econ.studio
Diamond Overlapping Generations Model
Section 6 of 16
Section 6

The Household Problem

Objective

A representative young agent born at tt maximises lifetime utility:

Ut=u(c1,t)+β u(c2,t+1),β=11+ρ,U_t = u(c_{1,t}) + \beta\, u(c_{2,t+1}),\qquad \beta = \frac{1}{1+\rho},
eq:utility

subject to the two budget constraints — one for each period of life.

Budget constraints

c1,t+st=wt,c_{1,t} + s_t = w_t,
eq:young-budget
Young: consume c1,tc_{1,t} and save sts_t out of wage income.
c2,t+1=(1+rt+1) st.c_{2,t+1} = (1 + r_{t+1})\, s_t.
eq:old-budget
Old: consume the proceeds of savings.

Combining the two by eliminating sts_t gives the consolidated lifetime budget constraint:

c1,t+c2,t+11+rt+1=wt.c_{1,t} + \frac{c_{2,t+1}}{1 + r_{t+1}} = w_t.
eq:lifetime-budget
Present value of lifetime consumption equals lifetime wage income. Discount the second-period claim at the gross return.

The optimisation problem

The young agent's problem is therefore:

max⁡c1,t, c2,t+1  u(c1,t)+β u(c2,t+1)s.t.c1,t+c2,t+11+rt+1=wt.\max_{c_{1,t},\, c_{2,t+1}}\; u(c_{1,t}) + \beta\, u(c_{2,t+1}) \quad \text{s.t.}\quad c_{1,t} + \frac{c_{2,t+1}}{1 + r_{t+1}} = w_t.
eq:household-program

First-order conditions

Form the Lagrangian L=u(c1,t)+βu(c2,t+1)+λ[wt−c1,t−c2,t+1/(1+rt+1)]\mathcal{L} = u(c_{1,t}) + \beta u(c_{2,t+1}) + \lambda [w_t - c_{1,t} - c_{2,t+1}/(1+r_{t+1})]:

  1. Step 1
    ∂L∂c1,t=u′(c1,t)−λ=0\frac{\partial \mathcal L}{\partial c_{1,t}} = u'(c_{1,t}) - \lambda = 0

    FOC in c1,tc_{1,t}.

  2. Step 2
    ∂L∂c2,t+1=βu′(c2,t+1)−λ1+rt+1=0\frac{\partial \mathcal L}{\partial c_{2,t+1}} = \beta u'(c_{2,t+1}) - \frac{\lambda}{1+r_{t+1}} = 0

    FOC in c2,t+1c_{2,t+1}.

  3. Step 3
    ⇒λ=u′(c1,t)=β(1+rt+1)u′(c2,t+1)\Rightarrow \lambda = u'(c_{1,t}) = \beta(1+r_{t+1}) u'(c_{2,t+1})

    Eliminate λ\lambda.

Rearranging the last line gives the **Euler equation** for the Diamond OLG model:

  u′(c1,t)=β(1+rt+1)u′(c2,t+1)  \boxed{\; u'(c_{1,t}) = \beta(1+r_{t+1}) u'(c_{2,t+1}) \;}
eq:euler-diamond
Marginal utility today equals discounted marginal utility tomorrow times the gross return — the canonical intertemporal trade-off, restricted to a two-period life.

The savings function

Solving Eq. (eq:euler-diamond) together with the lifetime budget gives consumption and savings as functions of (wt,rt+1)(w_t, r_{t+1}). We carry out this step for CRRA utility in the next section. The resulting object — the **savings function**

st=s(wt,rt+1)s_t = s(w_t, r_{t+1})
eq:savings-function

— is the bridge between household optimisation and aggregate capital accumulation. Note its arguments: today's wage (which the young observe) and tomorrow's interest rate (which they must forecast, perfectly under our perfect-foresight assumption).