Convex Optimization · Problem 4.58

Duality — Boyd & Vandenberghe

Mathematics — Optimization · optimal-consumption

Problem

Optimal consumption. In this problem we consider the optimal way to consume (or spend) an initial amount of money (or other asset)

k_0

over time. The variables are

c_0, \ldots, c_T

, where

c_t \geq 0

denotes the consumption in period

t

. The utility derived from a consumption level

c

is given by

u(c)

, where

u : \mathbf{R} \to \mathbf{R}

is an increasing concave function. The present value of the utility derived from the consumption is given by

U = \sum_{t=0}^T \beta^t u(c_t),

where

0 < \beta < 1

is a discount factor. Let

k_t

denote the amount of money available for investment in period

t

. We assume that it earns an investment return given by

f(k_t)

, where

f : \mathbf{R} \to \mathbf{R}

is an increasing, concave investment return function, which satisfies

f(0) = 0

. For example if the funds earn simple interest at rate

R

percent per period, we have

f(a) = (R/100)a

. The amount to be consumed, i.e.,

c_t

, is withdrawn at the end of the period, so we have the recursion

k_{t+1} = k_t + f(k_t) - c_t, \quad t = 0, \ldots, T.

The initial sum

k_0 > 0

is given. We require

k_t \geq 0

t = 1, \ldots, T+1

(but more sophisticated models, which allow

k_t < 0

, can be considered). Show how to formulate the problem of maximizing

U

as a convex optimization problem. Explain how the problem you formulate is equivalent to this one, and exactly how the two are related. Hint. Show that we can replace the recursion for

k_t

given above with the inequalities

k_{t+1} \leq k_t + f(k_t) - c_t, \quad t = 0, \ldots, T.

(Interpretation: the inequalities give you the option of throwing money away in each period.) For a more general version of this trick, see exercise 4.6.

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