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# Exercise 4 - Growth

This week's exercise is about growth, savings, and death. The first part can be solved with pen and paper, and the second part is about verifying the solution in the simulation.

When modeling growth through the birth of new agents, one consequently should also model death. Thus, we will make our agents mortal with a life-span of exactly 500 days. Overall, the population will still grow as long as the birth rate exceeds the death rate of 1/500 = 0.2%.

Furthermore, we add retirements to the model and disallow the agents to work as soon as they turn 400 days old. So if they still want to enjoy some utility during retirement, they must put aside some savings while working, and spend these savings in retirement.

This task can be solved entirely with pen and paper and good reasoning. Assuming that the agents maximize total life-time utility, how much should they save every day while working? And how much should of their savings should they spend in retirement?

You should maximize total life-time utility, i.e.:

$max \sum_{i=1}^{500} u(h_{l,i}) + u(x_{p,i})$

You can assume constant prices and a constant income stream $w_i = w$ during for as long as the agent works (i.e. for the first 400 days). This reduces the problem to maximizing utility from potatoe consumption $x_{p,i}$:

$max \sum_{i=1}^{500} u(x_{p,i})$ subject to the budget constraint $\sum_{i=1}^{500} p x_{p,i} = \sum_{i=1}^{400} w_i = 400 w$

Once you have solved this simple form of the problem, you derive a simple yet optimal savings heuristic for workers that do not receive a dividend. In a second step, try to adjust the heuristic to incorporate dividends, i.e. adding an income stream that continues in retirement. Ideally, this decision rule is robust against fluctuations in income, so you cannot simply assume that total life-time income is 400 times the latest income.