Nov 17

Consumption-Savings Problem

Consumption-Savings Problem

Here we look at the solution of

Example 7.5 (mt3042 Optimization Guide by M. Baltovic)

Suppose a consumer, let's call her Ms Thrifty, has an initial wealth of w. In any period t=1,...,T, denote the wealth she has at the start of the period by w_t, so that w_1=w. She can choose to spend c_t\in[0,w_t], giving her a utility of u(c_t)=c_t. However, the wealth w_t-c_t she does not consume in that period will be stored in a bank account where it earns an interest rate of r>0 by the beginning of the next period. Thus, she will enter the next time period t+1 with wealth w_{t+1}=(1+r)(w_t-c_t).

Actions of Ms Thrifty can be identified with her spendings \{c_1,...,c_T\} and the same spendings are her rewards. Thus, a strategy is a sequence of spendings \{c_1,...,c_T\} and the value of that strategy is the sum of rewards V(\{c_1,...,c_T\})=c_1+...+c_T. To obtain the solution, we use backwards induction.

Period T. The reward c_T\in[0,w_T] is obviously maximized at c_T=w_T so that

(1) c_T=w_T=(1+r)(w_{T-1}-c_{T-1}).

Period T-1. According to the Bellman equations, at each state s we need to select an action a that maximizes the sum

(2) r(s,a)+V(remainder).

I deliberately don't use the full notation because I prefer the following verbal description: r(s,a) is the immediate reward for taking action a at state s (for taking one step leading us to the next state). Whatever is the next state, remainder means an optimal strategy leading us from that state to the final destination. For period T-1 we use (1) to write (2) as

(3) c_{T-1}+c_T=c_{T-1}+(1+r)(w_{T-1}-c_{T-1})=-rc_{T-1}+(1+r)w_{T-1}.

Since c_{T-1}\in[0,w_{T-1}], (3) is maximized at c_{T-1}=0, giving

(4) V(\{0,c_{T}\})=(1+r)w_{T-1}.

Period T-2. Here (2) is

c_{T-2}+V(\{0,c_{T}\}) (using (4))

=c_{T-2}+(1+r)w_{T-1} (using w_{T-1}=(1+r)(w_{T-2}-c_{T-2}))

=c_{T-2}+(1+r)^{2}(w_{T-2}-c_{T-2})=[ 1-(1+r)^2]c_{T-2}+(1+r)^2w_{T-2}.

This is maximized at c_{T-2}=0 and the value function for three last actions is V(\{0,0,c_T\})=(1+r)^{2}w_{T-2}. This easily generalizes leading to the solution V(\{0,...,0,c_T\})=(1+r)^{T-1}w_1=(1+r)^{T-1}w.

Ms Thrifty fasts for T-1 periods, then gorges in the last period and dies a happy death. Math is a cruel science.

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