econ.studio
Diamond Overlapping Generations Model
Section 8 of 16
Section 8

Derivations

Step 1 — Euler equation under CRRA

Start with CRRA felicity u(c)=c1−θ/(1−θ)u(c) = c^{1-\theta}/(1-\theta), so u′(c)=c−θu'(c) = c^{-\theta}. Substituting into the Euler equation (Section 6) gives:

  1. Step 1
    u′(c1,t)=β(1+rt+1)u′(c2,t+1)u'(c_{1,t}) = \beta (1 + r_{t+1}) u'(c_{2,t+1})

    Generic Euler equation.

  2. Step 2
    c1,t−θ=β(1+rt+1) c2,t+1−θc_{1,t}^{-\theta} = \beta (1 + r_{t+1})\, c_{2,t+1}^{-\theta}

    Substitute CRRA marginal utility.

  3. Step 3
    (c2,t+1c1,t)θ=β(1+rt+1)\left( \frac{c_{2,t+1}}{c_{1,t}} \right)^{\theta} = \beta (1 + r_{t+1})

    Move the ratio to one side.

  4. Step 4
    c2,t+1c1,t=[β(1+rt+1)]1/θ\frac{c_{2,t+1}}{c_{1,t}} = \left[\beta (1 + r_{t+1})\right]^{1/\theta}

    Raise both sides to the power 1/θ1/\theta.

The consumption growth rate over the lifecycle is [β(1+rt+1)]1/θ[\beta(1+r_{t+1})]^{1/\theta}. Larger β\beta (more patient) or larger 1+rt+11+r_{t+1} (better return) ⇒ higher second-period consumption relative to first.

Step 2 — Combine with the budget constraint

Substitute the lifetime budget c1,t+c2,t+1/(1+rt+1)=wtc_{1,t} + c_{2,t+1}/(1+r_{t+1}) = w_t:

  1. Step 1
    c2,t+1=[β(1+rt+1)]1/θ c1,tc_{2,t+1} = [\beta(1+r_{t+1})]^{1/\theta}\, c_{1,t}

    From the Euler equation.

  2. Step 2
    c1,t+[β(1+rt+1)]1/θ c1,t1+rt+1=wtc_{1,t} + \frac{[\beta(1+r_{t+1})]^{1/\theta}\, c_{1,t}}{1+r_{t+1}} = w_t

    Substitute into the lifetime budget.

  3. Step 3
    c1,t[1+β1/θ(1+rt+1)(1−θ)/θ]=wtc_{1,t} \left[ 1 + \beta^{1/\theta} (1+r_{t+1})^{(1-\theta)/\theta} \right] = w_t

    Factor out c1,tc_{1,t} and simplify the second term.

  4. Step 4
    c1,t=wt1+β1/θ(1+rt+1)(1−θ)/θc_{1,t} = \frac{w_t}{1 + \beta^{1/\theta} (1+r_{t+1})^{(1-\theta)/\theta}}

    Closed-form young-period consumption under CRRA.

Savings are st=wt−c1,ts_t = w_t - c_{1,t}, giving the **CRRA savings function**:

  st=β1/θ(1+rt+1)(1−θ)/θ1+β1/θ(1+rt+1)(1−θ)/θ wt  \boxed{\; s_t = \frac{\beta^{1/\theta} (1+r_{t+1})^{(1-\theta)/\theta}}{1 + \beta^{1/\theta} (1+r_{t+1})^{(1-\theta)/\theta}}\, w_t \;}
eq:crra-savings
Savings are a fraction of wage income. The fraction depends on patience β\beta, the EIS 1/θ1/\theta, and the future return rt+1r_{t+1}.

Step 3 — The sign of the interest-rate effect

Differentiating the savings function with respect to rt+1r_{t+1} reveals the central ambiguity of intertemporal choice:

CaseSign of ∂s/∂r\partial s / \partial rInterpretation
θ<1\theta < 1 (high EIS)>0> 0Substitution effect dominates: higher rr ⇒ save more.
θ=1\theta = 1 (log utility)00Effects exactly cancel: savings independent of rr.
θ>1\theta > 1 (low EIS)<0< 0Income effect dominates: higher rr ⇒ save *less* (target wealth lower).
The savings response to the interest rate decomposes into a substitution and an income effect.

Step 4 — Log utility: the workhorse special case

Set θ→1\theta \to 1. Then u(c)=ln⁡cu(c) = \ln c and the savings function collapses to a clean expression independent of rt+1r_{t+1}:

  1. Step 1
    st=β1(1+rt+1)01+β1(1+rt+1)0wts_t = \frac{\beta^1 (1+r_{t+1})^0}{1 + \beta^1 (1+r_{t+1})^0} w_t

    Plug θ=1\theta = 1 into the CRRA savings function.

  2. Step 2
    st=β1+βwts_t = \frac{\beta}{1 + \beta} w_t

    Simplify — the interest rate vanishes.

  st=β1+β wt  \boxed{\; s_t = \frac{\beta}{1+\beta}\, w_t \;}
eq:log-savings
With log utility, agents save a *constant fraction* β/(1+β)\beta/(1+\beta) of their wage. This is the formula behind every closed-form Diamond OLG result you will see in macroeconomics texts.

Step 5 — Summary of the household solution

ObjectFormula
sts_tβ1+β wt\dfrac{\beta}{1+\beta}\, w_t
c1,tc_{1,t}11+β wt\dfrac{1}{1+\beta}\, w_t
c2,t+1c_{2,t+1}β(1+rt+1)1+β wt\dfrac{\beta(1+r_{t+1})}{1+\beta}\, w_t
UtU_tln⁡c1,t+βln⁡c2,t+1\ln c_{1,t} + \beta \ln c_{2,t+1}
Closed-form consumption and savings for log utility (θ=1\theta = 1).