Government in the Standard Solow Model
Jorge F. Chavez∗
October 30, 2012
1
Basic setup
Time is discrete. The economy is inhabited by Lt households which growth at a rate n every
period. That is Lt+1 = (1 + n)Lt . The technology is of the Cobb-Douglas type with constant
returns to scale and besides capital and labor, includes government spending1 .
Yt ≡ F (Kt , Lt , Gt ) = Ktα Lγt Gβt
where α, β, γ > 0 and α + β + γ = 1 (CRS assumption). Here, the productive government
assumption comes into play sice it increases the output of the firm.
The resource constraint now includes Gt :
Ct + It + Gt ≤ Yt ≡ Ktα Lγt Gβt
2
The case of a lump-sum tax
Suppose first that the government poses a lump-sum tax (a tax that you pay in fix amount,
“just because you are born”). Each person pays the same amount, independent of income.
For simplicity we will assume that the government runs a balanced budget: there is no debt
and government spending Gt is financed entirely by aggregate tax reveneus Tt :
Gt = Tt
The Ricardian Equivalence guarantees that we do not need to worry about any other way to
finance government spending (why?).
Let Ytd = Yt − Tt denote disposable income after the lump sum tax is collected. We will be
using the long-run Keynesian consumption function:
Ct = (1 − s)Ytd
∗
1
e-mail:j.chavez-cotrado@warwick.ac.uk
One possibility would be to think of Gt as public infrastructure
1
where (1 − s) is the marginal propensity to consume and s is the usual saving rate.
The aggregate stock of capital follows the usual law of motion:
Kt+1 = It + (1 − δ)Kt
Using the no population growth assumption (L0 = 1 and n = 0) and the fact that savings are
a constant portion of disposable income (which is equivalent to assume that consumption is a
constant proportion of disposable income) then:
(1 + n)kt+1 = s(yt − τt ) + (1 − δ)kt
where τt = Tt /Lt . Then:
kt+1 =
s(ktα τtβ − τt ) + (1 − δ)kt
(1 + n)
Note that we could assume that individuals pay exactly the same amount of tax every period,
that is τt = τ . This is equivalent at assuming that Tt grows at the same rate as population
to that the ratio Tt /Lt remains constant. Hence τ here would be just another parameter and
therefore the non-linear difference equation is well specified.2 The steady-state condition is then:
s(k α τ β − τ ) = δk
Remark 2.1. Because Gt = Tt enters in the production function, the above setting cannot
yield the standard setting with no taxes. That is, we cannot easily shut-down taxes in the above
model to get the standard version of the Solow model with Cobb-Douglas production function.
Remark 2.2. How would the steady state condition would change if we use an income tax
instead of a lump-sum tax scheme?
2
Otherwise the sequence {τ } would act as an exogenous forcing term for the path of kt+1 and its treatment
would make things a little bit more complicated.
2