Life Cycle Consumption Problem with Assets

Case Study Contents

Problem Statement

In the Life Cycle Consumption Problem, the lifetime budget constraint restricted consumption in equating the present value of consumption to the present value of wage income. The Life Cycle Consumption Problem with Assets extends the Life Cycle Consumption Problem in that the consumer is allowed to have assets, i.e., the consumer can borrow money and repay it in future periods or save money and spend it in other periods. The objective of the life cycle consumption problem with assets is to determine how much one can consume in each period so as to maximize utility. The model includes a constraint on the minimum asset level; if the minimum asset level is zero, the consumer is not allowed to borrow.

Mathematical Model

To formulate the life cycle problem with assets, we start with the same notation as the formulation for the life cycle consumption problem and add notation for assets.

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

\(w(p,n)\) = wage income function
\(r\) = interest rate
\(\beta\) = discount factor
\(a_{min}\) = minimum asset level

Decision Variables
\(c_p\) = consumption in period \(p\), \(\forall p \in P\)
\(a_p\) = assets available at the beginning of 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 \(\sum_{p \in P} \beta^{p-1} u(c_p)\)

The life cycle consumption model with assets tracks the assets available at the beginning of each period. The constraint on consumption now is defined in terms of assets (wealth) and consumption. The wealth at the beginning of period \(p+1\) equals the wealth at the beginning of period \(p\) plus the net savings in period \(p\) (wage income minus consumption), multiplied by the return \(R\) on savings (where \(R = 1 + r\)). There is one constraint for each period:

\[a_{p+1} \leq R(a_p + w(p,n) - c_p), \forall p \in P.\]

The model assumption is that initial wealth is zero \((a_1 = 0)\) and that terminal wealth is non-negative \((a_{n+1} \geq 0)\).

There is a minimum asset level, \(a_{min} \leq 0\). If \(a_{min} = 0\), then no borrowing is allowed. Otherwise, an individual can borrow as long as s/he can repay the amount before period \(n\). Therefore, there is one lower bound constraint for each period:

\(a_p \geq a_{min}\)

Also, the amount consumed in each period should be non-negative:

\(c_p \geq \epsilon, \forall p \in P.\)

Demo and Examples

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

GAMS Model

$Title Life Cycle Consumption - with explicit modeling of savings

Set p period /1*15/ ;

Scalar B discount factor /0.96/;

Scalar i interest rate /0.10/ ;

Scalar R gross interest rate ;

R = 1+i ;

Scalar amin minimum asset level /0/ ;

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

Parameter w(p) wage income in period p ;
w(p) = ((15 - p.val)*p.val) / 15 ;

Parameter lbnds(p) lower bounds of consumption
/ 1*15 0.0001 / ;

Positive Variable c(p) consumption expenditure in period p ;

Variable a(p) assets (or savings) at the beginning of period p ;

Variable Z objective ;

budget(p) lifetime budget constraint ,
obj objective function ;

budget(p) ..
a(p+1) - R*(a(p) + w(p) - c(p)) =l= 0 ;

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

Model LifeCycleConsumptionSavings /budget, obj/ ;

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

a.lo(p) = amin ;

a.fx('1') = 0 ;

Solve LifeCycleConsumptionSavings using nlp maximizing Z ;

Optimization Category (Linear Programing, Integer, MIP and etc.):