One (relatively minor) conclusion from Olivier Blanchard's excellent, thoughtful, and provocative American Economic Association Presidential Address "Public Debt and Low Interest Rates" https://www.aeaweb.org/aea/2019conference/program/pdf/14020_paper_etZgfbDr.pdf is that, in the near-canonical Diamond (1965) Overlapping-Generations model he considers, whether a marginal increase in public debt raises welfare—whether an economy is "dynamically inefficient"—depends not just on the relationship of the rate of interest on safe government bonds to the growth rate but on both that interest rate's and the expected rate of profit's relationship to the economy's growth rate.

In this model, one important effect—perhaps the principal effect—of government debt is to directly shift consumption away from the young who buy the debt to use it as a savings vehicle and to the old who hold and sell the debt and . I follow and agree with Blanchard’s correct conclusion that this direct-transfer effect can be welfare-raising or welfare-lowering. It will be welfare-raising as long as the economy is dynamically-inefficient, defined as having its safe rate of interest is less than its growth rate. So far, so good.

In this near-canonical model, a second effect springs from government debt’s potential role in crowding-out capital investment. To the extent that it does so, and to the extent with that crowding-out raises profits received by old wealthholders and diminishes wages earned by younger workers, this second factor-price effect can also be welfare-raising or welfare-lowering: it also transfers from the young, who earn less in wages, to the old, whose profits are increased because the firms they own pay less in wages. It will be welfare-raising, Blanchard concludes in his presidential address, only under a stronger condition than that required for the direct-transfer effect: not just the safe rate of interest on government bonds but the risky rate of profit on capital investment must be lower than the economy’s growth rate.

This piece tries to dot some of the i's and cross some of the t's in Blanchard's argument.

### The Two-State Log-Utility Cobb-Douglas Production Case¶

Simplify the already-simple near-canonical model. Express all variables in per-worker values. Require that there be neither population growth nor technological progress. Require that there be time-separable log utility. Assume Cobb-Douglas production. And add a wedge between the expected profit rate and the safe government bond rate generated by two-state aggregate uncertainty about returns to capital.

Begin with zero initial government debt. Then consider an infinitesimal increase $dD$ in this debt, to then be rolled over indefinitely.

Assume that the economy is populated by people who live for two periods, working when young, receiving profits from their savings when old, and consuming in both periods. Their utility function is:

(1) $V = (1-\beta)U(C_y) + \beta{U(C_o)} (1-\beta)ln(C_y) + \beta{ln(C_o)}$

where $C_y$ is consumption when young, and $C_o$ is consumption when old. Maximizing utility produces this consumption first-order condition for an agent young ("y") in period t and old ("o") in period t+1:

(2) $\frac{(1-\beta)}{C_{y,t}} = {R^f}_t E_t \left[ \frac{\beta}{C_{o,t+1}} \right]$

Now add uncertainty: in each period there is a shock $\epsilon$ to the production function. Thus the poroduction function is:

(3) $Y_{t+1} = (1 + \epsilon_{t+1})F(K_{t+1}) = (1 + \epsilon_{t+1})(K_{t+1})^\alpha$

Because this is a Cobb-Douglas production function, the profits and thus the consumption of the old are simply a fraction $\alpha$ of the volume of production:

(4) $C_{o,t+1} = R_{t+1}K_{t+1} = {\alpha}(1 + \epsilon_{t+1})F(K_{t+1}) = {\alpha}(1 + \epsilon_{t+1})F({\beta}W_t)$

And from the assumptions of Cobb-Douglas production and log utility, the consumption of the young is a fraction $(1-\beta)(1-\alpha)$ of the volume of production:

(5) $C_{y,t} = (1-\beta)W_t = (1-\beta)(1-\alpha)(1 + \epsilon_{t})F(K_{t})$

Under these assumptions, the risky interest rate must then satisfy:

(6) $\frac{(1-\beta)}{C_{y,t}} = E_t \left[ \frac{\beta{R_{t+1}}}{C_{o,t+1}} \right]$

Now consider the two-state-shock world: in each period the shock to the economy $\epsilon$ can take on one of two values with equal probability: $+\sigma$ or $-\sigma$:

Derive the one-period safe interest rate as of time t, after the resolution of period-t uncertainty:

(7) $\frac{(1-\beta)}{C_{y,t}} = {R^f}_t \left[ \frac{1}{2}\frac{\beta}{{\alpha}(1 + \sigma)F({\beta}W_t)} + \frac{1}{2}\frac{\beta}{{\alpha}(1 - \sigma)F({\beta}W_t)} \right]$

(8) $\frac{1}{W_t} = {R^f}_t \frac{\beta}{{\alpha}F({\beta}W_t)} \left[ \frac{1}{1 - \sigma^2} \right]$

(9) ${R^f}_t = \frac{{\alpha}F({\beta}W_t)}{{\beta}W_t}\left[ 1 - \sigma^2 \right] = \frac{\alpha[1-\sigma^2]}{({\beta}W_t)^{1-\alpha}}$

Derive the one-period risky interest rate as of time t, after the resolution of period-t uncertainty:

(10) $E_t(R_{t+1}) = \frac{{\alpha}F({\beta}W_t)}{{\beta}W_t} = \frac{\alpha}{({\beta}W_t)^{1-\alpha}}$

Thus the one-period risk spread as of time t, after the resolution of period-t uncertainty:

(11) $\frac{E_t(R_{t+1})}{{R^f}_t} = \frac{1}{1 - \sigma^2}$

This completes our framework.

