Macro-Economic Modelling 3

Elements of Fiscal Policy

Governments cannot create resources out of thin-air: they, as any other economic agent, are subject to a budget constraint. In particular, the present value of government's spending \(G\) cannot exceed the present value of tax revenues \(T\) net initial government debt/wealth outstanding \(D\).
The stock of debt \(D\) increases (decreases) as a function of primary deficits (surpluses) \(G-T\) plus interest payments \(r(t)D(t)\), with \(r(t)\) the istantaneous real interest rate. Thus, \(D\) is governed by \begin{equation} \label{eq:gov_debt_stock_eom} \dot{D}(t) = G(t)-T(t) + r(t)D(t). \end{equation} Denoting the discount factor \(e^{-R(t)}=\exp\left(-\int_{0}^t r(t')dt'\right)\), the government budget constraint reads \begin{equation} \label{eq:gov_BC} \int_0^\infty e^{-R(t)}G(t)dt \le \int_0^\infty e^{-R(t)}T(t)dt - D(0). \end{equation} This constraint can equivalently be written as \begin{equation} \label{eq:gov_BC2} \lim_{t\rightarrow\infty}e^{-R(t)}D(t) \le 0, \end{equation} implying that the present value of government debt must be non-positive.
This condition, often called the "No-Ponzi" condition, states that the present value of government debt far in the future must be non-positive. Economically, it rules out a scenario where the government runs a "Ponzi scheme" by perpetually rolling over its debt—paying off old debt and interest simply by issuing new debt, without ever raising future taxes or cutting spending. If this condition did not hold, the government's debt would be growing faster than its ability to service it (as measured by the interest rate), which is not sustainable.
Notice that nothing prevents the government from never repaying its debt (it can continously roll it over) nor from continuously expanding it, as long as its growth rate is less than the real interest rate.

Exercise 1
Integrate \eqref{eq:gov_debt_stock_eom} to find an expression for \(D(t)\).
To start, realise the integrating factor is \(\exp(-\int r(t)dt)=e^{-R(t)}\). Multiplying through one gets \begin{equation} \begin{split} e^{-R(t)}\frac{dD(t)}{dt}-e^{-R(t)}r(t)D(t) &= e^{-R(t)}G(t)-e^{-R(t)}T(t)\\ \int_0^t e^{-R(t')}\frac{dD(t')}{dt'}-e^{-R(t')}r(t')D(t') dt' &= \int_0^t e^{-R(t')}\Bigl(G(t')-T(t')\Bigr) dt'\\ e^{-R(t)}D(t)-D(0) &= \int_0^t e^{-R(t')}\Bigl(G(t')-T(t')\Bigr) dt' \end{split}. \end{equation} Therefore: \begin{equation} D(t) = e^{R(t)}D(0)+\int_0^t e^{R(t)-R(t')}\Bigl(G(t')-T(t')\Bigr) dt'. \end{equation}

◻

Exercise 2
Show that \eqref{eq:gov_BC2} follows directly from \eqref{eq:gov_BC}.
To start, realise (cf. Exercise 1) that integration of \eqref{eq:gov_debt_stock_eom} yields \begin{equation} D(t) = e^{R(t)}D(0)+\int_0^t e^{R(t)-R(t')}\Bigl(G(t')-T(t')\Bigr) dt' \end{equation} which can be equivalently written as \begin{equation} \begin{split} e^{-R(t)}D(t) - D(0) &= e^{-R(t)}\int_0^t e^{R(t)-R(t')}\Bigl(G(t')-T(t')\Bigr) dt'\\ &= \int_0^t e^{-R(t')}\Bigl(G(t')-T(t')\Bigr) dt'\\ \end{split}. \end{equation} Taking the limit \(t\rightarrow\infty\) one obtains: \begin{equation} \lim_{t\rightarrow\infty}e^{-R(t)}D(t) - D(0) = \int_0^\infty e^{-R(t)}\Bigl(G(t)-T(t)\Bigr) dt, \end{equation} which plugged into \eqref{eq:gov_BC} gives the desired result \eqref{eq:gov_BC2}.

