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 Overlapping-Generations model, 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 the near-canonical model he presents:

1. What he calls the "partial-equilibrium" transfer effect raises welfare if the rate on safe government bonds is less than the growth rate.
2. What he calls the "general-equilbrium" factor-price effect working through crowding-out raises the welfare of the average agent only when the risky profit rate is less than the growth rate.

I may well be being stupid here—my estimate of the chance I am being stupid is about 75%. But I have a conceptual problem here that I have been unable to resolve...

## My Conceptual Problem¶

Back up: Olivier Blanchard‘s presidential address has left me puzzled in one particular: I am not able to follow all his conclusions as to what interest rate should be used to discount the different effects of government debt.

Blanchard considers a near-canonical model: the Diamond Overlapping Generations model. In this model, one important effect—perhaps the principal effect—of government debt is to directly shift consumption to the old who hold and sell the debt and away from the young who buy the debt to use it as a savings vehicle. 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, in that 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 will be welfare-raising, Blanchard concludes, only under a much 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. And here, while I can follow Blanchard’s algebra—it certainly looks correct to me—I have a conceptual difficulty...

Here is my conceptual difficulty in the case in which there is uncertainty and thus a gap between the risk-free interest rate at time t ${R_{t+1}}^f$ and the expected profit rate $E_t\left[R_{t+1} \right]$:

### Utility-Metric Valuations of the Gains and Losses¶

Consider the utility values of the four cash flows associated with the direct-transfer and factor price effects:

(1) $-E \left[ U'(C_{y,t})dD \right]$ :: the direct-transfer reduction in consumption when young because you purchases the rolled-over debt.

(2) $E \left[ {\beta}U'(C_{o,t+1})dD \right]$ :: the direct-transfer boost to consumption when old because you receive interest and principal from the maturing debt.

(3) $E \left[ U'(C_{y,t})dW_t \right]$ :: the factor-price effect reduction in consumption (remember! $\frac{dW_t}{dD} < 0$) when young because the reduced stock of capital lowers your bargaining power in the labor market.

(4) $-E \left[ {\beta}U'(C_{o,t+1})dW_{t+1} \right]$ :: the factor-price effect boost to consumption (remember! $\frac{dW_t}{dD} < 0$) when old because the reduced stock of capital lowers' the next generation's bargaining power in the labor market.

### The Direct-Transfer Effect¶

The direct-transfer effect boosts utility when (2) outweighs (1), which requires:

(5) $-E \left[ U'(C_{y,t})dD \right] + E \left[ {\beta}U'(C_{o,t+1})dD \right] > 0$

Since $dD$ is fixed and not stochastic, we can then write:

(6) $\left[ -E \left[ U'(C_{y,t}) \right] + E \left[ {\beta}U'(C_{o,t+1}) \right] \right] dD > 0$

(7) $\left[ \frac{-E \left[ U'(C_{y,t})\right]}{{\beta} E \left[ U'(C_{o,t+1}) \right]} + 1 \right] dD > 0$

And since the first term inside the brackets is simply the risk-free rate ${R_{t+1}}^f$, the direct-transfer effect boosts utility if and only if:

(8) $\left[ -{R_{t+1}^f} + 1 \right] dD > 0$

### Blanchard's Analysis of the Factor-Price Effect¶

Now consider the factor-price effect. Since $W_{t+1}$ is stochastic, we cannot move it through the expectations operator $E$ in (5) to arrive at (6), and so our analog of (7) is:

(9) $\left[ -\frac{E \left[ U'(C_{y,t})dW_{t}\right]}{{\beta} E \left[ U'(C_{o,t+1})dW_{t+1} \right]} + 1 \right] dD > 0$

If $dW_{t}$ were not stochastic and $dW_{t+1}$ were— if $dW_{t} = dW$ and $dW_{t+1} = (1+\epsilon_{t+1})dW$—then we could get:

(10) $\left[ -\frac{E \left[ U'(C_{y,t})\right]dW}{ {\beta} E \left[ U'(C_{o,t+1})(1+\epsilon_{t+1}) \right]dW} + 1 \right] = \left[ -\frac{E \left[ U'(C_{y,t})t\right]}{ {\beta} E \left[ U'(C_{o,t+1})(1+\epsilon_{t+1}) \right]} + 1 \right] > 0$

And since $\frac{E \left[ U'(C_{y,t})\right]}{ {\beta} E \left[ U'(C_{o,t+1})(1+\epsilon_{t+1}) \right]} = E_t\left[ R_{t+1} \right ]$, it would then be the case that:

(11) $\left[ -E_t\left[{R_{t+1}}\right] + 1 \right] dD > 0$

which is what Blanchard asserts.