### Intergenerational Transfer Scenarios¶

We consider intergenerational transfer scenarios:

(a) Transfer a fixed amount from the young to the old: appropriately evaluated from the standpoint of those young at time t at the period-t safe interest rate ${R^f}_t$

(b) Make a risky investment that costs a fixed amount when young and returns an amount correlated with consumption when old: appropriately evaluated from the standpoint of those young at time t at the period-t risky interest rate $E_t(R_{t+1})$.

(c) The factor-price crowding-out effect in Blanchard's presidential address. Evaluated from the individual's standpoint—after the resolution of the period-t shock to the economy—this looks exactly like a risky investment: your profits in the future will be higher by an amount correlated with consumption-when-old at the cost of a known and certain reduction in your consumption resources today, and hence is appropriately evaluated at the period-t risky interest rate $E_t(R_{t+1})$

(d) But I also want to think about another case": The "social insurance" case—a transfer proportional to your wage and consumption when young, to finance a fixed payment to the old. As this is a negative-beta asset, it is properly evaluated at an interest rate lower than the period-t safe rate if we evaluate it before period-t uncertainty is resolved. But if it is evaluated after period-t uncertainty is resolved, it is appropriately evaluated at the period-t safe interest rate

### Stepping Back from Period t to Unconditional Expectations¶

And things get (more) complicated. Analysis as of time t knowing the one-period risky and safe interest rates as of time t is well and good. However, from a broader social-welfare perspective the risky interest rate as of time t $E_t(R_{t+1}) = \frac{{\alpha}F({\beta}W_t)}{{\beta}W_t}$ is itself stochastic—a function of all those previous $\epsilon$ shocks up to time t that determine $Y_t$ both directly and by feeding into the value of $K_t$. The safe interest rate as of time t ${R^f}_t = (1-\sigma^2)E_t(R_{t+1})$ is also stochastic: it also depends on everything that affects the risky interest rate, plus it depends on the variance of the shock.

The natural next step, therefore, is to cease conditioning on $\left[ \epsilon_t, \epsilon_{t-1}, \epsilon_{t-2}... \epsilon_{t-∞} \right]$ and take unconditonal expectations. In which case things get messy... or messier...

From a social-welfare every-generation-counts-equally standpoint, it is sufficient to compare the average utility loss to the young with the average utility gain to the old: we are aggregating over both generations and ages. Thus we do not have to keep track of the fact that relatively-poor young with low consumption because of a low wage when young and thus with a high marginal utility of wealth when young will probably still be poor (but less poor) when old because they have little capital and hence have a high (but less high) marginal utility of wealth when old.

Proper analysis this thus requires calculating the unconditional expected utility cost of fixed and proportional-to-current consumption transfers:

(a) From an unconditional-expectations standpoint, the expected marginal utility of the young which multiplies the amount of a fixed direct-transfer is:

(12) $E \left[ \frac{1-\beta}{(1-\beta)W} \right]$

The expected marginal utility of the old which multiplies the amount of a fixed direct-transfer is:

(13) $E \left[ \frac{\beta(1-\alpha)}{{\alpha}W} \right]$

Since from an unconditional-expectations standpoint the distribution of W relevant when young is the same as the distribution relevant when old, a direct transfer will raise expected well-being if:

(14) $E \left[ \frac{1}{W} \right] < \left[ \frac{\beta(1-\alpha)}{\alpha} \right] E \left[ \frac{1}{W} \right]$

(15) $\frac{\alpha}{\beta(1-\alpha)} < 1$

The question then is: how does this relate to $E \left[ {R^f}_t \right]$ ? Which depends on $\sigma$, doesn't it?

(c) From an unconditional-expectations standpoint, a proportional transfer raises well-being if the proportional rise in consumption for the old times $\beta$ is greater than the proportional decline in consumption for the young. The ratio of consumption of the old to the consumption of the young is:

(16) $\frac{\alpha}{(1-\beta)(1-\alpha)}$

Thus the unconditional expected utility gain to the old will be greater than the loss to the young when

(17) $\frac{\alpha(1-\beta)}{\beta(1-\alpha)} < 1$

#### MEMO: The No-Shock Steady State¶

If we consider the steady-state of the model without any stochastic shocks, we have:

(18) $W = (1-\alpha)K^{\alpha}$

(19) $RK = {\alpha}K^{\alpha}$

(20) $W = (1-\alpha)\frac{RK}{\alpha}$

(21) $\frac{W}{K} = \frac{(1-\alpha)R}{\alpha}$

And we also have:

(22) $K = {\beta}W$

(23) $\frac{W}{K} = \frac{1}{\beta}$

So:

(24) $\frac{1}{\beta} = \frac{(1-\alpha)}{\alpha}R$

(25) $\frac{\alpha}{\beta(1-\alpha)} = R$

is the safe and the risky interest rate in the steady-state without uncertainty.