## Introduction¶

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.

## Brad DeLong: The Finger-Exercise Cobb-Douglas Production Case¶

### 1. The Zero-Uncertainty 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 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:

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

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:

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

Now consider a government that issues a small amount of debt $\delta_0$ to the young in period 0, repays ${R^f}_t\delta_0$ to the now-old in period 1 and issues $\delta_0$ to the new-young in period 1, and so forth, each period paying off its old debt at the safe rate and issuing new debt.

Then as long as it is always the case that $1 > {R^f}_t$:

• Each generation is indifferent between the no-debt and the debt world.

• The government has extra resources to spend on public goods, in amount $\delta_0$ in period 0 and in amount $(1 - {R^f}_t)\delta_0$ in all subsequent periods t.

• The government does not have to levy any taxes to rollover its debt.

Now let's add in a Cobb-Douglas production function, where a single worker's productivity is:

(1.3) $Y_t = A K_t^\alpha$

The wage rate is:

(1.4) $w_t = (1 - \alpha) A K_t^\alpha$

The profit rate is:

(1.5) $R^f_t = \alpha A K_t^{\alpha -1}$

Using:

(1.6) $C_{y,t} = w_t - K_{t+1}$

(1.7) $C_{o,t+1} = A K_{t+1}^\alpha$

Substituting into the first order condition:

(1.8) $\frac{(1-\beta)}{w_t - K_{t+1}} = {R^f}_t \left[ \frac{\beta}{A K_{t+1}^\alpha} \right]$

(1.9) $\frac{(1-\beta)}{w_t - K_{t+1}} = \alpha A K_t^{\alpha -1} \left[ \frac{\beta}{A K_{t+1}^\alpha} \right]$

(1.10) $\frac{(1-\beta)}{w_t - K_{t+1}} = \frac{\alpha\beta}{K_{t+1}}$

(1.11) $( 1 - \beta ) K_{t+1} = \alpha \beta (w_t - K_{t+1})$

(1.12) $K_{t+1} = \frac{ \alpha \beta w_t}{1 + \alpha \beta - \beta}$

In [ ]:



Now add uncertainty, and also add Cobb-Douglas production. There will then be a wedge between the expected profit rate and the safe government bond rate. Assume that in each period output is determined by (a) the form of the production function $F()$, (b) a random shock $\epsilon_{t}$, and (c) the last period's investment in capital by the then-young $K_t$.

Capital is the difference between the wages the then-young received, what they invested in debt, and what they consumed when young. In the zero-debt case:

(2.1) $K_{t} = W_{t-1} - C_{y,t-1}$

With a production function inclding a random shock of:

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

Cobb-Douglas guarantees us that the profits of the old will be a share $\alpha$ of production:

(2.3) $\pi_t = \alpha{Y_t}$

And the wages of the young will be a share $1 - \alpha$ of production:

(2.4) $W_t = (1- \alpha){Y_t}$

From the assumptions of Cobb-Douglas production and log utility, in the zero-debt case the consumption of people when young is a fraction $(1-\beta)(1-\alpha)$ of the volume of production:

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

The capital stock is:

(2.6) $K_{t+1} = \beta(1-\alpha)(1 + \epsilon_{t})F(K_{t})$

And, in the zero-debt case, the consumption of people when old in period t+1 is simply the profits they receive:

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

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

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

### 3. Brad DeLong: The Two-State Log-Utility Cobb-Douglas Production Case¶

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:

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

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

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

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

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

(3.5) $E_t(R_{t+1}) = E_t \left[\frac{1}{2}\frac{\alpha{F(K_{t+1})}(1+\sigma)}{K_{t+1}} + \frac{1}{2}\frac{\alpha{F(K_{t+1})}(1-\sigma)}{K_{t+1}} \right]$

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

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

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

This completes our framework.

