Intertemporal Choice

Life Cycle Models, Labor Supply, and the Overlapping Generations Model

Alan Lujan

Johns Hopkins University

February 3, 2025

The Two-Period Life Cycle Model

The Fisher Problem

Irving Fisher’s two-period consumption model: a consumer lives for two periods with no uncertainty.

Lifetime value:

\[V(c_1, c_2) = u(c_1) + \beta \, u(c_2)\]

where \(\beta \in (0,1)\) is the time preference factor.

Assumptions:

Marginal utility is positive: \(u'(\cdot) > 0\)
Diminishing marginal utility: \(u''(\cdot) < 0\)

Budget Constraints

The consumer begins with bank balances \(b_1\) and earns income \(y_1\) in period 1, \(y_2\) in period 2.

Dynamic budget constraint (DBC):

\[b_2 = (b_1 + y_1 - c_1) R\]

Intertemporal budget constraint (IBC):

\[c_1 + \frac{c_2}{R} = b_1 + \underbrace{y_1 + \frac{y_2}{R}}_{\equiv \, h_1 \text{ (human wealth)}}\]

The PDV of lifetime spending equals the PDV of lifetime resources.

The Lagrangian

\[\max_{\{c_1, c_2\}} \; u(c_1) + \beta \, u(c_2) \quad \text{s.t.} \quad c_2 = (b_1 + y_1 - c_1)R + y_2\]

The Lagrangian is:

\[\mathcal{L} = u(c_1) + \beta \, u(c_2) + \lambda\left[c_2 - (b_1 + y_1 - c_1)R - y_2\right]\]

First order conditions:

\[\frac{\partial \mathcal{L}}{\partial c_1} = u'(c_1) + R\lambda = 0\]

\[\frac{\partial \mathcal{L}}{\partial c_2} = \beta \, u'(c_2) + \lambda = 0\]

The Euler Equation

From the FOCs: \(\lambda = -\beta \, u'(c_2)\). Substituting into the first condition:

\[u'(c_1) - R \, \beta \, u'(c_2) = 0\]

\[\boxed{u'(c_1) = R \, \beta \, u'(c_2)}\]

The intertemporal price \(R\) determines the tradeoff: giving up one unit of \(c_1\) yields \(R\) units of \(c_2\).

Code: Symbolic Euler Equation

Intuition: Perturbation Argument

At the optimum \((c_1^*, c_2^*)\), reduce \(c_1\) by \(\epsilon\) and invest it:

\[\Delta \text{Utility} \approx \underbrace{u'(c_1^*)\epsilon}_{\text{loss today}} - \underbrace{\beta \, u'(c_2^*) R\epsilon}_{\text{gain tomorrow}}\]

If \(\Delta \neq 0\), we have a contradiction: the consumer could do better.

\(\Delta > 0\): consume more today
\(\Delta < 0\): save more today
\(\Delta = 0\): optimality \(\Rightarrow\) the Euler equation

CRRA Utility and Consumption Growth

With constant relative risk aversion (CRRA) utility,

\[u(c) = \frac{c^{1-\rho}}{1-\rho}, \qquad u'(c) = c^{-\rho}\]

the Euler equation becomes:

\[c_1^{-\rho} = R\beta \, c_2^{-\rho} \quad \Longrightarrow \quad \frac{c_2}{c_1} = (R\beta)^{1/\rho}\]

The intertemporal elasticity of substitution is \(1/\rho\):

\[\frac{d \log(c_2/c_1)}{d \log R} = \frac{1}{\rho}\]

The Consumption Function

Using the Euler equation and the IBC:

\[c_1 = \frac{b_1 + h_1}{1 + R^{-1}(R\beta)^{1/\rho}}\]

Log utility (\(\rho = 1\)) simplifies to:

\[c_1 = \frac{b_1 + h_1}{1 + \beta}\]

The marginal propensity to consume out of wealth is \(\frac{1}{1+\beta}\).

Code: Consumption Response to R

Code: Fisher Diagram

Code: Effect of Income Timing

Fisherian Separation

The consumption growth profile

\[\frac{c_2}{c_1} = (R\beta)^{1/\rho}\]

depends only on \(R\) and \(\beta\), not on the income profile \((y_1, y_2)\).

Two consumers with the same lifetime wealth but different income timing have identical consumption paths.

