Dissertation

N−1

f(x) dx ≈ "0.5 × {f(a) + f(b)} +

Za_b
^Xi=1 _{f(a + iΔ)}# _{× Δ (2.32)}

where Δ = (b − a)/N, which implies that the domain [a, b] has been divided into N equally

spaced intervals.

As defined earlier, a stationary competitive equilibrium in the current framework is

characterized by a cross-sectional consumption-saving profile and optimal retirement age,

an aggregate capital stock, labor supply and a labor-to-retiree ratio, a real rate of return

and wage rate, and an accidental bequest that solve the household’s optimization problem,

satisfy the aggregation conditions, equilibrate the factor markets, balance the social secu-

rity budget and also satisfy the bequest-balance condition. Therefore, computation of a

stationary competitive equilibrium can be broken down into the following two steps:

1. Solve for the household optimum for a given set of factor prices, a given labor-to-retiree

ratio, a given accidental bequest and given values for the other model parameters.

2. Using the aggregation conditions, find the factor prices, the labor-to-retiree ratio and

the accidental bequest consistent with the household optimum. The general equilib-

rium is obtained when the factor prices, the labor-to-retiree ratio and the accidental

bequest values computed from the household optimum match the given factor prices,

the labor-to-retiree ratio and the accidental bequest values that were used to compute

the household optimum.

To sequentially accomplish these two steps, I define a contraction mapping algorithm as

follows:

• Step 1: Set the model parameters to some values and guess some values for the factor

prices, the labor-to-retiree ratio and the accidental bequest (label as vector xin).

31

• Step 2: Create a grid of retirement ages, compute the optimal consumption plans

at each point on the grid using the trapezoidal approximations to the integrals, and

then find the age at which the value of discounted life-cycle utility is maximized for

the given xin vector and the parameter values.

• Step 3: Compute the aggregate capital stock, labor supply, the labor-to-retiree ra-

tio and the accidental bequest that are consistent with the optimal retirement ages

obtained in Step 2 using the trapezoidal approximations to the integrals.

• Step 4: Compute the market-clearing factor prices consistent with the aggregate

capital stock and labor supply obtained in Step 3, and store the factor prices, the

labor-to-retiree ratio and the accidental bequest in a vector xout.

• Step 5: Compute the percentage difference between vectors xin and xout, and store

it in a vector diff.

• Step 6: If the 2-norm of the vector diff is greater than some tolerance parameter

Tol, then update xin using the rule xin = xin.*(1+step), where step is given by step = diff/9, and repeat steps 2 through 5.^11,12

• Step 7: If the 2-norm of the vector diff is lesser than some tolerance parameter Tol,

then terminate the algorithm.

2

_i <sup>.

12</sup>This implies that the algorithm updates the guessed vector by a factor that depends on the divergence

between the guess and the feedback.