(4.1b) $K_{t+1}^* = \beta(1-\alpha)(1 + \epsilon_{t})F(K_{t}^{*})$

(4.1) $K_{t+1} = \beta(1-\alpha)(1 + \epsilon_{t})F(K_{t}) - \delta$

(4.2b) $R_{t+1}^* = (1+\epsilon_t)\frac{\alpha}{F(K_{t+1}^{*})}$

(4.2) $R_{t+1} = (1+\epsilon_t)\frac{\alpha}{F(K_{t+1}^*- \delta)}$

(4.3) $\frac{R_{t+1}}{R_{t+1}^*} = \frac{F(K_{t+1}^{*})}{F(K_{t+1}^* - \delta)}$

(4.4) $\tau_t = \frac{R_{t+1}}{R_{t+1}^*} - 1$

## Brad DeLong: The Finger-Exercise Constant Young Marginal Utility Case¶

### 1. The Zero-Uncertainty Case ¶

#### 1.1 The Safe Interest Rate ¶

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 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:

(1.1.1) $V = C_{yt} + \beta \left(C_{ot+1} - \frac{C_{ot+1}^2}{2} \right)$

where $C_{yt}$ is consumption when young, and $C_{ot+1}$ 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:

(1.1.2) $0 = -1 + \beta{R^f}_{t}(1 - C_{ot+1})$

(1.1.3) ${R^f}_t = \frac{1}{\beta (1 - C_{ot+1}) }$

(1.1.4) $C_{ot+1} = 1 - \frac{1}{\beta {R^f}_t}$

Now consider a government that issues a small amount of debt $\delta$ to the young in period 0, repays ${R^f}_t\delta$ to the now-old in period 1 and issues $\delta$ to the new-young in period 1, and so forth, each period paying off its old debt at the safe rate and issuing new debt.

Then as long as it is always the case that $1 > {R^f}_t$:

• Each generation is indifferent between the no-debt and the debt world.

• The government has extra resources to spend on public goods, in amount $\delta$ in period 0 and in amount $(1 - {R^f}_t)\delta$ in all subsequent periods t.

• The government does not have to levy any taxes to rollover its debt.

## Brad DeLong: The Finger-Exercise Constant Young Marginal Utility Case¶

### 1. The Zero-Uncertainty Case ¶

#### 1.1 The Safe Interest Rate ¶

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 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:

(1.1.1) $V = C_y + \beta{\ln(C_o)}$

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:

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

Now consider a government that issues a small amount of debt $\delta$ to the young in period 0, repays ${R^f}_t\delta$ to the now-old in period 1 and issues $\delta$ to the new-young in period 1, and so forth, each period paying off its old debt at the safe rate and issuing new debt.

Then as long as it is always the case that $1 > {R^f}_t$:

• Each generation is indifferent between the no-debt and the debt world.

• The government has extra resources to spend on public goods, in amount $\delta$ in period 0 and in amount $(1 - {R^f}_t)\delta$ in all subsequent periods t.

• The government does not have to levy any taxes to rollover its debt.

### 1.2 Capital ¶

Now add a Cobb-Douglas production function. Capital is the difference between the wages the then-young received, what they invested in debt, and what they consumed when young. In the zero-debt case:

(1.2.1) $K_{t} = W_{t-1} - C_{y,t-1} - \delta$

With a production function:

(1.2.2) $Y_{t} = (K_{t})^\alpha$

Cobb-Douglas guarantees us that the profits of the old will be a share $\alpha$ of production:

(1.2.3) $\pi_t = \alpha{Y_t}$

And the wages of the young will be a share $1 - \alpha$ of production:

(1.2.4) $W_t = (1- \alpha){Y_t}$

The maximization problem is to maximize expected utility by maximizing:

(1.2.5) $E_t[V] = E_t\left[ C_{y,t} + \beta\ln(C_{o,t+1}) \right] = E_t\left[ (1-\alpha)Y_t - K_{t+1} - \delta + \beta\ln\left[\alpha K_{t+1}^{\alpha} + R^f_t \delta \right] \right]$

