**Case Study Contents**

## Problem Statement

The **life cycle consumption problem** is a generalization of the three-period life cycle problem in that the number of periods can vary from 1 to \(n\). Since the number of periods is variable in the general life cycle problem, the model takes as input a *wage function* instead of a set of discrete values. The wage function returns the wage for the current period \(p\) as a function of the total number of periods \(n\). The objective of the **life cycle consumption problem** is to determine how much one can consume in each period so as to maximize utility subject to the lifetime budget constraint.

## Mathematical Formulation

The formulation for the general life cycle problem generalizes the formulation of the three-period life cycle problem from three periods to \(n\) periods.

**Set**

P = set of periods = \({1..n}\)

**Parameters**

\(w_p\) = wage income in period \(p\), \(\forall p \in P\)

\(r\) = interest rate

\(\beta\) = discount factor

**Decision Variables**

\(c_p\) = consumption in period \(p\), \(\forall p \in P\)

**Objective Function**

Let \(c_p\) be consumption in period \(p\), where "life" begins at \(p=1\) and continues to \(p=n\). Let \(u()\) be the utility function and let \(u(c_p)\) be the utility value associated with consuming \(c_p\). Utility in future periods is discounted by a factor of \(\beta\). Then, the objective function is to maximize the total discounted utility:

maximize \(\sum_{p \in P} \beta^{p-1 }u(c_p)\)

**Constraints**

The main constraint in the life cycle model is the *lifetime budget constraint*, which asserts that, over the life cycle, the present value of consumption equals the present value of wage income. From above, \(r\) is the interest rate; therefore, \(R = 1 + r\) is the gross interest rate. If I invest 1 dollar in this period, then I receive \(R\) dollars in the next period. The expression for the present value of the consumption stream over the life cycle is

\[\sum_{p \in P} \frac{c_p}{R^{p-1}}.\]

Similarly, the expression for the present value of the wage income stream over the lifecycle is

\[\sum_{p \in P} \frac{w_p}{R^{p-1}}.\]

The lifetime budget constraint states that the present value of the consumption stream must equal (or be less than) the present value of the wage income stream:

\[\sum_{p \in P} \frac{c_p}{R^{p-1}} \leq \sum_{p \in P} \frac{w_p}{R^{p-1}}.\]

To avoid numerical difficulties, we add constraints requiring the consumption variables to take a non-negative value:

\(c_p \geq 0.0001, \forall p \in P\)

To solve the three-period life cycle consumption problem, we need to specify a utility function and the values of the parameters. As in the case of the three-period life cycle problem, the solution of the general life cycle consumption problem specifies the amount that Joey should consume in each period to maximize his utility.

## Demo

To solve your own life cycle consumption problems, check out the Life Cycle Consumption demo.

## GAMS Model

$Title Life Cycle Consumption

Set p period /1*10/ ;

Scalar B discount factor /0.96/;

Scalar i interest rate /0.10/ ;

Scalar R gross interest rate ;

R = 1+i ;

$macro u(c) (-exp(-c))

Parameter w(p) wage income in period p ;

w(p) = ((10 - p.val)*p.val) / 10 ;

Parameter lbnds(p) lower bounds of consumption

/ 1*10 0.0001 / ;

Positive Variables

c(p) consumption expenditure in period p ,

PVc present value of consumption expenditures ,

PVw present value of wage income ;

Variable Z objective ;

Equations

defPVc definition of PVc ,

defPVw definition of PVw ,

budget lifetime budget constraint ,

obj objective function ;

defPVc ..

PVc =e= sum(p, c(p) / power(R, p.val - 1)) ;

defPVw ..

PVw =e= sum(p, w(p) / power(R, p.val - 1)) ;

budget ..

PVc =l= PVw ;

obj ..

Z =e= sum(p, power(B, p.val - 1)*u(c(p))) ;

Model LifeCycleConsumption /defPVc, defPVw, budget, obj/ ;

c.lo(p) = lbnds(p) ;

Solve LifeCycleConsumption using nlp maximizing Z ;