## A few Keynesian model with multipliers

A few is more than two but not a lot. As noted below, I don’t have the gift which it takes to be simple. I really can’t manage 2 period models which illustrate my confused thoughts.

My last post (which was messy especially the math in plain ascii and spelling) noted that the government spending multiplier is greater than one if government spending is investment in productive public capital when private investment is stuck at the irreversible investment zero lower bound. I went on to vaguely argue that the multiplier is still greater if private investment is positive and public capital isn’t too close a substitute for private capital.

This time I will assume that public spending is not productive. It might be pure waste or it might provide utility for citizens as entertainment or something. I will assume that it won’t affect the marginal utility of private consumption. The model will work as if it is pure waste (but I will assume that it isn’t in some welfare comparisons). However, I will also assume that, even in the liquidity trap, private investment is positive

The bottom lines

There is a multiplier much larger than one in the liquidity trap. Partly this reflects capital formation corresponding to increased expected government consumption. In balanced uh shrinking the multiplier remains greater than one. The effect of government consumption on Net national product is one, the effect on Gross national product is greater than one.

ìEven if the welfare benefits of the government consumption are just equal to the disutility of the effort of producing it (which is less than the market price) government spending causes higher expected welfare, because it causes capital accumulation which is useful when the economy emerges from the liquidity trap.

Private investment is determined by a first order condition equating the marginal product of capital and the user cost of capital (r + delta) where r is the real interest rate and delta is the rate of depreciation. Again I am going to assume extreme price stickiness and in fact price rigidity so wages and prices are fixed forever. Of course this means that the real interest rate is equal to the nominal interest rate, so the zero lower bound is that r is greater than or equal to zero.

I am going to try to micro found a model of the liquidity trap. The idea generally is that in the liquidity trap investment is very low even if r= 0. I really have only one free parameter to make this true — delta. So I will assume that, in normal times depreciation = delta and and in bad times depreciation = Delta with Delta> delta.

Hmm how does this relate to stories about what went wrong ? Well one way of modelling a downturn is that the quality of projects of entrepreneurs shifts down. This means that more of the money invested in projects is lost because the project fails. Oh that looks like depreciation. The loss of capital isn’t due to things physically wearing out or becoming technologically obsolete, but because they were bad technology to begin with. But the equation is the same. Another way of looking at things is panic about panics — investors fear something like a run which will cause solvent firms to be illiquid and go bankrupt. This matters, because bankruptcy is costly as money goes to lawyers fighting over different claims on the firms assets. If this cost is proportional to the firms assets (K) then it is like depreciation. OK with that feeble effort at justification, I assume that, from time to time, deprecation switches from delta to Delta or from Delta to delta. I am also going to assume that such switches are unpredictable and rare so tjhey always come as a surprise, so I will approximate fully rational expectations as the assumption that depreciation will be the same next period as it is this period. Note time is discrete.

production is standard with GDP

Y = Kt^(alpha)Lt^(1-alpha)

where Kt is capital at time t and Lt is labor at time t. There is no population growth or technological progress. Lt is the natural (non accelerating inflation) level of employment when the economy is not in a liquidity trap (that is the monetary authority targets this level if it can). Otherwise it is whatever labor is demanded at the fixed real wage.

There is one good which can be consumed or used as capital. A really crazy and non standard assumption is that there is no cost of converting the good from consumption good to capital and back. In serious models, there must be some adjustment cost. Otherwise investment gross of depreciation bounces up and down (and is often negative). In fact, the model will show absurdly high effects of fiscal policy on investment. As with the assumption of total wage and price rigidity, I am trying to illustrate effects with a very simple very extreme case. The price of the good is p which I will set to one, so w is the real wage.

Consumers maximize

1) sum (1-rho)^(t)ln(Ct) + h(Lt)

where rho is a rate of impatience, Ct is consumption at time t and Lt is labor at time t. They won’t actually choose Labor beause there is excess supply of labor at the rigid real wage. This is standard in new Keynesian models (although next periods nominal wage depends on the marginal disutility of labor except in the extreme case in which it is totally rigid as I am assuming).

This just means

2) C_(t+1)/Ct = (1+r_(t+1))(1-rho)

r_(t+1) is real interest paid at time t+1. When the economy is in the liquidity trap, r is zero. In the non liquidity trap steady state r = rho. delta is less than rho which is less than Delta.

There is excess supply of goods at the rigid price. I will just assume that for any depreciation, any equilibrium Kt and any fiscal policy, the price is higher than marginal cost, so supply meets demand at that price. This is a standard assumption in new Keynesian models (although typically the future price depends on current demand). In the liquidity trap the real interest rate r = 0. This means the user cost of capital is Delta. The real wage is w (recall I use the price of the good as numeraire). Even though output is demand limited, firms choose the capital labor ratio to minimize costs so

3) (alpha)/(1-alpha)(L/K) = Delta/w

when in the liquidity trap and

4) (alpha)/(1-alpha)(L/K) = = (rt+delta)/w

when the economy is not in the liquidity trap.