(1.2.6) $0 = \frac{dE_t[V]}{dK_{t+1}} = -1 + \beta\left[ \frac{\alpha^2K_{t+1}^{\alpha-1}}{\alpha K_{t+1}^{\alpha} + R^f_t \delta} \right]$

(1.2.7) $\alpha K_{t+1}^{\alpha} + R^f_t \delta = \beta \alpha^2 K_{t+1}^{\alpha-1}$

And so the capital stock is:

(2.7) $K_{t+1} = \beta\alpha$

From the assumptions of Cobb-Douglas production and log utility, in the zero-debt case the consumption of people when young is a fraction $(1-\beta)(1-\alpha)$ of the volume of production:

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

And, in the zero-debt case, the consumption of people when old in period t+1 is simply the profits they receive:

(2.9) $C_{o,t+1} = \alpha(1 + \epsilon_{t+1})(\beta\alpha)^\alpha$

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

(2.10) $R_{t+1} = (1 + \epsilon_{t+1})\alpha(\beta\alpha)^{\alpha-1}$

And the expected risky rate is always:

(2.11) $E_tR_{t+1} = \beta^{\alpha-1}\alpha^{\alpha}$

### 1.2 Capital ¶

Now add a Cobb-Douglas production function. There will then be a wedge between the expected profit rate and the safe government bond rate. Assume that in each period output is determined by (a) the form of the production function $F()$, (b) a random shock $\epsilon_{t}$, and (c) the last period's investment in capital by the then-young $K_t$.

Capital is the difference between the wages the then-young received, what they invested in debt, and what they consumed when young. In the zero-debt case:

(2.1) $K_{t} = W_{t-1} - C_{y,t-1}$

With a production function inclding a random shock of:

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

Cobb-Douglas guarantees us that the profits of the old will be a share $\alpha$ of production:

(2.3) $\pi_t = \alpha{Y_t}$

And the wages of the young will be a share $1 - \alpha$ of production:

(2.4) $W_t = (1- \alpha){Y_t}$

The maximization problem is to maximize expected utility by maximizing:

(2.5) $E_t[V] = E_t\left[ C_{y,t} + \beta\ln(C_{o,t+1}) \right] = E_t\left[ (1-\alpha)Y_t - K_{t+1} + \beta\ln\left[\alpha(1 + \epsilon_t)K_{t+1}^{\alpha}\right] \right]$

(2.6) $0 = \frac{dE_t[V]}{dK_{t+1}} = -1 + E_t \left[ \frac{\beta\alpha^2(1 + \epsilon_t)K_{t+1}^{\alpha-1}}{\alpha(1 + \epsilon_t)K_{t+1}^{\alpha}} \right] = -1 + \frac{\beta\alpha}{K_{t+1}}$

And so the capital stock is:

(2.7) $K_{t+1} = \beta\alpha$

From the assumptions of Cobb-Douglas production and log utility, in the zero-debt case the consumption of people when young is a fraction $(1-\beta)(1-\alpha)$ of the volume of production:

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

And, in the zero-debt case, the consumption of people when old in period t+1 is simply the profits they receive:

(2.9) $C_{o,t+1} = \alpha(1 + \epsilon_{t+1})(\beta\alpha)^\alpha$

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

(2.10) $R_{t+1} = (1 + \epsilon_{t+1})\alpha(\beta\alpha)^{\alpha-1}$

And the expected risky rate is always:

(2.11) $E_tR_{t+1} = \beta^{\alpha-1}\alpha^{\alpha}$

### 3. Brad DeLong: The Two-State Case¶

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:

(3.1) $1 = {R^f}_t E_t\left[ \frac{\beta}{C_{o,t+1}} \right] = {R^f}_t E_t\left[ \frac{\beta}{\alpha(1 + \epsilon_{t+1})(\beta\alpha)^{\alpha}} \right]$