### My Intuition and My Conceptual Difficulty¶

My conceptual difficulty is that $W_t$ is stochastic: $W_t = (1+ \epsilon_t)W$, and so I get:

(12) $\left[ -\frac{E \left[ U'(C_{y,t})(1+\epsilon_{t})\right]}{ {\beta} E \left[ U'(C_{o,t+1})(1+\epsilon_{t+1}) \right]} + 1 \right] > 0$

which I cannot reduce to (11), because $\frac{E \left[ U'(C_{y,t})(1+\epsilon_{t})\right]}{ {\beta} E \left[ U'(C_{o,t+1})(1+\epsilon_{t+1}) \right]} ≠ E_t\left[ R_{t+1} \right ]$

My intuition tells me that the appropriate rate to discount the factor-price effect cannot be the expected risky profit rate, for to receive the higher profits when old you have not made a fixed investment but instead made something like a contribution to a social insurance scheme, and the utility cost of such a contribution is clearly lower than its expected value. Thus this factor-price effect might be a good deal even when it does not look not a profitable bargain from a what-is-the-risky-rate-of-profit? perspective.

### The Two-State Log-Utility Case¶

Consider a near-canonical model: the Diamond Overlapping-Generations model

In this model, one important effect—perhaps the principal effect—of government debt is to directly shift consumption to the old who hold and sell the debt and away from the young who buy the debt to use it as a savings vehicle. 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, in that 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 as it transfers from the young, who earn less in wages, to the old, who pay less in wages when they hire people to work in their firms with their capital. It will be welfare-raising, Blanchard concludes, 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.

Let me set out a simple case, with neither population growth nor technological progressm with log utility, and with uncertainty and thus a wedge begtween the expected profit rate and the safe government bond rate generated by a shock to productivity in each period that can take on two values: the productivity of capital in period t can be, with 50% probability for each, either $R_t = (1 + \epsilon_t)R(K_t)$ or $R_t = (1 - \epsilon_t)R(K_t)$.

Let me do the log-utility case:

(1) $u(C) = \ln(C)$

with rate of time-discount $\beta$:

(2) $U_t = E_t \left[ u\left(C_{y,t}\right) + {\beta}u\left(C_{o,t+1}\right) \right] = E_t \left[ \ln(C_{y,t}) + \beta\ln(C_{o,t+1}) \right]$

and uncertainty:

(3) $R_t = (1+\epsilon)R(K_t)$ with $P=0.5$; $R_t = (1-\epsilon)R(K_t)$ with $P=0.5$

Then log utility gives us:

(4) $K_{t+1} = \frac{\beta}{1+\beta}W_t$

And consumption spending when young and old are:

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

(6) $C_{o,t+1} = \frac{{\beta}R(K_{t+1})W_t}{1+\beta}$

Marginal utility when young is:

(7) $u'(C_{y,t}) = \frac{1}{C_{y,t}} = \frac{1+\beta}{W_t}$

And marginal utility when old is:

(8) $u'C_{o,t+1} = E\left[\frac{1}{C_{o,t+1}}\right] = \frac{1}{2}\frac{1}{(1+\epsilon)R(K_{t+1})K{t+1}} + \frac{1}{2}\frac{1}{(1-\epsilon)R(K_{t+1})K{t+1}}$

(9) $u'C_{o,t+1} = \frac{1+\beta}{\beta(1-\epsilon^2)R(K_{t+1})W_t}$

These together imply the safe government-bond rate of interest:

(10) ${R^f}_{t+1} = (1-\epsilon^2)R(K_{t+1})$

Now let us consider a government that issues debt at the safe rate and rolls it over, paying out the surplus or taxing to cover its deficit. Suppose the government raises its debt from 0 by an amount $dD$. This has a negative direct-transfer effect on the utility derived when young because the young commit cash to buying the debt of:

(11) $-E \left[ u'(C_{y,t})dD \right] = -\frac{1+\beta}{W_t}dD$

This has a positive direct-transfer effect on the utility derived when old because the old receive interest and principle (plus or minus surplus transfers or taxes) of:

(12) $E \left[ {\beta}u'(C_{o,t+1})dD \right] = \frac{1+\beta}{(1-\epsilon^2)R(K_{t+1})W_t}dD$

Now let us consider the factor-price effect of this infinitesimal debt increase. Via crowding-out of capital it lowers the bargaining power and thus the wage of the young by an amount $dW_t$, with utility effect:

(13) $E \left[ \frac{(1+\beta)dW_t}{W_t} \right]$

And this factor-price effect reduces the wages paid by the old by an amount $dW_{t+1}$, with utility effect:

(14) $-\beta E \left[ u'C_{o,t+1} \right] = -\frac{1}{2}\frac{(1+\beta)dW_{t+1}}{(1+\epsilon)R(K_{t+1})W_t} - \frac{1}{2}\frac{(1+\beta)dW_{t+1}}{(1-\epsilon)R(K_{t+1})W_t}$

From (11) and (12), our condition for the direct-transfer effect to raise well-being becomes:

(15) $-\frac{1+\beta}{W_t} + \frac{1+\beta}{(1-\epsilon^2)R(K_{t+1})W_t} > 0$

(16) $-(1-\epsilon^2)R(K_{t+1}) + 1 > 0$

which is that the risk-free rate is less than one—i.e., that the risk-free rate is less than the zero growth rate of the economy.

If $W_t$ and $dW_{t+1}$ are not stochastic—if wages are set before that period's productivity shock is revealed—then the factor-price effect condition derived from (13) and (14) also becomes (16), and the relevant rate is the risk-free rate.

If by contrast wages are stochastic—if wages are a constant fraction of output, say, so that the productivity shock affects wages so that: $W_t = (1 ± \epsilon_t)W(K_t)$ and $dW_{t+1} = (1 ± \epsilon_t)\frac{dW(K_{t+1})}{dD}dD$, then (13) becomes

(17) $\frac{1}{2}\frac{(1+\beta)(1 + \epsilon_t)dW(K_t)}{(1 + \epsilon_t)W(K_t)} + \frac{1}{2}\frac{(1+\beta)(1 - \epsilon_t)dW(K_t)}{(1 - \epsilon_t)W(K_t)} = \frac{(1+\beta)dW(K_t)}{W(K_t)}$

and (14) becomes:

(18) $-\frac{1}{4}\frac{(1+\beta)(1+\epsilon)dW(K_{t+1})}{(1+\epsilon)R(K_{t+1})(1+\epsilon)W(K_t)} - \frac{1}{4}\frac{(1+\beta)(1-\epsilon)dW(K_{t+1})}{(1-\epsilon)R(K_{t+1})(1+\epsilon)W(K_t)} -\frac{1}{4}\frac{(1+\beta)(1+\epsilon)dW(K_{t+1})}{(1+\epsilon)R(K_{t+1})(1-\epsilon)W(K_t)} - \frac{1}{4}\frac{(1+\beta)(1-\epsilon)dW(K_{t+1})}{(1-\epsilon)R(K_{t+1})(1-\epsilon)W(K_t)}$

Simplifying:

(19) $\left[ -\frac{1}{4}\frac{1}{R(K_{t+1})(1+\epsilon)W(K_t)} - \frac{1}{4}\frac{1}{R(K_{t+1})(1+\epsilon)W(K_t)} -\frac{1}{4}\frac{1}{R(K_{t+1})(1-\epsilon)W(K_t)} - \frac{1}{4}\frac{1}{R(K_{t+1})(1-\epsilon)W(K_t)} \right] \left[(1+\beta)dW(K_{t+1}) \right]$

(20) $-\left[ \frac{1}{4}\frac{1}{(1+\epsilon)} +\frac{1}{4}\frac{1}{(1+\epsilon)} +\frac{1}{4}\frac{1}{(1-\epsilon)} +\frac{1}{4}\frac{1}{(1-\epsilon)} \right] \left[ \frac{(1+\beta)dW(K_{t+1})}{R(K_{t+1})W(K_t)} \right]$

(21) $\frac{(1+\beta)dW(K_{t+1})}{(1-\epsilon^2)R(K_{t+1})W(K_t)}$

The condition for the factor-price effect on utility to be positive is then:

(22) $\frac{(1+\beta)dW(K_t)}{W(K_t)} - \frac{(1+\beta)dW(K_{t+1})}{(1-\epsilon^2)R(K_{t+1})W(K_t)} > 0$

(23) $dW(K_t) - \frac{dW(K_{t+1})}{(1-\epsilon^2)R(K_{t+1})} > 0$

And since $dW(K_t) < 0$:

(24) $(1-\epsilon^2)R(K_{t+1}) - \frac{dW(K_{t+1})}{dW(K_t)} < 0$

This looks, again, like the risk-free rate. I think the intuition is that two equally-risky cash flows are, in fact, properly discounted at the risk-free rate.

Thus I have been unable to reproduce Blanchard's conclusion that the proper discount rate to apply in assessing the factor-price effect is not the risk-free government-bond rate but the expected risky profit rate. As Robert Waldmann says, Blanchard is correct for the case he considers—which is of a generation of workers who work with the average capital stock and receive the average wage. But the effect on the average is not the average of the effects: the factor-price effect has a social insurance component, charging the young less for the transfer when the young are poor and when as a result their marginal utility of consumption when young is low.

Of course, the chance I have not made an algebraic error somewhre here is 0.1, and my assessment of the chance that my conclusion is correct is only 0.25...

# More Incomplete Notes and Chicken-Scratchings¶

$-E \left[ U'(C_{y,t})dD \right]$ :: direct-transfer loss when young

$E \left[ {\beta}U'(C_{o,t+1})dD \right]$ :: direct-transfer gain when old

$E \left[ U'(C_{y,t}dW_t \right]$ :: factor-price loss when young

$-E \left[ {\beta}U'(C_{o,t+1})dW_{t+1} \right]$ :: factor-price gain when old

$Y_t = (1 + \epsilon_t){K_t}^{\alpha}$ ::production function

$W_t = (1 - \alpha)(1 + \epsilon_t){K_t}^{\alpha}$ :: wage

$C_{o,t+1} = \alpha(1 + \epsilon_{t+1}){K_{t+1}}^{\alpha} + D$ :: consumption when old

$C_{y,t} = (1 - \alpha)(1 + \epsilon_t){K_t}^{\alpha} - D_t - K_{t+1}$ :: consumption when young

$U(C) = \frac{C^\gamma}{\gamma}$ :: utility function

$U'(C) = C^\gamma$ :: marginal utility

$dW_{t+1} = {\alpha}(1 - \alpha)(1 + \epsilon_{t+1}){K_{t+1}}^{\alpha - 1}dK_{t+1}$

Evaluate at D=0...

$-E \left[ {\beta}(\alpha(1 + \epsilon_{t+1}){K_{t+1}}^{\alpha})^{\gamma} {\alpha}(1 - \alpha)(1 + \epsilon_{t+1}){K_{t+1}}^{\alpha - 1}dK_{t+1} \right]$ :: factor=price gain when old

$-E \left[ {\beta}{\alpha}^{\gamma+1}(1 - \alpha)(1 + \epsilon_{t+1})^{\gamma}{K_{t+1}}^{\alpha^{\gamma}} (1 + \epsilon_{t+1}){K_{t+1}}^{\alpha - 1}dK_{t+1} \right]$ :: factor=price gain when old

In brief, a marginal change $dD$ in a transfer program from young to old has, in Blanchard's model, two effects: a direct-transfer partial-equilibrium effect $dU_a$ and a factor-price general-equilibrium effect $dU_b$:

$\frac{dU}{dD} = \frac{dU_a}{dD} + \frac{dU_b}{dD}$

The partial-equilbrium direct-transfer effect reduces consumption when young $C_y$ by:

$\frac{dC_y}{dD} = -1$

while raising consumption when old by:

$\frac{dC_y}{dD} = 1$

This direct-transfer partial-equilibrium effect will be a good thing if:

$U'(C_y)\frac{dC_y}{dD} + {\beta}U'(C_o)\frac{dC_o}{dD} > 0$

$-U'(C_y) + {\beta}U'(C_o) > 0$

$1 > \frac{U'(C_y)}{{\beta}U'(C_o)}$

And since:

$R = \frac{U'(C_y)}{{\beta}U'(C_o)}$

This direct-transfer partial-equilibrium effect will be a good thing if:

$R < 1$

To the extent that dD crowds out capital investment, there is a factor-price general-equilibrium effect alters factor prices, and so also reduces consumption when young and raises consumption when old:

$\frac{dW}{dD} = -\frac{KdR}{dD}$

$\frac{dC_y}{dD} = -\frac{KdR}{dD}$

$\frac{dC_o}{dD} = \frac{KdR}{dD}$

This factor-price general-equilibrium effect will be a good thing if:

$-U'(C_y)\frac{KdR}{dD} + {\beta}U'(C_o)\frac{KdR}{dD} > 0$

$-U'(C_y) + {\beta}U'(C_o) > 0$

$1 > \frac{U'(C_y)}{{\beta}U'(C_o)}$

And since:

$R = \frac{U'(C_y)}{{\beta}U'(C_o)}$

This factor-price general-equilibrium effect will be a good thing if:

$R < 1$

Now move into the uncertainty case, in which the production function at any time t is:

$Y_t = (1 + \epsilon_t)F(K_t,1)$

And the rate of profit is:

$R_t = R(K_t, \epsilon_t) = (1+\epsilon_t)R(K_t)$

Stochastic shocks to the system are:

• $K_t$, which is the amount invested by those young in period t-1
• $\epsilon_t$, the productivity shock in period t
• $\epsilon_{t+1}$, the productivity shock in period t+1

Under uncertainty, the magnitude of the transfer is not stochastic, so the partial-equilibrium direct-transfer effect is a good thing if:

$E \left[ U'(C_{y,t}) + {\beta} U'(C_{o,t}) \right] > 0$

Define the one-period risk-free rate as of time t by:

$U'(C_{y,t}) = {R^f}_t{\beta} \left[ E_t \left[ U'(C_{o,t+1}) \right] \right]$

At time t, the young invest enough in capital to set:

$[ U'(C_{y,t}) = E_t \left[ {\beta}(1+\epsilon_{t+1})R(K_{t+1})U'(C_{o,t+1}) \right]$

So:

$E \left[ E_t \left[ {\beta}(1+\epsilon_{t+1})R(K_{t+1})U'(C_{o,t+1}) \right] + {\beta} U'(C_o) \right] > 0$

As of time t, $K_{t+1}$ is known, so we can move terms that depend on it outside of the inner expectation:

$E \left[ {\beta}R(K_{t+1}) E_t \left[ (1+\epsilon_{t+1})U'(C_{o,t+1}) \right] + {\beta} U'(C_{o,t+1}) \right] > 0$

But I have found no flaw either in Blanchard’s algebra or in his modeling strategy.

This leads me to expect that the reasoning behind my current visualization of the Cosmic All is in error.

If anyone could teach me whether my thought is correct thought in accord with the Group of 17, and how to correct my thought to be in accord if it is not, I would be very grateful...

## Marching Through Algebra¶

So let me march through algebra, so that others can follow (and, I hope, correct) my thinking...

### The Certainty Case¶

Consider the Diamond Overlapping-Generations Model—DOG—with two period-lived agents who work, collect a wage W, invest in capital K, and consume $C_y$ ("y" for "young") when young; and who collect profits RK, equal to the profit rate R times their capital K, and consume $C_o$ ("o" for "old") when old. Let the government embark on a transfer program from the young to the current old of amount D. Let the rate of time discount be $\beta$ for a time-separable utility function:

(1) $U = U_y + {\beta}U_o$ :: Utility

The budget constraint: consumption when young $C_y$ is equal to the wage W minus capital investment K minus the amount of the transfer program D; consumption when old $C_w$ is equal to capital investment K times the rate of profit R plus the amount of the transfer program D:

(2) $C_y = W - K - D ; C_o = RK + D$ :: Budget Constraint

There is a standard labor-and-capital production function, with one unit of labor assumed available:

(3) $Y = F(K,1)$ :: Production Function

And factors of production are paid their marginal products:

(4) $R = \frac{dF(K, 1)}{dK}; W = \frac{dF(K, 1)}{dL}$ :: Rate of Profit and Level Wages

The utility-maximizing agents will then set their consumption levels to satisfy:

(5) $U'(C_y) = {\beta}RU'(C_o)$ :: First-Order Condition

Now consider an infinitesimal expansion dD in the transfer program. What is its effect on utility? Blanchard separates the effect on utility into two terms—a direct-transfer effect ("a") and a factor-price crowding-out effect ("b"), which he calls "partial-equilibrium" and "general-equilibrium" effects:

(6) $\frac{dU}{dD} = \frac{dU_a}{dD} + \frac{dU_b}{dD}$ :: Separate Out Partial- and General-Equilibrium Effects

The partial-equilibrium effect is straightforward: the transfer reduces utility when young and raises utility when old:

(7) $\frac{dU_a}{dD} = -U'(C_y) + {\beta}U'(C_o)$ :: Partial-Equilibrium Effect

The general-equilibrium effect is that the increased transfer, to the extent that it crowds out investment, raises profits when old and lowers wages when young:

(8) $\frac{dU_b}{dD} = U'(C_y)\frac{dW}{dD} + {\beta}U'(C_o)K\frac{dR}{dD}$ :: General Equilibrium Effect

Using the first-order condition (5) to evaluate the direct-transfer partial-equilibrium effect $U_a$:

(9) $\frac{dU_a}{dD} = -{\beta}RU'(C_o) + {\beta}U'(C_o)$

(10) $\frac{dU_a}{dD} = {\beta}(1-R)U'(C_o)$

Hence it is appropriate evaluate costs and benefits of the partial-equilibrium effect using the profit rate, and when the profit rate is less than one—and in more general models less than the growth rate—the effect is to raise utility.

Now turn to the general-equilibrium effect:

Since there is one unit of labor and K units of capital, the slope fothe factor-price frontier is:

(11) $\frac{dW}{dR} = -K$ :: Factor-Price Frontier

Substituting (11) into (8):

(12) $\frac{dU_b}{dD} = \left[ -U'(C_y)K + {\beta}U'(C_o)K \right] \frac{dR}{dD}$

Substituting the first-order condition (5) into 12:

(13) $\frac{dU_b}{dD} = \left[ -{\beta}RU'(C_o)K + {\beta}U'(C_o)K \right] \frac{dR}{dD}$

And regrouping:

(14) $\frac{dU_b}{dD} = (1-R)U'(C_o)K\frac{dR}{dD}$

finds that an increase in the transfer program raises utility via the general-equilibrium crowding-out factor-price effect if the profit rate is less than one—less than the grow=th rate in more general models. Hence it is appropriate to evaluate costs and benefits of the general-equilibrium effect using the profit rate.

### Uncertainty Case¶

But what happens when there is uncertainty?

Suppose that each generation the production function is subject to a productivity shock:

(15) $Y_t = (1+\epsilon_t) F(K_t, 1)$

The first-order condition for utility maximization then becomes:

(16) $U'(C_{y,t}) = {\beta}E\left[ R_{t+1}(1+\epsilon_{t+1})U'(C_{o,t+1}) \right]$

When there is uncertainty we can calculate a shadow risk-free rate. Note that this is a "shadow" rate because there is no asset other than capital and hence zero supply of a risk-free asset:

(17) ${R^f}_{t+1}E[U'(C_{o,t+1})] = E[R_{t+1}(1+\epsilon_{t+1})U'(C_{o,t+1})]$

And we can transform (17) to get:

(18) ${R^f}_{t+1} = R_{t+1} \left(1 + \frac{E[U'(C_{o,t+1})\epsilon_{t+1}]}{E[U'(C_{o,t+1})]} \right)$

Next, as before, divide the effect of an increase in the transfer program on utility into (a) direct-transfer and (b) factor-price effects:

(19) $\frac{dU_t}{dD} = \frac{dU_{a,t}}{dD} + \frac{dU_{b,t}}{dD}$

The expected-utility impact of the direct-transfer effect (a) is:

(20) $\frac{dU_{a,t}}{dD} = -E[U'(C_{y,t})] + E[{\beta}U'(C_{o,t+1})]$

The expected-utility impact of the factor-price effect (b) is:

(21) $\frac{dU_{b,t}}{dD} = E\left[U'(C_{y,t})\frac{dW_t}{dD}\right] + E\left[{\beta}U'(C_{o,t+1})K_{t+1}(1+\epsilon_{t+1})\frac{dR_{t+1}}{dD}\right]$