5) K_(t+1) = It+(1-Delta)Kt or It+(1-delta)Kt

depending on whether the economy is in the liquidity trap.

In the liquidity trap, there is a balanced shrinking equilibrium in which consumption and capital shrink by a factor (1-rho) each period. Gross investment is (Delta-rho)Kt The ratio of gross investment to capital is fixed. So is the ratio of consumption to capital. Therefore with no government spending (G=0) the ratio of output to capital is constant. That ratio

6) C/K + (Delta-rho) = (Delta(1-alpha)/[(w)alpha])^(1-alpha)

I will assume that Y/K = 1/3 which is very roughly correct for the USA.

This guessing and checking shows that there is a balanced shrinking equilibrium. it is, in fact, the unique solution to the reresentative consumer’s sad maximization problem. Other paths which satisfy the Euler equation (2) and the cost mimization equation 4 either imply negative capital at high t (which is impossible) or violate the strange looking transversality condition for r=0 that lim_(t->infinity) Kt = 0.

Note that Lt is proportional to Kt so employment decreases so long as the economy is in the liquidity trap. Now a tiny touch of realism implies that wages can’t really remain fixed as unemployment reaches say 99% but in the model they do.

This can, in theory continue forever. I want to assume that there is some hope, so each period there is a tiny chance that depreciation will fall from Delta to delta. At that point the economy becomes a Ramsey Cass Koopmans model. L is given at the non accelerating inflation level. This is much higher than the liquidity trap level (which shrinks down to zero over time). This means that r is much greater than zero. Welfare increases in K.

OK back to the liquidity trap. Assume in time tau the government suddenly starts spending money and taxing labor income so government spending rises from zero to G. It doesn’t matter if the tax is distortionary, because employment is limited by demand for labor. Also there is no capital income to tax. I assume throughout that the government runs a balanced budget so there is no public debt (it doesn’t matter since the assumptions about intertemporal utility maximization imply Ricardian equivalence). It is known that government spending will remain G. It is assumed that G is pure waste.

I gues sand will check that this has no effect on consumption. However, output will be higher in tau+1 and subsequently. This means that capital must be higher to satisfy the cost minimization equation (3). So investment is higher too. First Itau is increased by 3G (recall 3 is K/Y). but also keeping that high ratio requires investment of Delta3G each period starting tau+1. So I_(tau) is (1+Delta)3G higher than I_(tau-1). Then for t > tau, It is Delta3G higher than I_(tau-1). The short term multiplier is 4 + 3Delta and the long term multiplier is (1+3Delta).

The totally huge short term multiplier is silly. It comes from the assumtion of no costs of adjusting K. I said that investment would jump around in a crazy way. The long term multiplier is reasonable. It is also sort of the moral equivalent of one. The reason is that while *Gross* domestic product increases by (1+3Delta)G *Net* domestic product increases by only G (which is then wasted)

Notably if governemtn spending is totally wasted, this makes people worse off. Consumption stays the same and leisure falls. In contrast, the policy can make things better when in the liquidity trap if goverenmtn spending provides utility even if one unit of government spending provides less utility than one unit of private spending. Also the idea that the unemployed enjoy their leisure rather than suffer from a loss of a sense of self worth is crazy.

Now when depreciation surprisingly falls to delta at time T, the stimulus is switched off. It is only reasonable to assume that this is after one period of pointless “stimulus”. The policy maker is as surprised as anyone else. When the economy is not in a liquidity trap, one period of Government spending mostly crowds out investment. Consumption is reduced by rhoG and investment on the order of (1-rho) G. However, the capital stock from the period of liquidity trapping is increased by 3G so K_(T+1) is increased by 3G(1-delta)(1-Delta) – G + rhoG . With this and the reduction in consumption to first order in rhom welfare is increased as it would be by a windfall of 3G(1-delta)(1-Delta) – G. Government spending during the liquidity trap has increased the welfare of consumers after it ends. Their working and saving to feed the government beast caused them to produce more and accumulate more capital. This is useful when things return to normal.

There is a multiplier much larger than one in the liquidity trap. Partly this reflects capital formation corresponding to increased expected government consumption. In balanced uh shrinking the multiplier remains greater than one. The effect of government consumption on Net national product is one, the effect on Gross national product is greater than one.

Even if the welfare benefits of the government consumption are just equal to the disutility of the effort of producing it (which is less than the market price) government spending causes higher expected welfare, because it causes capital accumulation which is useful when the economy emerges from the liquidity trap.

[…] think it is possible to write down a model of an economy with a Nash equilibrium that is to be stuck in sustained depression. This can happen with intertemporal optimization all the […]