(3.2) $1 = {R^f}_t \left[\frac{\beta}{2(\alpha(1 + \sigma)(\beta\alpha)^{\alpha})}+ \frac{\beta}{2(\alpha(1 - \sigma)(\beta\alpha)^{\alpha})} \right]$

(3.3) $1 = {R^f}_t \left[\frac{\beta}{(\alpha(1 - \sigma)^2(\beta\alpha)^{\alpha})}\right]$

(3.4) ${R^f}_t = \beta^{\alpha-1}\alpha^{1+\alpha} \left[ 1-\sigma^2 \right]$

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

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

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

This completes our framework.

## Robert Waldmann: Dynamic Inefficiency¶

In a world with no stochastic shocks, there is dynamic efficiency so long as the profit and safe interest rate $R > 1 + g$.

If $1 + g > R$, there is dynamic inefficiency, in the sense that a Pareto improvement can be achieved through a reduction in the stock of capital. Simply have the government issue and then rollover debt, and spend the money on useful things. In the semi-canonical Diamond model:

1. If the money is distributed to all, that is a Pareto improvement.
2. Agents have expanded opportunities—they can, when young, invest in debt rather than lower-yielding capital, and that is a second Pareto improvement.
3. Moreover, the reduction in the captial stock shifts factor prices toward profits received by the old and away from wages received by the young, which is a Diamond-model Pareto improvement as well.

However, when there is a wedge between the (risky) rate of profit $R$ and the (safe) rate of interest $R^f$ there are at least two real interest rates. Which of these rates matters for dynamic inefficiency? This matters, for, as noted by Blanchard, in the United States typically the safe interest rate $R^f$ has been less than $1 + g$, but the risky profit rate $R$ has been greater than $1 + g$.

It turns out that it is the safe rate that matters. The presence of a profit rate $R > 1 + g$ along with an interest rate $1 + g > R^f$ means that achieving a Pareto improvement via a reduction in the capital stock is not trivially simple—as it is in the no-wedge case—but such a policy exists.

For simplicity, assume trend growth $g = 0$. Furthermore, assume ${R^f}_t < 1$ for every t. (Note that this is a strong assumption: $R^f$ varies with the capital stock $K$, and this requires that there be no state of the world in which the capital stock is so low that the economy becomes—temporarily—dynamically efficient.) Consider a government that issues and rollsover debt $D$, paying each period's safe rate ${R^f}_t < 0$, and taxes or transfers to keep D constant:

1. Since ${R^f}_t < 1$ for all t, the state transfers wealth to its citizens each period—a Pareto improvement.

2. Moreover, $D$ crowds out K, so there is reduced production. Consumers lose ${R_t}{\Delta}K_t$, where ${\Delta}K$ is the reduction in $K$ due to $D$, and gain ${R^f}_tD$. Since they have a revealed preference for ${R^f}_tD$, this is a Pareto improvement as well.

3. However, as Blanchard stresses, this reduction ${\Delta}K$ in the capital stock also has a general equilibrium effect on factor prices: a higher rate of profit $r$ and lower wages $W$. Blanchard demonstrates that the sign of this effect on the welfare evaluated after the resolution of period t-1 uncertainty but before thje resolution of period t uncertainty of the median generation which works with median K using median technology is the same as the sign of: $E_{t-1}(R_t - 1)$.

It is this third effect that leads Blanchard to conclude that dynamic inefficiency depends not just on the (safe) interest rate relative to the growth rate but also on the (risky) profit rate relative to the growth rate.

This is correct if the only policy levers available to the government are the issuing and redemption of debt and associated lump-sum taxes and transfers. However, if the state can also impose a profits tax and a wage subsidy, then the third effect exists only if its existence pleases the government. If the existence of this general-equilibrium factor-price effect does not please the government, it can neutralize it. It is a feasible (balanced budget) tax-and-transfer policy to ensure that citizens face the same $W_t$ and $R_t$ that they would have faced under zero-debt laissez-faire. In this case effects (1) and (2) are still present: citizens still get a subsidy each period, and citizens still have access to a high-value safe saving vehicle in which they chose to invest.

