1 Model economy

Time is discrete, t = 0, 1, . . ..

1.1 Demographics

The economy is populated by overlapping generations (or cohorts) who live up to I periods. The size of a cohort that was born at t = g and that has reached age h is denoted by x_{g,h}. For each individual of a cohort, survival is stochastic and s_{h} denotes the survival probability for an individual of age h to be still alive at age h + 1. Therefore, we have x_{g,h+1} = s_{h}x_{g,h}. We write the cumulative probability of

an individual of age h to be still alive at age h′ as S_{h}^{h′} = ∏_{j=h}^{h′−1} s_{j}. At each t, a new cohort x_{t,0} of size x_{0} is born.

1.2 Preferences

There is one good at every date t, used for consumption and investment. At time t, an individual from generation g with age h̄ has time-separable preferences over consumption according to

U_{g,h̄} = ∑_{j=h̄}^{I} β^{j−h̄}S_{h}^{j}u(c_{g,j})

where the instantaneous utility function is u(c_{g,h}) = \frac{c_{g,h}^{1−σ}−1}{1−σ}. At time t total consumption in the

economy is c_{t} = ∑_{h} c_{t−h,h}x_{t−h,h}.

Labor supply is fixed, but age-dependend. An individual of age h supplies e_{h} efficiency units of labor. Total labor supply in the economy at time t is n_{t}^{s} = ∑_{h} e_{h}x_{t−h,h}.

Individuals can hold assets to save or borrow. We denote by a_{g,h} the asset holdings of an individual from generation g of age h. Each individual starts life with zero assets, i.e. (a_{g,0} = 0), and cannot have negative assets at the certain death age I + 1, i.e. (a_{g,I+1} ≥ 0). The gross return from holding one unit of the asset from period t to t + 1 is r_{t}. An individual receives wage w_{t} per efficiency unit of labor supplied. Remember that in our formulation we have t = g +h. Therefore, the budget constraint writes as

a_{g,h+1} = (1 + r_{t−1})a_{g,h} + e_{h}w_{t} + b_{g,h} − c_{g,h},

where b_{g,h} denotes accidental bequests (to be specified). Aggregate asset holdings in the economy are denoted by a_{t} = ∑_{h} a_{t−h,h}x_{t−h,h}. With q_{t} denoting the time zero price of the time t good, we can establish a present value budget constraint:

∑_{j=h}^{I} q_{j} [(1 + r_{g+j−1})a_{g,j} + e_{j}w_{g+j} + b_{g,j} − a_{g,j+1} − c_{g,j}] ≥ 0  (PV-BC)

1.3 Technology

Output is produced by final good producers with inputs capital and labor according to y_{t} = f(k_{t}, z_{t}n_{t}). They rent capital from households at interest rate r_{t} and also bear the depreciation cost δk_{t}.

1.4 Equilibrium

Definition 1 (Competitive equilibrium). Define ℳ = {(g, h)|g ∈ {−I, 1, …, ∞}, h ∈ {max[0, −g], 1, …, I}}. Given k_{0} and {x_{g,h}}_{(g,h)∈ℳ} , a competitive equilibrium consists of an allocation {c_{g,h}, a_{g,h}}_{(g,h)∈ℳ} , {k_{t}}_{t=1}^{∞}, {n_{t}}_{t=0}^{∞}, and prices {r_{t}, w_{t}}_{t=0}^{∞} such that

1. For each individual in cohort g and is now of age h̄, {c_{g,h}, a_{g,h}}_{h̄=max[0,−g]}^{I} solves

max_{{c̃_{g,h},ã_{g,h}}_{h̄=max[0,−g]}^{I}} ∑_{j=h̄}^{I} β^{j−h̄}Sj ; h̄u(c̃_{g,j})

s.t. ∀h ∈ {max[0, −g], 1, …, I}

∑_{j=h̄}^{I} q_{j} [(1 + r_{g+j−1})ã_{g,j} + e_{j}w_{g+j} + b_{g,j} − ã_{g,j+1} − c̃_{g,j}] ≥ 0  (λ_{g})
ã_{g,0} = 0, ã_{g,I+1} ≥ 0

where λ_{g} is the Langrange multiplier on the budget constraint for cohort g. Notice that we use g + h = t in the individual’s problem. Aggregation over individuals and cohorts gives

n_{t}^{s} = ∑_{h} e_{h}x_{t−h,h}
a_{t} = ∑_{h} a_{t−h,h}x_{t−h,h}
c_{t} = ∑_{h} c_{t−h,h}x_{t−h,h}

2. For the final goods producers {n_{t}}_{t=0}^{∞} and {k_{t}}_{t=0}^{∞} solve at any t

max_{ñ_{t},k̃_{t}} f(k̃_{t}, z_{t}ñ_{t}) − (r_{t} + δ)k̃_{t} − w_{t}ñ_{t}  (1)

