The Ramsey/Cass-Koopmans (RCK) Model with Government

This section solves the Ramsey/Cass-Koopmans (RCK) model with government.^[1]

no technological progress, population growth, or depreciation, and a

continuum of consumers with mass 1 distributed on the unit interval

as per Aggregation For Dummies (Macroeconomists).

Consider first the case where government is financed by constant

lump-sum taxes of amount $\Tax$ per period, and spending is at rate

$\mathbf{x}$ per period, and suppose the government has a balanced budget

requirement so that $\mathbf{x} = \Tax$ in every period and there is no

government debt. We suppose further that government spending yields

no utility. The individual’s optimization problem (leaving out the $i$

subscripts that we used in the previous section, but understanding that

they are implicitly present) is now

subject to the household DBC

which has Hamiltonian representation

The first Hamiltonian optimization condition requires $\partial \Ham_{t}/\partial c_{t} = 0$:

The second Hamiltonian optimization condition requires:

Finally, the household’s behavior must satisfy a transversality

constraint, which is equivalent to the intertemporal budget constraint:

which says that the present discounted value of consumption must equal

the current net physical wealth plus human wealth minus the PDV of

taxes.

Now consider the problem from the standpoint of a social planner who has the same utility function as the

individual consumers. If the social planner wants to spend a constant

amount $\mathbf{x}$ per period, the social planner’s budget constraint is

which reflects the fact that the social planner divides total net output

between consumption and government spending. This leads to Hamiltonian

yielding the first order condition

Now recall from the section on Decentralizing the Ramsey/Cass-Koopmans Model

that

where the gross return on capital $\hat{r}_t$ is equal to the net

return $r_t$ plus the depreciation rate.

Thus, the social planner’s DBC is:

which is equivalent to the household’s budget constraint

(2) when $a_{t}=k_{t}$ and $\Tax=\mathbf{x}$. As discussed

in Decentralizing the Ramsey/Cass-Koopmans Model, $a_t=k_{t}$ must hold in equilibrium for identical households,

and $\Tax = \mathbf{x}$ was the balanced budget assumption that we started

off with.

Note that the $\dot{c}_t=0$ locus in the phase diagram is unchanged by

changing $\Tax$ and $\mathbf{x}$. However, the $\dot{k}_{t}=0$ locus is

shifted down by amount $\Tax = \mathbf{x}$.

Now what happens if the government does not face a balanced budget

requirement? Specifically, suppose we continue to have the same

constant amount of spending per period but now want to consider the

effect of allowing taxes to vary over time, which we denote by a

subscript on $\Tax_{t}$. Suppose $d$ is the level of government bonds

(debt); the government’s Dynamic Budget Constraint is

which says that debt must rise by the amount by which spending exceeds

taxes.

The government’s IBC will be the integral of its DBC:

and we assume ${d}_{0}=0$ so that the government starts out with no debt

(to maintain comparability with the previous example).

The DBC of the idiosyncratic family also changes. They can now own

either capital $k_{t}$ or government debt ${d}_{t}$. If the family is to be

indifferent between the two forms of assets, the interest rate must be

the same.

Now the family’s IBC becomes

Note: Nowhere in this equation does the time path of taxes matter;

all that matters is the PDV of taxes. And the time path of taxes

also does not enter the $\dot{c}/c$ equation. Thus, the path

of consumption over time is unaffected by the path of taxes over time!

This is not so surprising when you realize that it is simply the

Ricardian equivalence proposition in this perfect foresight framework.

However, now consider the case where there is a tax on capital income

at rate $\tau$. Furthermore, for simplicity suppose that the

government rebates all of the tax revenue in a lump sum per capita,

and suppose depreciation $\depr=0$. Thus the household budget

constraint becomes

where $\mathbf{z}_{t}$ is the per-capita size of the lump-sum rebates,

The household’s Hamiltonian becomes

The crucial difference between this situation and the previous one is

that now the effective rate of return on saving has been decreased, so

that $\partial \Ham_{t}/\partial a_{t}$ is now $r_{t}(1-\tau)$ rather

than $r_{t}$. Ultimately this produces a consumption Euler equation

of

which implies that the economy will be in equilibrium at

so that the equilibrium level of the marginal product of capital is higher,

and the capital stock must therefore be lower, than before the capital

taxation was instituted.

Notice, however, that because the taxes are being rebated, the aggregate

budget constraint does not change when the tax is imposed:

Thus the social planner will choose exactly the same amount of

consumption as before the tax was instituted.

The crucial point is that if an individual household saves more and

thus causes next year’s capital stock to be a bit higher, the

personal benefit to that household is essentially zero. The higher

taxes that the household will pay next year will be distributed to the

entire population in a lump sum, so the saver will get nothing. The

higher saving of this individual household is basically a positive

externality from the point of view of the other consumers in the

economy. However, if the social planner forces the economy as a whole

to save more, the social planner receives all of the extra tax

revenue.