This is a Pareto improvement associated with a reduced capital stock. The existence of this Pareto improvement shows that a sufficient condition for dynamic inefficiency is ${R^f}_t < 0$ for all t.

### Robert Waldmann: Relaxing the $R^f$ always less than $g$ Constraint¶

More generally, if $E({R^f}_t) < 0$, the government can increase money metric welfare averaged over generations.

OK the point (if any) of this note is to worry about the complexity of the (always feasible) tax and transfer policy described above. Use the notation:

• $K_t$ is capital at time t given the debt D and the taxes and transfers up until t.
• ${K^*}_t$ is the capital which there would be with laissez-faire.
• $R_t$ is the profit rate with the debt D
• ${R^*}_t$ is the profit rate under laissez-faire.
• ${\tau}_t$ is the profit tax rate

We require: $(1-{\tau}_t){R^*}_t) = R_t$

So: $(1-{\tau}_t) = \frac{R_t}{{R^*}_t}$

Assume: $Y_t = A_t(K_{t-1})^{\alpha}$

Here it is important that the shock is a stochastic Hicks-neutral technology shock.

This mean that $\frac{R_t}{{R^*}_t} = \left(\frac{K_{t-1}}{{K^*}_{t-1}}\right)^{\alpha-1}$.

The ratio does not depend on $A_t$ because the shocks are Hicks neutral.

(In addition, to get a simple formula for ${R^f}_)t$,I also need that the utility from consumption when old yields constant relative risk aversion: for example. $u(C_o) = \ln(C_o)$. This implies that ${R^f}_t$ depends only on $K_t$—is a constant times $K_t^{\alpha-1}$. The safe rate is lower than the expected value of the risky return, so the constant is less than ${\alpha}E(A_t)$.)

$K_{t-1}$ is known by agents at time (beause they are choosing it). Assume ${K^*}_{t-1}$ can be figured out, and is, by the extremely arithmetically-inclined state and its citizens. The state must do this period-by-period to find the ${\tau}_t$ sequence which guarantees a Pareto improvement...

### Robert Waldmann: A Simple Special Case¶

There is a special case in which the required ${\tau}_t$ is constant, and the effect of debt and of the profits-tax-and-wage-subsidy on the expected welfare of a generation taken when the generation is young is constant. Consider:

$U = C_y + \ln(C_o)$

In this case, all variation in $Y_t$ is absorbed by $C_{yt}$. Then everything is very simple. There are closed-form solutions...

## Brad DeLong: Intergenerational Transfer Scenarios¶

Dear Larry—

In response to your question, this is what I think Robert is getting at in thinking about Olivier's presidential address. And I think Robert is right. But I may well be wrong here. My intuitions are (often) flawed, and there are things I think are true that I have not been able to math out satisfactorily...

Back up: In the semi-canonical DIamond (1965) OLG model without a government, there is one and only one asset that a representative agent can use to shift consumption over time: investing in capital.

Once you introduce a government, however, there are more options. The government can make or grease transfers between the old and the young at any one moment in time. These look to agents like assets with different return profiles.

With a constant capital share $\alpha$ and labor share $1-\alpha$, the government can:

1. Impose a lump-sum tax $\delta$ on the young and pay it to the old, thus, from the standpoint of each generation when young, providing a $\beta=0$ transfer from present to future .

2. Impose a lump-sum tax $\delta$ on the old and pay it to the young, thus, from the standpoint of each generation when young, providing a $\beta=0$ transfer from the future to the present .

3. Impose a proportional tax at rate $\tau$ on the labor income of the young and provide a proportional subsidy at rate $\frac{\tau(1-\alpha)}{\alpha}$ to the capital income of the old, thus, from the standpoint of each generation when young providing a $\beta=1$ transfer from present to future.

