I cannot follow Olivier's algebra here https://www.aeaweb.org/aea/2019conference/program/pdf/14020_paper_etZgfbDr.pdf going from (4) to (5):

What am I doing wrong? Going from (4) to (4a):

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

Where the hell does the extra $R_t$ in the denominator of Blanchard's (5) come from?

But I Have a Deeper Problem...¶

And I belive that I am probably being very stupid here. But I cannot figure out where...

I understand how, given the multiplicative nature of the outcomes shock in period t+1, the general equilibrium-effect gain when you are old from the higher rate of profit dR that arises out of the transfer program dD is perfectly correlated with your consumption when old. I get how it is thus worth less then the product of its expected value and the expected value of the marginal utility of consumption. I get how it is appropriately discounted not at the risk-free but at the risky rate.

But here is the rub: the same multiplicative nature of the outcome shock means that the general equilibrium-effect reduction in your wage when you are young is greatest when your wage is relatively high— which is when you are relatively rich, and your marginal utility of consumption is relatively low. So the certain-equivalent cost value of what the general equilibrium-effect makes you give up when young is less than its expected value times your expected marginal utility when you are young.

It cannot be right to discount at the risky rate the trade-off involved in giving up a risky cash flow today that is correlated with your wealth now for a risky cash flow correlated with your future consumption, can it??

Thus I still cannot see where the conclusion that you discount the general equilibrium-effect at the risky rate comes from...

OK: Time to March Through the Algebra. Wish Me Luck!¶

Send out the dogs if I do not report back before tomorrow...

Clear the decks. Let us shift to my preferred notation...

Diamond overlapping-generations model (y, o)...

One unit of labor...

Two states of the world as far as technology is concerned: a high productivity shock and a low productivity shock case (+, -)...

Production function: $Y = K^{\alpha}(1±\epsilon)$

Utility function: $U = -e^{-{\gamma}C_y} + \beta \left[ -e^{-{\gamma}C_o} \right]$

Budget constraint young: $C_y = W_y - K_o$

Budget constraint old: $C_o = (K_o)^{\alpha}(1±\epsilon) - W_o$

Wage: $W = Y = (1 - \alpha)K^{\alpha}(1±\epsilon)$

Profit Rate: $r = \frac{dY}{dK} = {\alpha}K^{\alpha-1}(1±\epsilon)$

Utility should thus depend on:

• whether productivity was ± when one was young (and thus got a high/low wage), and * on whether productivity was ± when one was old (and thus got a high/low rate of profit), but
• CARA should mean that the capital stock when old should not depend on one's wealth and thus on the wage when young, but be constant across states of the world...

Thus productivity when young should be a shift only in the level of utility; and all economic decisions should only depend on the two possible states of productivity when one is old, and the interaction with risk aversion...

Let me see if I can still do this...

I cannot follow Olivier's algebra here https://www.aeaweb.org/aea/2019conference/program/pdf/14020_paper_etZgfbDr.pdf going from (4) to (5):

What am I doing wrong? Going from (4) to (4a):

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

Where the hell does the extra $R_t$ in the denominator of Blanchard's (5) come from?

But I Have a Deeper Problem...¶

And I belive that I am probably being very stupid here. But I cannot figure out where...

I understand how, given the multiplicative nature of the outcomes shock in period t+1, the general equilibrium-effect gain when you are old from the higher rate of profit dR that arises out of the transfer program dD is perfectly correlated with your consumption when old. I get how it is thus worth less then the product of its expected value and the expected value of the marginal utility of consumption. I get how it is appropriately discounted not at the risk-free but at the risky rate.

But here is the rub: the same multiplicative nature of the outcome shock means that the general equilibrium-effect reduction in your wage when you are young is greatest when your wage is relatively high— which is when you are relatively rich, and your marginal utility of consumption is relatively low. So the certain-equivalent cost value of what the general equilibrium-effect makes you give up when young is less than its expected value times your expected marginal utility when you are young.

It cannot be right to discount at the risky rate the trade-off involved in giving up a risky cash flow today that is correlated with your wealth now for a risky cash flow correlated with your future consumption, can it??

Thus I still cannot see where the conclusion that you discount the general equilibrium-effect at the risky rate comes from...

FOC for CARA¶

$\frac{dU}{dC_y} = {\gamma}e^{-{\gamma}C_y}$

$\frac{dU}{dC_o} = {\beta}{\gamma}e^{-{\gamma}C_o}$

$\frac{dU}{dK} = \frac{dU}{dC_o}\frac{dC_o}{dK} - \frac{dU}{dC_y}$ (because $\frac{dC_y}{dK} = -1)$

$\frac{dC_o}{dK} = R(1±{\epsilon})$

$C_o = KR(1±{\epsilon})$

Utility¶

{\alpha}K^{\alpha}

Indexing states of the world by "++" subscripts—"++" meaning a high productivity shock when young and a high productivity shock when old, etc., and setting out utilities in those four states of the world:

$U = \frac{U_{++} + U_{+-} + U_{-+} + U_{--}}{4}$

$U = -\frac{1}{4} \left[ e^{-{\gamma}(W_{+y} - K)} + {\beta}e^{-{\gamma}({\alpha}K^{\alpha}(1+{\epsilon}))} \right] -$
$\frac{1}{4} \left[ e^{-{\gamma}(W_{+y} - K)} + {\beta}e^{-{\gamma}({\alpha}K^{\alpha}(1-{\epsilon}))} \right] -$
$\frac{1}{4} \left[ e^{-{\gamma}(W_{-y} - K)} + {\beta}e^{-{\gamma}({\alpha}K^{\alpha}(1+{\epsilon}))} \right] -$
$\frac{1}{4} \left[ e^{-{\gamma}(W_{-y} - K)} + {\beta}e^{-{\gamma}({\alpha}K^{\alpha}(1-{\epsilon}))} \right]$

Regrouping into (1) terms that do not depend on investment, (2) terms that depend on investment but not risk, and (3) terms that depend on investment and risk:

$U = \left[ -\frac{e^{-{\gamma}W_{+y}}}{2} - \frac{e^{-{\gamma}W_{-y}}}{2} \right] + \left[ - e^{{\gamma}K} - {\beta}e^{-{\gamma}{\alpha}K^{\alpha}} \right] + \left[ - \frac{{\beta}e^{-{\gamma}{\alpha}K^{\alpha}\epsilon}}{2} - \frac{{\beta}e^{{\gamma}{\alpha}K^{\alpha}\epsilon}}{2} \right]$

Taking the first-order condition for investment and the capital stock:

$0 = \frac{dU}{dK} = -{\gamma}e^{{\gamma}K} + {\beta}{\gamma}{\alpha}^{2}K^{\alpha-1}e^{-{\gamma}{\alpha}K^{\alpha}} + {\beta}{\gamma}{\alpha}^{2}{\epsilon}K^{\alpha-1}\left[\frac{e^{-{\gamma}{\alpha}K^{\alpha}{\epsilon}} - e^{{\gamma}{\alpha}K^{\alpha}{\epsilon}}}{2}\right]$

Eliminating $- {\gamma}$:

$0 = \frac{dU}{dK} = e^{{\gamma}K} - R{\beta}e^{-{\gamma}RK} +$ $\frac{R{\beta}{\epsilon}\left[e^{{\gamma}KR\epsilon} - e^{-{\gamma}KR\epsilon}\right] }{2}$