3. All markets clear, i.e., ∀t

(i) Goods: y_{t} = c_{t} + i_{t} (ii) Assets: a_{t} = k_{t} (iii) Labor: n_{t}^{s} = n_{t}

1.5 Solving the individual’s problem

The Lagrangian for an individual from cohort g who is now of age h̄ writes as

ℒ = ∑_{j=h̄}^{I} β^{j−h̄}Sj ; h̄u(c_{g,j}) + λ_{g} ∑_{j=h̄}^{I} q_{g+j} [(1 + r_{g+j−1})a_{g,j} + e_{j}w_{g+j} + b_{g,j} − a_{g,j+1} − c_{g,j}]  (2)

The first order condition for cohort g’s consumption at age h, c_{g,h} is

\frac{β^{h−h̄}Sh ; h̄c_{g,h}^{−σ}}{λ_{g}} = q_{g+h}  (3)

The first order condition with respect to asset holdings a_{g,h+1} is

−q_{g+h} + (1 + r_{g+h})q_{g+h+1} = 0

Combining the two FOCs gives the consumption Euler equation

βs_{h}(1 + r_{g+h}) (\frac{c_{g,h+1}}{c_{g,h}} )^{−σ} = 1  (EE)

More general, we have a relation between consumption at age h relative to consumption at base age h̄:

c_{g,h} = c_{g,h̄} (β^{h−h̄}Sh ; h̄ ∏_{j=h̄}^{h−1}(1 + r_{g+j}))^{\frac{1}{σ}}  (4)

Now we plug (3) and (4) into the present value budget constraint (PV-BC) which will be binding. Asset holdings cancel out except for a_{g,h̄} and a_{g,I}, where the latter will be zero. We obtain

\frac{c_{g,h̄}^{−σ}}{λ_{g}} \{a_{g,h̄} + ∑_{l=h̄}^{I} \frac{e_{l}w_{g+l} + b_{g,l} − c_{g,h̄} (β^{l−h̄}Sl ; h̄ ∏_{j=h̄}^{l−1}(1 + r_{g+j}))^{\frac{1}{σ}}}{∏_{j=h̄}^{l−1}(1 + r_{g+j})} \} = 0  (5)

Because of our choice of the utility function c_{g,h} > 0 ∀h, and with (PV-BC) binding, λ_{g} > 0 usually holds. Then by solving for c_{g,h̄} and by virtue of (4) we have a closed form expression for the entire consumption path (for a given path of prices):

c_{g,h} = \{∑_{l=h̄}^{I} \frac{(β^{l−h} \frac{Sl ; h̄}{Sh ; h̄} \frac{∏_{j=h̄}^{l−1}(1+r_{g+j})}{∏_{j=h̄}^{h−1}(1+r_{g+j})} )^{\frac{1}{σ}}}{∏_{j=h̄}^{l−1}(1 + r_{g+j})} \}^{−1} \{a_{g,h̄} + ∑_{l=h̄}^{I} \frac{e_{l}w_{g+l} + b_{g,l}}{∏_{j=h̄}^{l−1}(1 + r_{g+j})}\} , ∀h ∈ \{h̄,, …, I\} (6)

The second curly bracket is the present value of the individual’s lifetime income. The first curly bracket is a function of prices and survival probabilities (and the discount factor) and shows which fraction of the lifetime income will be consumed in a given period.

1.6 Solving the firms problems

For the final goods producers the problem is static. The first order conditions are f_{1}(.) ≡ \frac{∂f(.)}{∂k_{t}} = r_{t} +δ and f_{2}(.) ≡ \frac{∂f(.)}{∂n_{t}} = w_{t}.

1.7 Steady state

Definition 2 (Steady state). A steady state is a competitive equilibrium in which

1. the demographic structure is constant/ time-invariant, i.e. x_{g,h} = x̄_{h}, ∀g,

2. labor productivity is constant, i.e., z_{t} = z̄, ∀t,

3. the capital-output ratio is constant, i.e., \frac{k_{t}}{y_{t}} = \frac{k̄}{ȳ}, ∀t. Or should rather capital be constant? I guess the two are synonymous since with an invariant age structure labor input will be constant, and with constant labor productivity output will be constant.

The first order conditions for the final goods producers imply - taking into account the CRS property of the production function - that the interest rate and the wage are constant in steady state. Consequently, if we let each cohort start with zero assets, i.e., a_{g,0} = 0 ∀g, the paths for consumption and asset holdings are identical for each cohort. Since the population shares x̄_{h} are time-invariant, it follows that aggregate asset holdings (and aggregate consumption) are constant for all t. The market clearing condition for assets tells us subsequently that capital is constant in steady state.