This is a pervasive feature of models combining:

Perfect foresight
No liquidity constraints

Income, Substitution, and Human Wealth Effects

When \(R\) increases, three forces operate on \(c_1\):

Substitution effect: Future consumption is cheaper \(\Rightarrow\) save more (\(c_1 \downarrow\))
Income effect: Higher return on savings \(\Rightarrow\) richer (\(c_1 \uparrow\))
Human wealth effect: PDV of future income \(h_1\) falls (\(c_1 \downarrow\))

Summers (1981) argues the human wealth effect dominates quantitatively: for most consumers, the bulk of lifetime income is future labor income, so higher \(R\) substantially reduces its present value.

Code: Decomposing the Effect of R

Consumption and Labor Supply

Setup: Consumption and Leisure

Utility depends on both consumption \(c_t\) and leisure \(z_t\): \(u(c_t, z_t)\)

Constraints:

Time allocation: \(\ell_t + z_t = 1\)
Labor income: \(y_t = W_t(1 - z_t)\)
Expenditure: \(x_t = c_t + z_t W_t\)

FOC: \(W_t = u^z / u^c\) (wage = MRS between leisure and consumption)

Stylized Facts on Labor Supply

Wages in the US have risen by a factor of 2 to 4 over the life cycle (youth to middle age), yet labor supply barely changes.

Over long periods, hours worked have remained roughly stable despite large wage increases (Ramey and Francis, 2009)
Across countries with vastly different per capita income, variation in leisure is small relative to variation in wages
Cross-sectionally, vast wage differences produce only small leisure differences

These facts motivate the Cobb-Douglas specification: the expenditure share on leisure should be constant as wages rise.

Cobb-Douglas Preferences

Assume Cobb-Douglas aggregation inside an outer function \(f\):

\[u(c_t, z_t) = f\!\left(c_t^{1-\alpha} z_t^{\alpha}\right)\]

This implies \(z_t W_t = c_t \eta\) where \(\eta = \alpha / (1-\alpha)\).

Key property: the share of expenditure on leisure is constant as wages rise.

Utility simplifies to \(f\!\left((W_t/\eta)^{-\alpha} c_t\right)\).

Code: Cobb-Douglas Leisure Derivation

Two-Period Model with Labor

With CRRA outer utility \(f(\chi) = \chi^{1-\rho}/(1-\rho)\):

\[\frac{c_2}{c_1} = (R\beta)^{1/\rho} \left(\frac{W_2}{W_1}\right)^{-\alpha(1-\rho)/\rho}\]

Log utility (\(\rho = 1\)): consumption growth \(c_2/c_1 = R\beta\) (no wage effect)

Labor supply: \(\displaystyle\frac{1-\ell_2}{1-\ell_1} = \frac{R\beta \, W_1}{W_2}\) (work harder when wages are higher)

Log Utility Consumption Level

With log utility, the IBC with labor becomes:

\[c_1(1+\beta)(1+\eta) = W_1 + R^{-1}W_2 \equiv h_1\]

Solving for consumption:

\[c_1 = \frac{h_1}{(1+\beta)(1+\eta)}\]

The \((1+\eta)\) factor captures the expenditure share allocated to leisure. Compared to the pure consumption model, the MPC out of human wealth is lower because part of each dollar funds leisure.

Code: Consumption-Labor Tradeoff

The Labor Supply Puzzle

With \(R\beta = 1\) and \(\ell_1 = 1/2\):

\[\ell_2 = \frac{2\omega + 1}{2(1+\omega)}\]

where \(\omega = W_2/W_1 - 1\) is wage growth.

If \(\omega = 2\): \(\ell_2 = 5/6 \approx 0.83\)

The model predicts middle-aged people work 67% more than young people. This is inconsistent with the data: labor supply is about the same for 55-year-olds as for 25-year-olds.

Code: Predicted vs Actual Labor Supply

Occupational Variation

One response: assume \(R\beta / \Omega = 1\) fixes aggregate labor supply, where \(\Omega = W_2/W_1\). Then \((1-\ell_2)\Gamma_i = (1-\ell_1)\) for occupation-specific wage growth \(\Omega_i = \Omega \Gamma_i\).

Plausible values of \(\Gamma_i\) range from \(0.5\) (manual laborers) to \(1.5\) (doctors):

\(\Gamma = 0.5 \Rightarrow \ell_2 = 0\) (zero hours!)
\(\Gamma = 1.5 \Rightarrow \ell_2 = 2/3\) (much harder in middle age)

Empirically, cross-occupation variation in middle-age labor supply is small. The theory drastically overpredicts: a “small intertemporal elasticity of labor supply.”