4. Impose a proportional tax at rate $\frac{\tau(1-\alpha)}{\alpha}$ on the capital income of the old and provide a proportional subsidy at rate $\tau$ to the labor income of the young, thus, from the standpoint of each generation when young providing a $\beta=-1$ transfer from future to present.

(1) has a direct-transfer effect that is welfare-increasing if $R^f < g$; it also crowds-out capital investment, and so has a general-equilibrium factor-price effect (3).

(2) has a direct-transfer effect that is welfare-increasing if $R^f > g$ for everyone except those old at time 0, who are taxed and receive nothing in return; it also crowds-in investment, and so has a general-equilibrium factor-price effect (4).

(3) is welfare-increasing if $R < g$; under log-utility its general-equilibrium effects cancel out.

(4) is welfare-increasing if $R > g$ for everyone except those old at time 0, who are taxed and receive nothing in return; under log-utility its general-equilibrium effects cancel out.

(One of) Olivier's (many) points is that a government can issue debt, spend it on today's old, and then looking forward thereafter having the effect of doing a partial-equilibrium direct-transfer effect (1) and a general-equilibrium factor-price effect (3), and thus the welfare effect of this policy on future generations has two terms one of which is appropriately discounted at r-safe and the other of which is appropriately discounted at r-risky.

Robert's point is that the government can issue debt, thus doing a partial-equilibrium direct-transfer effect (1) and a general-equilibrium factor-price effect (3), and also do (4) to neutralize the (3). Thus all that is left from this joint debt-and-wage-subsidy-funded-by-capital-tax policy is (1), which is appropriately discounted at the risk-free rate.

Hence there is a Pareto-improving policy the government can undertake if $R^f < g$ that reduces the economy's capital stock; in this sense an $R^f < g$ economy is "dynamically inefficient". But this policy is more complex than simply introducing debt, and then letting the young invest in it...

There is also something that I am confused about: Olivier does his welfare analysis from the standpoint of a generation young in period t after period-t uncertainty has been resolved. From that standpoint (3) and (4) involve a certain transfer from or to the young in period t, and an uncertain transfer to or from the old in period t+1, and thus are appropriately discounted at the risky interest rate. But from behind the veil of ignorance, all parts of (3) and (4) are correlated with lifetime wealth, and it is my intuition—which I have not been able to math up—that the appropriate discount rate should be somewhere between $R^f$ and $R$ here...

There is also the following point: the gap between $R^f$ and $R$ in the world is not a rational representative agent's discount for risk. The gap is as wide as it is in the real world because of (a) financial market frictions, including (b) the fact that index funds are not yet the marginal investor, that these days you need not 10 but 50 individual stocks to be diversified, and thus that bsnkruptcy-fearing holders of individual stocks do not extend themselves in equities. If one had a model in which the factors of production were labor, capital, and risk-bearing capacity, then the natural conclusion would be that dynamic efficiency depends on $R^f$ alone.

There is also the fact that Diamond (1965) is only semi-canonical. Olivier said at lunch—with the eyes of the smart young whippersnapper Ben Schoeffer (Ph.D. Harvard 2005), who had never been taken through the Diamond model in lecture, glazing over as we talked—that his estimate was that only 1 in 4 recent Ph.D.'s were taught the Diamond model. For the other 3 in 4, it is all Ramsey—and, of course, for Ramsey with a representative agent there is nothing that a government can do that the RA cannot, and so dynamic inefficiency is simply impossible as no rational RA would ever hold too much capital.

As I said, this is what I think, but I also think that I can provide no warranty of correctness and intelligence: Olivier has been thinking about this more intensively and harder than I have, the issues are complex, and a rational Bayesian would bet that Olivier is right if (when?) he disagrees with any of this...

## Brad DeLong: 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) $\frac{\alpha(1-\beta)}{\beta(1-\alpha)} < 1$

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.