#### The Direct-Transfer Partial-Equilibrium Term¶

Substitute the risk-free rate definition (17) into the first order condition (16) and then into (20) to assess the partial-equilibrium (a) welfare term:

(22) $\frac{dU_{a,t}}{dD} = -{\beta}\left[{R^f}_{t+1}E[U'(C_{o,t+1})]\right] + {\beta}E[U'(C_{o,t+1})]$

Discover that the partial-equilibrium channel raises utility if the risk-free rate is less than one—less than the economy's growth rate in more general models:

(23) $\frac{dU_{a,t}}{dD} = {\beta}(1-{R^f}_{t+1})E\left[E[U'(C_{o,t+1})]\right]$

Thus the right interest rate to use for the partial-equilibrium term is the risk-free rate ${R^f}_{t+1}$.

#### The General-Equilibrium Term¶

Begin by using the factor-price frontier relationship:

(24) $\frac{dW_t}{dR_t} = (1+\epsilon_t)K_t$

to get:

(25) $\frac{dU_{b,t}}{dD} = -E\left[U'(C_{y,t})\frac{dW_t}{dD}\right] + {\beta} E\left[ U'(C_{o,t+1})\frac{dW_{t+1}}{dD} \right]$

(26) $\frac{dU_{b,t}}{dD} = -E\left[U'(C_{y,t})(1+\epsilon_{t})K_t\frac{dR_t}{dD}\right] + {\beta} E\left[ U'(C_{o,t+1})K_{t+1}(1+\epsilon_{t+1})\frac{dR_{t+1}}{dD} \right]$

The sign of (26) is the sign of:

(27) ${\beta}\frac{ E\left[ (1+\epsilon_{t+1})U'(C_{o,t+1})K_{t+1}\frac{dR_{t+1}}{dD} \right] }{E\left[U'(C_{y,t})(1+\epsilon_{t})K_t\frac{dR_t}{dD}\right]} - 1$

Blanchard conditions on specific values of $K_t$ and $\epsilon_t$. That makes the denominator of (27) fixed, and also makes $K_{t+1}$ and$R_{t+1}$ non-stochastic, and so, for each possible fixed $K_t$ and $\epsilon_t$, we can write:

(28) ${\beta}\frac{ E \left[ (1+\epsilon_{t+1})U'({C_{o,t+1}} | (\epsilon_t)(K_t)) \right] }{U'({C_{y,t} | (\epsilon_t)(K_t)})(1+\epsilon_{t})} \frac{({K_{t+1}} | (\epsilon_t)(K_t)) \left( \frac{dR_{t+1} | (\epsilon_t)(K_t)}{dD} \right)}{K_t\frac{dR_t}{dD}} - 1$

And from (16) we can get that the sign of $1 - (R_{t+1} | (\epsilon_t)(K_t))$ is the sign of:

(29) ${\beta} \frac{E \left[ (1+\epsilon_{t+1})U'(C_{o,t+1} | (\epsilon_t)(K_t)) \right] }{U'(C_{y,t} | (\epsilon_t)(K_t))} - 1$

And it is here that I am stuck. I see no way to get a conclusion that the signs of (27) and (28) are the same.

#### And a Typo¶

In Blanchard's going from his (4) to his (5):

I think he has picked up an extra $R_t$ in the denominator:

(4a) $dU_{bt} = \left[ \frac{\beta\alpha}{\eta} \right] E \left[ \left( R_{t+1}R_t-\frac{R_{t+1}}{R_t}R_t \right)U'(C_{2,t+1}) \right]dK$

to (4b):

(4b) $dU_{bt} = \left[ \frac{\beta\alpha}{\eta} \right] E \left[ \left( R_{t+1}R_t-R_{t+1} \right) U'(C_{2,t+1}) \right] dK$

to (4c):

(4c) $dU_{bt} = \left[ \frac{\beta\alpha}{\eta} \right] E \left[ \left( R_{t+1}(R_t-1) \right) U'(C_{2,t+1}) \right] dK$

to (5a):

(5a) $dU_{bt} = \left[ \frac{\beta\alpha}{\eta} \right] E \left[ \left( R_{t+1} \right) U'(C_{2,t+1}) \right] (R_t-1) dK$

And Blanchard has an extra $R_t$ in the denominator.