◻


A more practical way to assess whether governments respect their budget constraint is to analyse the concept of fiscal sustainability, typically by examining the evolution of the debt-to-GDP ratio, \(d(t) = D(t)/Y(t)\). Letting \(g\) be the growth rate of the economy, the change in the debt ratio is approximately \(\dot{d}(t) \approx (r(t) - g(t))d(t) + (G(t)-T(t))/Y(t)\). This equation reveals a crucial dynamic: if the real interest rate \(r\) exceeds the economy's growth rate \(g\), the government must run sufficiently large primary surpluses (\(T > G\)) to stabilise or reduce its debt-to-GDP ratio. If \(r < g\), the government can run persistent primary deficits as the economy 'grows out' of its debt. The 'No-Ponzi' condition in \eqref{eq:gov_BC2} essentially rules out a situation where debt grows faster than the economy's ability to finance it (pegged to \(r\)) indefinitely.

Ricardian Equivalence

At this point, a natural question to ask is: is there any real implication of the government's choice to finance itself via debt or via taxes? To answer this question let us consider the Ramsey NGM model . In addition, let us further assume that:
  1. The government satisfies its budget constraint with equality
  2. The financing cost of households and the government is the same (\(r_G=r_H=r\))
  3. Taxes are lump-sum
  4. Taxes do not enter directly into the households' utility function
The household's budget constraint requires that the present value of consumption does not exceed after-tax income (wage \(W\) minus tax \(T\)) plus any initial wealth (capital plus debt): \begin{equation} \int_0^\infty e^{-R(t)}C(t)dt\le \int_0^\infty e^{-R(t)}\Bigl(W(t)-T(t)\Bigr)dt + K(0)+D(0). \end{equation} Accounting for \eqref{eq:gov_BC} with assumption 1 then implies that \begin{equation} \label{eq:ricardian_HH_budget_constraint} \int_0^\infty e^{-R(t)}C(t)dt\le \int_0^\infty e^{-R(t)}\Bigl(W(t)-G(t)\Bigr)dt + K(0). \end{equation} Looking at \eqref{eq:ricardian_HH_budget_constraint} it is apparent that what enters the budget constraint of households is only the present value of government spending \(G\), regardless of how this was financed. Therefore, one finds an important result: only government spending impacts consumption and capital accumulation, not whether this spending is financed via debt or taxes. Or put differently, additional government debt is equivalent to present or future additional taxation.
This equivalence principle was first proposed in the late 19th century by Ricardo [1], reason for which it goes by the name of Ricardian equivalence, but the derivation in the context of the NGM is due to Barro [2].
This result rests however on a number of assumptions which fundamentally determine the extent to which Ricardian equivalence may hold in practice: let's consider each in turn.
  1. The NGM comes with the assumption of infinitely lived households. This is clearly not true in reality, but it can be shown that such a world is equivalent to one in which finitely-lived households care about the wellbeing of the next generation [3] as shown in [2]. Moreover, albeit finite, the average houshold lifespan is substantially longer than the typical government financing schedule [4], implying that Ricardian equivalence may be at least a good approximation.
  2. The model assumes perfect capital markets where households and the government can borrow and lend at the same interest rate. In practice, many households face borrowing constraints (or liquidity constraints). For these households, a debt-financed tax cut is not neutral; it provides immediate liquidity, enabling them to increase consumption today even if they are aware of future tax liabilities. Their inability to borrow against future income means their consumption is tied to their current disposable income, a direct violation of the RE premise.
  3. The Ricardian proposition relies on the use of lump-sum taxes, which do not affect the relative prices of goods or factors of production. Real-world taxes, such as those on labor income, capital, or consumption, are distortionary. A government's choice of financing, therefore, involves not just the level of taxes but also their timing and type. A tax cut on labor income today, financed by a future increase in capital taxes, will have significant real effects as households alter their labor-supply and saving decisions to respond to the changing incentives. The timing of distortionary taxes is not neutral.
  4. The baseline model is deterministic. Introducing uncertainty, regarding future income, government spending, or the timing and incidence of future taxes, complicates the household's problem. If government debt increases, households may become more uncertain about their future tax burden. This can lead to precautionary saving, altering consumption paths in a way that deviates from the simple RE prediction. The equivalence may also fail if households perceive that future tax liabilities are less certain than the current tax cut.
  5. The model assumes government debt is risk-free and will be repaid. The equivalence proposition rests on the idea that government bonds are a perfect substitute for future tax liabilities. However, in the real world, government bonds carry risk, particularly the risk of sovereign default. If households perceive that debt issued today might be defaulted on tomorrow rather than paid for with taxes, they will not treat that debt as a future tax liability. In this case, a debt-financed tax cut is seen as a simple windfall, not a delayed tax bill, and Ricardian equivalence will fail.
Empirical Evidence and Policy Implications
Given these assumptions, a crucial question is whether Ricardian equivalence holds empirically. The empirical literature however, offers mixed results.
Some studies [6][7], often examining household responses to large-scale tax cuts or rebates, reject the pure Ricardian proposition. Evidence suggests that a fraction of households, particularly those who are liquidity-constrained or have lower incomes, exhibit 'Keynesian' behavior by increasing consumption in response to a tax cut. However, the response is typically not one-for-one; many households do save a portion of the tax cut, indicating that the Ricardian channel is at least partially active.
Others, while akowledging that the pure equivalent statement is obviously false, provide evidence that the proposition holds as a good approximation for the real world [5].
The debate over Ricardian equivalence has profound policy implications. If the equivalence were to hold, it would imply that deficit-financed fiscal stimulus (e.g., tax cuts) is ineffective at changing aggregate demand, as households would simply save the extra income to pay for future taxes. Conversely, the empirical failure of RE suggests that such policies can at times be tool for short-term economic stabilisation, precisely because a portion of the population will spend, not save, the additional disposable income. This validates the use of fiscal policy for demand management, but also underscores that its effects are contingent on the financial health and planning horizons of households.

