rate (r).
Given these three conditions we can proceed.
The government’s CURRENT period budget constraint is:
G = T + B Æ this is just a restatement of condition (3).
The government’s FUTURE period budget constraint will include future spending,
future taxes and repayment with interest of the output (in the form of consumption goods)
that was borrowed in the previous period.
G′ + (1 +r)B = T′
Aside- this is still a TWO period economy so anything borrowed in the first period must
be paid back with interest in the second period- there can not be any second period
borrowing since there is no third period in which to repay.
Rearrange this equation Æ
B = T′ - G′
Substitute this equation into the current period constraint and rearrange.
G + G′ = T + T′
1+r 1+r Notice that the future variables, G′ and T′ are discounted down to the present value so we
can everything into one equation.
This is done by dividing by 1/(1+r).
The left hand side of the equation demotes the present values of government purchases
and the right hand side is the present value of taxes.
Now that the government is in the model the equilibrium will entail:
1) Consumer choices c, c′ and s
2) The government choice of G ,G′. T and T′.
Notice that we have the first period bond market open. This market is where some of the
expenditure of second period is financed. Consumers “spend” their saving in the second
This bond market is open to individuals and governments alike. Both groups compete for
saving in the first period.
Therefore: private saving = total supply of bonds
(1) S = B
Now recall that saving is what is left over after consumption and taxes are deducted from
(2) S = Y – C –T
From the government’s first period budget constraint we know:
(3) B = G-T
By substitution of (3) into (1)
(4) S = G-T By substitution of (4) into (2)
G – T = Y – C –T
Æ (5) Y = C+ G
This is a very, very important result. Why? Because it shows that when the market for
goods C and G is in equilibrium that the bond market is also in equilibrium!!
Notice we went from (1) to (5).
This is a theorem that you will run across in macroeconomic theory. It is a theorem that
essentially says that the timing of tax changes will not affect consumption decisions,
because people will foresee any tax changes as changes in their lifetime income and will
Whether this holds empirically is another matter and we will not discuss that here. We are
however going to outline the idea of Ricardian equivalence.
This treatment is not quite as involved as the one in the text.
Suppose our economy is made up of m identical consumers. Each consumer pays taxes
in the present t and future period t′ . Therefore mt = T and mt′ = T .
Recall the government’s budget constraint:
G + G′/ (1+r) = T + T′/(1+r)
Using the definition stated above we can write this as:
G + G′/ (1+r) = m[ t + t′/(1+r) ]
Now here’s the trick to understanding Ricardian equivalen