Code: Occupational Labor Supply

The Overlapping Generations Model

OLG Model: Demographics

The Diamond (1965) OLG model following Samuelson (1958):

Two generations alive at any point: young (age 1) and old (age 2)
Young population: \(\mathcal{N}_t = \mathcal{N}_0 N^t\) where \(N = 1+n\)
Households work only when young, earning \(Y_{1,t}\); no income when old (\(Y_{2,t+1} = 0\))
They consume part of first-period income and save the rest

OLG Model: Markets and Production

Assets of the young fund next period’s capital: \(K_{t+1} = \mathcal{N}_t a_{1,t}\)
The old own the capital stock and consume everything (no bequest motive)
Aggregate production: CRS technology \(Y = F(K,L)\) with perfect competition
No depreciation

Production and Factor Prices

Cobb-Douglas production function:

\[F(K, L) = K^\varepsilon L^{1-\varepsilon} \quad \Rightarrow \quad f(k) = k^\varepsilon\]

Factor prices equal marginal products (where \(R_{t+1} = 1 + r_t\)):

\[W_t = (1-\varepsilon) k_t^\varepsilon\]

\[r_t = \varepsilon k_t^{\varepsilon - 1}\]

where \(k_t = K_t / \mathcal{N}_t\) is capital per young worker.

Individual Optimization

The young consumer maximizes \(u(c_{1,t}) + \beta \, u(c_{2,t+1})\).

Euler equation: \(u'(c_{1,t}) = \beta R_{t+1} u'(c_{2,t+1})\)

With log utility: \(c_{1,t} = W_{1,t}/(1+\beta)\) and \(a_{1,t} = W_{1,t} \cdot \beta/(1+\beta)\)

The saving rate \(\beta/(1+\beta)\) is constant (a strong prediction of log utility).

Capital Accumulation Dynamics

Since \(k_{t+1} = a_{1,t}/N\) (assets per young worker, adjusted for population growth):

\[k_{t+1} = \underbrace{\frac{(1-\varepsilon)\beta}{N(1+\beta)}}_{\equiv \, \mathcal{Q}} \cdot k_t^\varepsilon\]

This is a nonlinear first-order difference equation in \(k\).

Since \(\varepsilon < 1\), the mapping is concave and converges monotonically to a unique steady state.

Code: OLG Phase Diagram

Steady State

Setting \(k_{t+1} = k_t = \bar{k}\):

\[\bar{k} = \mathcal{Q} \, \bar{k}^{\,\varepsilon} \quad \Longrightarrow \quad \bar{k} = \mathcal{Q}^{1/(1-\varepsilon)}\]

Steady-state factor prices:

\[\bar{W} = (1-\varepsilon)\bar{k}^{\,\varepsilon}, \qquad \bar{r} = \varepsilon \bar{k}^{\,\varepsilon - 1}\]

Code: Comparative Statics

Code: OLG Steady State (SymPy)

Dynamic Efficiency and the Golden Rule

Per-capita steady-state consumption: \(\bar{c} = f(\bar{k}) - n\bar{k}\)

Golden Rule maximizes \(\bar{c}\): \(f'(\bar{k}^{**}) = n \;\Rightarrow\; \bar{k}^{**} = (n/\varepsilon)^{1/(\varepsilon - 1)}\)

Three capital levels to compare:

\(\bar{k}\): competitive equilibrium
\(\bar{k}^*\): social optimum (depends on \(\beth\))
\(\bar{k}^{**}\): golden rule (maximizes \(\bar{c}\))

Code: Golden Rule and Dynamic Efficiency

Code: Three Capital Levels

Key Takeaways (1/2)

Euler equation: \(u'(c_1) = R\beta \, u'(c_2)\) governs all intertemporal consumption decisions
Fisherian Separation: consumption growth depends on \(R\) and \(\beta\), not income timing
Labor supply puzzle: the model drastically overpredicts the response of labor supply to predictable wage variation

Key Takeaways (2/2)

OLG model: competitive equilibria can be dynamically inefficient; the Golden Rule \(f'(\bar{k}) = n\) maximizes steady-state consumption
Social optimum: there is no particular relationship between the competitive equilibrium and the social planner’s solution

Summary: How the Topics Connect

Each topic in this module builds on the Euler equation:

Fisher (2-period): introduces the fundamental tradeoff \(u'(c_1) = R\beta \, u'(c_2)\)
Labor supply: adds the leisure choice, revealing the model’s limits when confronted with cross-occupation data
OLG: embeds the individual Euler equation in general equilibrium, where capital accumulation can overshoot the efficient level