Optimal Fiscal Policy: Tax Smoothing

The discussion of Ricardian Equivalence is positive: it describes how households might react to government financing decisions. A related but normative question is how a government should optimally finance its spending over time, given that it must respect its intertemporal budget constraint. This leads to the theory of tax smoothing, most famously developed by Robert Barro [8]. The core idea is that governments should maintain a stable, or "smooth," tax rate over time, rather than allowing tax rates to fluctuate with temporary changes in government spending or economic output. The rationale for this stems from the deadweight loss of taxation. As discussed earlier, most real-world taxes are distortionary, they alter the incentives for households and firms, leading to inefficiencies. For example, a tax on labor income distorts the trade-off between working and leisure. Crucially, this deadweight loss (or "distortion cost") is generally assumed to be convex. We can formalise this idea by switching our setup slightly. The Ricardian Equivalence models used lump-sum taxes \(T(t)\), which have no distortionary cost. To model tax smoothing, we must assume taxes are distortionary. Let's make the following changes to the framework:
  1. Instead of lump-sum revenue \(T(t)\), the government now sets a proportional tax rate \(\tau(t)\) on a tax base, which we can assume is output (income) \(Y(t)\). Total tax revenue is \(T(t) = \tau(t)Y(t)\).
  2. The deadweight loss (or distortion cost) from this tax is a convex function of the tax rate. Let's represent this cost as a share of output, \(C(\tau(t))Y(t)\), where \(C\) is an increasing and convex function.
The government's goal is to choose a path of tax rates \(\{\tau(t)\}_{t=0}^\infty\) to minimise the total present value of distortion costs: \begin{equation} \min_{\{\tau(t)\}} \int_0^\infty e^{-R(t)} C(\tau(t))Y(t) dt \end{equation} This objective seems to imply setting \(\tau(t) = 0\) always. However, the government cannot do this. It must still satisfy its intertemporal budget constraint, which is the same as Equation (2) from before, but now with \(T(t) = \tau(t)Y(t)\) \begin{equation} \int_0^\infty e^{-R(t)} \tau(t)Y(t) dt = \int_0^\infty e^{-R(t)} G(t) dt + D(0) \end{equation} The left side is the present value of all future tax revenues. The right side is the present value of all future government spending plus any initial debt that must be repaid. This is the government's "bill". The problem is therefore to minimise the integral of a convex function \(C(\tau(t))\), subject to an integral constraint on its tax revenue \(\tau(t)Y(t)\).
For simplicity, let's assume the interest rate \(r\) and the tax base \(Y\) are constant. The government's problem is to choose \(\tau(t)\) to minimise \(\int_0^\infty e^{-rt} C(\tau(t)) dt\) subject to \(\int_0^\infty e^{-rt} \tau(t)Y dt = \text{Constant}\). Because the cost function \(C(\tau)\) is convex, any "variance" in the tax rate is costly. The most efficient way to raise the required revenue is to keep the tax rate perfectly constant over time: \(\tau(t) = \tau\) for all \(t\). A volatile path (e.g., \(\tau=50\%\) today, \(\tau=10\%\) tomorrow) and a smooth path (\(\tau=30\%\) always) might raise the same present-value revenue, but the smooth path will have a much lower total distortion cost due to convexity. This constant, optimal tax rate \(\tau^*\) would be the one that just satisfies the GBC: \begin{equation} \tau^* = \frac{\int_0^\infty e^{-R(t)} G(t) dt + D(0)}{\int_0^\infty e^{-R(t)} Y(t) dt} \end{equation} That is, the optimal policy is to set the tax rate equal to the total present value of all government liabilities (spending + initial debt) divided by the total present value of the national tax base (output). This is the formal logic of tax smoothing. The government runs deficits or surpluses in specific periods not to balance the budget annually, but to keep the tax rate \(\tau^*\) constant in the face of temporary fluctuations in \(G(t)\) or \(Y(t)\).
