### In Normal Times

In normal times, the fiscal policy multiplier μ applied to a government-purchases boost ΔG is effectively zero: the monetary authority has a view of the level of aggregate demand consistent with its price-level target, and will either offset any effect of expansionary fiscal policy on aggregate demand or rapidly take steps to depress the economy to reverse any rise in inflation produced by expansionary fiscal policy. In the multiplier equation in which expansionary fiscal policy raises current-year GDP Yn by:

(1) ΔYn = μΔG

In normal times μ=0, so ΔYn=0

In normal times, expansionary fiscal policy, will, however, raise the national debt by:

(2) ΔD = (1 - μτ)ΔG

which is less than ΔG because the government recaptures some of the cost of short-run expansionary fiscal policy via higher tax collections. This increase in the debt requires raising additional tax revenue to amortize it by:

(3) ΔT = (r-g)(1 - μτ)ΔG

Since raising an additional dollar in taxes carries with it a reduction in GDP via the Laffer parameter ξ, the effect of amortizing the debt is to reduce typical future-year real GDP Yf by:

(4) ΔYf = -ξ(r-g)(1 - μτ)ΔG

Taking the present value of (4) which applies to all future periods and adding it to (1) gives us our equation for the impact of temporary expansionary fiscal policy on the present value of future output:

(5) ΔV = [μ - ξ(1 - μτ)]ΔG

which, in normal times, with μ=0, will simply be:

(6) ΔV = - ξΔG

Expansionary fiscal policy as a stabilization policy tool in normal times is a bad idea.

### At the Zero Nominal Lower Bound

But what if we are at the zero nominal lower bound? What if the monetary authority wishes the flow of spending were higher, but cannot get there by reducing short-term safe nominal interest rates, and either cannot or will not but in any event does not target spending by itself? Then there will be a positive multiplier μ. The benefit-cost calculation (5) that is the effect of expansionary fiscal policy on the present value of output will turn out positive as long as:

(7) μ > ξ/(1 + ξτ)

For a a marginal tax-and-transfer share τ=0.33 and a Laffer parameter ξ=0.25, expansionary fiscal policy at the zero nominal lower bound is a good deal as long as μ > 0.25; for an ξ=0.5, expansionary fiscal policy at the zero nominal lower bound is a good deal as long as μ > 0.43; for an ξ=1.0, expansionary fiscal policy at the zero nominal lower bound is a good deal as long as μ > 0.75.

Even in the absence of any hysteresis effects--even in the absence of any long-run shadow cast on potential output by a short-run downturn--expansionary fiscal policy at the zero nominal lower bound looks likely to be a good deal even for very moderate values of the multiplier μ.

### Wedges Between the Cost of Funds and the Social Rate of Time Discount

The calculations so far have assumed that the government can borrow at the social rate of time discount r. What if it cannot? What if there is a wedge ρ between the social rate of time discount r and the government borrowing rate r+ρ? Then our equation (5) becomes:

(8) ΔV = [μ - ξ(1 - μτ)(r+ρ-g)/(r-g)]ΔG

And the benefit-cost test becomes impossible to pass for significant positive values of the wedge ρ: expansionary fiscal policy by the Greek or Spanish or Italian governments right now is a bad idea.

Conversely, suppose that there is a negative value of ρ: suppose government debt carries with it a liquidity or a safety premium so that the government can borrow for the long term at less than the social rate of time discount--as the U.S., Germany, and Japanese governments can do now. Then the cost terms in (8) shrink, and the benefit-cost test becomes even easier to pass. If the magnitude of the safety and liquidity discount is large enough, (8) can be positive even were the multiplier μ to be zero: in such circumstances, it would indeed be the case that--in the words of Alexander Hamilton--a national debt would be a national blessing.

### Hysteresis and Self-Financing Expansionary Fiscal Policy

Return to the case in which the Treasury real borrowing rate is the social rate of time discount, but allow for hysteresis effects: a current downturn in Yn today casts a shadow on potential output in a typical future years so that in the absence of Laffer parameter effects:

(9) ΔYf = ηΔYn

Then we find that the extra tax revenue raised from higher potential output due to a reduced shadow cast by the downturn more than covers the cost of amortizing the extra debt needed to finance expansionary fiscal policy if:

(10) ημτ > (r-g)(1 - μτ)

Or:

(11) r > g + ημτ/(1 - μτ)

(12) η > (r-g)(1/μτ - 1)

For a g=2.5%/year, an r=4%/year, a τ=0.33, and a μ=1.0, equation (11) is satisfied and short-run expansionary fiscal policy is self-financing for η>0.03. Even if only 1/30 of the downturn turns into a permanent shortfall in potential output, expansionary fiscal policy is self-financing--and austerity is self-defeating.

For a g=2.5%/year, an r=6%/year, a τ=0.33, and a μ=0.5, equation (11) is satisfied and short-run expansionary fiscal policy is self-financing for η>0.17. Even if only 1/6 of the downturn turns into a permanent shortfall in potential output, expansionary fiscal policy is self-financing--and austerity is self-defeating.

### Hysteresis and the Benefit-Cost Test

Even if expansionary fiscal policy now is not self-financing, the existence of hysteresis effects shrink the net costs and make it much easier to pass the benefit-cost test. Equation (5) then becomes:

(12) ΔV = [μ(1 + η(1 + ξτ)/(r-g)) - ξ(1 - μτ)]ΔG

For μ=0.5, η=0.05, τ=1/3, g=2.5%/year and r=6%/year...

- Expansionary fiscal policy passes its benefit-cost test as long as raising $1.00 in extra tax revenue reduces incomes by less than $10.00...
- Compare to the $0.25 estimates of Diamond and Saez and Romer and Romer...

### Conclusion

This seminar has simply been a matter of reduced-form arithmetic. If your model has a multiplier μ, a Laffer parameter ξ, a hysteresis effect η, and tax shares, GDP growth rates, and Treasury borrowing rates τ, g, and r, then for reasonable parameter values expansionary fiscal policy appears highly likely to be self-financing at the zero nominal lower bound in which there is a non-zero multiplier. Moreover, even for unreasonable parameter values expansionary fiscal policy appears highly likely to pass its benefit-cost test.

If you want to upset this result, you need to:

1, assert that there is or soon will be a large positive wedge ρ between the Treasury borrowing rate and the social rate of time discount; 2. assert that the multiplier μ is effectively zero even at the zero nominal lower bound on interest rates; or 3. assert that there are no hysteresis effects η--that the future path of potential output is invariant to the size of today’s downturn.

In a world in which the CBO and the FRB have already marked down their estimates of potential output in 2020 by more than 3% as a result of the financial crisis and recession and jobless recovery that began in 2008, the last of these seems the least credible of the three. And the first two seem not credible at all.