# A Two Keynesian Analysis of Multipliers.

First “two Keynesian” is a phrase which only I use. It means a new Keynesian model with two periods the present and next period and forever after when the economy is in steady state. I recently noticed that Krugman’s models, which are much simpler than models I can think up and which have clarified my thought , have this form.

Before going on, I note that I just can’t do it. I can’t present a model in which all of the assumptions which go without saying are unstated. So my “simplest” effort is pages long – only the gifted can make it simple.

The bottom line point is that in a simplest new Keynesian model, the fiscal multiplier is one only if government spending is not at all productive (it can cause pleasure or be pure waste)

OK… so new Keynesian assumptions include rational expectations, clearly defined objectives with consumers who care about nothing other than consumption leisure and real money holdings and firms which maximize shareholder value and, finally, some sort of nominal rigidity. There is some appeal to imperfect competition, but in practice, it is assumed that wages and prices are changed only occasionally and that in between such adjustments firms and workers meet demand at fixed wages (w) and prices (p). I am going to consider the very simple very extreme case of wages and prices which are fixed forever. The analysis will be a limiting case of not so totally unsophisticated new Keynesian analysis as a couple of standard parameters go to zero.

I guess New Keynesians are willing to admit that there are liquidity constraints, but I will assume that all agents are free to borrow or lend as much as they choose at the safe interest rate and that none ever go bankrupt. This means that, with the simple preferences and rationality and all that, new Keynesian models have Ricardian equivalence. A temporary lump sum tax cut (or increase) has no effect on anything. Because of this and too make things clear, I will assume that the public budget is balanced so spending is equal to tax revenues each period. There is no technological progress or population growth.

Monetary policy consists of the one period nominal interest rate i (this is the safe rate as all debtors are perfect credit risks). The real interest rate is equal to the nominal interest rate, since prices are fixed forever. So I won’t consider money in the utility function or, in fact, consider money balances at all. The utility function 1) is logarithmic in consumption (Ct) and additively separable in consumption and labor (Lt) with a discount facter B.

1) U = sum (ln(Ct) + h(lt))B^t

h'() h(Lt) doesn’t matter, because workers supply as much labor is demanded at the given forever wage. Their only choice is between consumption and saving.

When the economy is not in a liquidity trap, i is set to make the unemployment rate equal the natural rate of unemployment. Given the assumption of no inflation ever, the natural unemployment rate is zero (that is the first order condition w/(pCt) + h'(Lt) = 0). This implies an employment level and a ratio of GDP (Y) to capital (K).

In period 1 the economy is in a liquidity trap so the interest rate is 0 and unemployment is higher than 0. I will assume this is caused by a bad shock to the capital formation technology. In normal times (from period 2 on) it is the simplest possible (delta is a rate of depreciation)

For t> 1

Kt+1 – Kt = It -(delta)Kt

I is gross investment which can’t be negative.

In period 1 it is different

K2-K1 = (It)s – (delta) K1

where s is a positive number less than one which makes optimal investment at interest rate zero too low for full employment. In the simplest case, investment in period 1 is equal to zero.

Now consider government spending G. First and totally standardly, if G does not affect production, then an increase in G (dG) does not affect first period consumption. Guess and check. In the liquidity trap GDP rises to meet demand so if C is not affected GDP increases by delta G. If taxes in period 1 are increased by delta G then disposable income in all periods is not affected, so C is not affected (this is true even if taxes affect labor supply given the property that workers supply all that is demanded at the current nominal wage) . More generally, if taxes are non distortionary and consumers are rational and not liquidity constrained, it doesn’t matter when taxes are increased.

But that is assuming that G is complete waste (or useful only because it is fun and appears in the utility function not in any form of production). That’s extreme. The simplest alternative assumption is to assume that G forms public capital which is neither a complement to nor a substitute for private capital. This is simple but strange (it requires that the public capital generates revenue without requiring workers as public employment crowds out private sector employment except when in the liquidity trap). Notably the return on public capital can be very low compared to the return on private capital in the full employment steady state. Let’s say delta G of public capital generates Rho dG of revenue per period. This revenue belongs to the citizen consumers. Period 1 Rho dG is discounted by the factor (1/(1+r)) = 1. From then on future revenue decreases by a factor (1-delta) as public capital depreciates and is discounted by the natural rate discount factor 1/(1+r*) so the present value of income is increased by [rho(1/[1-(1-delta)/(1+r*)])]dG. Given the assumptions about the utility function C1 is increased by (1-beta) [rho(1/[1-(1-delta)/(1+r*)])]dG and the multiplier is

1+ (1-beta) [rho(1/[1-(1-delta)/(1+r*)])]>1

A very strong assumption is was that investment was exactly zero in the liquidity trap because it was stuck at the irreversible investment zero lower bound. Except in that special case, higher output should cause higher investment as it causes a higher capital labor ratio. So the slightly greater than one new Keynesian multiplier is lower than the one which would be obtained from a less simple new Keynesian model.

This effect is much stronger if public capital is a complement to private capital. Investment in period 1 is chosen to satisfy a FOC based on the marginal product of capital in period 2 and the ZLB stuck interest rate.

And higher output doesn’t cause higher investment. If the government invests in something productive which is neither a substitute for or a complement to private capita., then the multiplier is slightly greater than one. If public capital (infrastructure say or subsidized human capital) is a complement for private capital the multiplier can be large. Both results hold even if optimal government spending is zero except when the economy is in a liquidity trap. Here the point is that productive investment causes increased income in the future and increased expected income causes higher consumption now.

This sounds right to me. (You are assuming full employment in period 2, right?)

Some comments:

1. If you increase G in period 1, and cut G by the same amount in period 2 (to pay off the debt, without raising taxes), so you are only preponing government spending, you get a multiplier of (roughly) two in the first period.

2. If you leave G unchanged in period 1, but cut G in period 2, you should also get a multiplier of plus one in period 1 and zero in period 2.

3. This shows it is the rate of change in G between periods 1 and 2, and not the levels of G, that give you a multiplier. A negative rate of change gives you a positive multiplier.

Oops. I think I misunderstood your 2 period assumption. The second period lasts forever, while the first period is short.

Regardless of how one calculates whatever, with a little effort and a lot of cooling it, a happy Thanksgiving may be had by all. Happy Thanksgiving!