This means the economic harm from a tax increases more than proportionally with the tax rate. For instance, doubling the tax rate will more than double the deadweight loss. Given this, the government can minimize the total present-value burden of tax distortions by keeping the tax rate constant. A volatile path of tax rates (e.g., 50% in one year and 10% in the next) will create a higher total deadweight loss over time than a stable 30% tax rate that raises the same total revenue in present value. A constant tax rate avoids the exceptionally harmful periods of very high taxation. How can the government achieve smooth tax rates when its spending (\(G(t)\)) is volatile (e.g., spiking during wars or recessions) and its tax base (\(Y(t)\)) also fluctuates? The answer is to use debt as a buffer: When spending is temporarily high (like during a war) or output is temporarily low (like in a recession): The government should not spike tax rates to cover the gap. Instead, it should keep tax rates stable and finance the shortfall by running a deficit (issuing new debt). When spending is temporarily low (during peacetime) or output is temporarily high (in an economic boom): The government should not cut taxes. It should maintain the same stable tax rate and use the resulting budget surplus to pay down the debt accumulated in the bad times. This theory provides a powerful normative benchmark for fiscal policy. It implies that deficits and surpluses are not inherently "good" or "bad". They are optimal tools for insulating the economy from the high distortion costs of volatile tax rates, allowing the government to finance temporary shocks by spreading the tax burden over time. This framework also generates empirically testable predictions: we should expect to see government debt-to-GDP ratios rise during major wars and deep recessions, and fall (or grow more slowly) during periods of peace and prosperity.

References and notes

[1] "Essay on the Funding System" in The Works of David Ricardo, David Ricardo, London: John Murray, 1888.

[2] "Are Government Bonds Net Wealth?", Robert J. Barro, Journal of Political Economy, Vol. 82, No. 6 (1974).

[3] That is they derive utility from the consumption level of the future generation.

[4] "Finite lifetimes and the effects of budget deficits on national saving", James Poterba and Lawrence Summers, Journal of Monetary Economics, 1987, vol. 20, issue 2, 369-391.

[5] "Ricardian Equivalence", Seater, John J, Journal of Economic Literature, American Economic Association, vol. 31(1), pages 142-190, March 1993.

[6] "Finite lifetimes and the effects of budget deficits on national saving", James M. Poterba, Lawrence H. Summers, Journal of Monetary Economics, Volume 20, Issue 2, September 1987, Pages 369-391.

[7] "Household Expenditure and the Income Tax Rebates of 2001.", Johnson, D. S., Parker, J. A., & Souleles, N. S., American Economic Review, 2006.

[8] "On the Determination of the Public Debt.", Robert Barro, Journal of Political Economy, 87(5), 940–971.


Back to Teaching