Dynamic Inefficiency

This is a post about macroeconomic theory. It is technical and I honestly don’t know how much is already in the literature. The aim is to address an important policy question — is public debt a burden on future generations.

It is possible in theory that the answer is no and that higher public debt causes permanently higher consumption and welfare. In such a case, it is clear that the low debt market outcome is inefficient, so this is called dynamic inefficiency. The standard result from simple models is that an economy is dynamically inefficient if r is less than g* where r is the real interest rate and g is the rate of GDP growth.

This formula isn’t very useful in the real world, because there isn’t one real interest rate — rather there is a low real interest rate on safe assets and higher rates on risky assets. The standard interpretation of “r” is that it refers to the ratio of total capital income to total capital. A lot of this r is not called interest at all since the claim on the income is equity not any kind of loan or bond.

The question of interest is whether increased public debt can cause increased welfare when the safe real interest rate (r1) is lower than g but the average return on capital (r2) is greater than g. I think the answer is yes, so I think it is plausible that, in the real world, greater public debt will cause permanently higher welfare.

I stress that I am not talking about any benefits of government spending which can be financed by the debt — the result would hold if the bonds are just given (that is the deficit is due to tax cuts) and the result is that people who weren’t alive at the time of the tax cut would benefit. Also it is assumed that wages and prices are flexible and markets clear (there is no unemployment) so the benefits of public debt have nothing to do with Keynesian stimulus.

This post was supposed to link to a pdf with models and proofs, but I can’t force myself to write the pdf without publicly promising it will exist. I will sketch the argument and 3 models after the jump.

update: the pdf with boring equations and a model which owes a lot to a comment by Nick Rowe is here

update 2: A new extended pdf with more boring equations is here

* in an earlier version this “r less than g” was rendered as “r” . html problem resolved thanks to Warren in comments.

All three models are overlapping generations model with capital (based on the Diamond model).

In model 1 r2>r1 because of idiosyncratic risk which can’t be diversified because of financial frictions. It is an odd sounding model but the math is simpler than in a more standard model (model 3). Agents work when young and divide their labor income between consumption and investment in their own little firm. I will assume this investment is in the form of materials which will be used in production and not fixed capital (in standard jargon that the capital will 100% depreciate after it is used for one period) Each old agent doesn’t work but does hire one young worker for his or her firm. The old consume all of their income then die. So far this is the standard Diamond model. I will assume that population doesn’t grow but that there is labor augmenting technological progress so effective labor grows.

The new assumptions are that the returns to each little firm are risky (the firm specific technology is stochastic) and that the single owner must bear this risk. The story is that these returns are not verifiable so the owner can’t sell shares. It is assumed that there is a large population and the returns on different small firms are uncorrelated so aggregate production (and the market clearing wage) are not stochastic. The average across firms rate of labor augmenting technological progress is g. The economy has a balanced growth path on which the capital stock, output and wages also grow at rate g.

Now introduce a government which issues bonds each of which is worth one unit of consumption good when it matures. Assume that, along the balanced growth path, the market clearing rate of return on these bonds r1 is less than g. The government chooses a ratio of public debt to GDP and maintains it. This means that the new bond issue raises more than enough to pay off the old bonds with interest. The extra revenue is equally divided among the old (as a pension).

The average return on capital invested in the little firms is r2>g. Because investment in one’s own little firm is risky r2 is definitely greater than r1.

The public debt has 3 effects. First there is the pension which is a pure gain for the agents. Second, there is a new savings vehicle (no one is forced to buy the government bonds). This provides an additional benefit for the agents — they could invest all of their savings in their own little firm and choose not to. Finally, public debt crowds out private investment and causes lower wages.

The reduction in wages has two effects on agents — they have a lower wage when young and pay a lower wage when they are old. Consider the balanced growth path in which total capital, GDP and wages grow by a factor 1+g each period. The crowding out effect is just a transfer from the young to the old. The assumption snuck in above that each old person hires one young person implies that the benefits are shared equally among the old. This means that the crowding out effect works like a safe asset which pays return 1+g > 1+r1. It is an additional benefit for the agents.

All three effects of government debt are beneficial so long as g>r1, so welfare is increased.

Note that GDP is reduced (since r2>0) and average consumption is reduced (since r2>g). The improvement in welfare is due to reduced risk bearing made possible by the availability of a safe savings vehicle.

The optimal balanced growth path is reached by choosing a debt to GDP ratio so high that r1=g. r1g* increased public debt causes increased welfare.

So much for idiosyncratic risk. A really standard model (model 3) has an aggregate technology shock which is the same for all firms. In such standard models, the number of firms doesn’t matter — the firms can be joint stock corporations. It doesn’t matter how fast capital depreciates either. The return on stock is higher than the safe return because the aggregate risk is undiversifiable (by definition).

These more standard assumptions do change the behavior of the model. The wage becomes stochastic as does aggregate GDP. For model 3 I assume that technology is multiplied by an iid stochastic term 1+g_t each period with the geometric mean of 1+g_t = 1+g.

The problem for the treasury is a bit trickier. I assume that it chooses to have its debt increase by a factor slightly slightly less than 1+g each period. With the assumption that the stochastic terms 1+g_t are iid means that the debt to GDP ration almost surely converges to zero.

In model 3 as in model 2, the crowding out causes lower wages benefit for the old is risky. Again this risk is just like the increased risk due to increased saving and investment. The qualitative results are the same as in model 2. The agents’ problem isn’t identical as the young bear some of the aggregate risk. For the same taste and technology parameters r2 is lower in the model with aggregate risk than in the model with idiosyncratic risk (because investors don’t bear all of the risk) but the welfare results given r1 r2 and g are the same.

Dynamic inefficiency has the standard alarming implications even if r2>g. It is possible for an intrinsically worthless asset which can’t be counterfeited (like money in the OLG model with money) to be valuable in each of the models. The value of this asset may depend on a sunspot.

Importantly, in balanced growth, aggregate capital income is greater than aggregate investment if r2>g. This means that Abel, Mankiw, Summers and Zeckhauser would conclude that such an economy is dynamically efficient. They might not be correct.

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