Question

Within the framework of the single period investment-consumption model, explain the optimal consumption/investment decision both with and without capital markets.

Answer 1

Consumer Utility

Normative financial economics concerns optimal decisions made by individuals, firms and/or institutions. In an important sense, much of the subject matter of investments deals with optimal choices of investment and consumption.

Thus far we assumed that the investor/consumer makes optimal choices from among alternative combinations of present and contingent future consumption opportunities. Initially, we suggested that the individual picks the combination that he or she likes best. This hardly offers much help. Imagine an Analyst saying to a client: "do what's best".

Our second characterization of investor behavior utilized the concept of indifference curves or, more generally, indifference surfaces. The conclusion was somewhat more elegant, although hardly more useful: pick the combination from the opportunity line (or plane or hyperplane) on the highest (best) indifference surface.

At this point, we have provided little help for the Analyst seeking to offer direction to individuals or institutions seeking advice on either the optimal amount to be invested or the particular investments that should be undertaken.

As we will see, an Analyst can provide useful advice concerning such decisions. Individuals may differ in preferences, circumstances, constraints and predictions. A rather rich body of analytic methods can be invoked to help take such differences into account. Such techniques provide a core set of normative methods for investment management.

Here we will deal with three aspects that may lead informed individuals to adopt different strategies: differences in preferences, differences in wealth and differences in predictions. We leave for later the analysis of differences in constraints, other circumstances, and so on.

A formal construct that helps to highlight the differences among utility-based, wealth-based and prediction-based investment decisions uses the concept of consumer utility and the assumption that the goal of the consumer is to maximize the expected value of such utility. In this scheme, (1) consumer utility summarizes an individual's preferences, (2) possible combinations of consumption are related to wealth, and (3) the probabilities utilized to compute expected utility can be considered predictions. In principle, one can thus determine the extent to which investment decisions differ due to differences in predictions as opposed to differences in preferences or differences in wealth. In practice, such a neat taxonomy is difficult to attain. Nonetheless, every investment decision should be scrutinized in an attempt to determine (as best possible) the role that each such type of difference plays.

1 Consumption with a Perfect Capital Market We consider a simple 2-period world in which a single consumer must decide between consumption c0 today (in period 0) and consumption c1 tomorrow (in period 1). Our consumer is endowed with money m0 today and m1 tomorrow. Consistent with his endowment, the consumer has the opportunity to borrow or lend b0 today at interest rate r. The equations governing the consumer’s feasible actions today and tomorrow are: c0 = m0 + b0 (1) c1 = m1 − (1 + r)b0. (2) It is understood that consumption in both periods must be nonnegative, which puts limits on how much the consumer may borrow or lend today, namely, −m0 ≤ b0 ≤ m1/1 + r. After substituting c0 − m0 for b0 in (2), the consumer’s budget constraint is c0 + c1 1 + r = m0 + m1 1 + r := W0. (3) We have written the budget constraint as an equality since we shall assume our consumer always prefers more consumption. We interpret the symbol W0 as the consumer’s present value of wealth. Note that the consumer’s future value of wealth W1 would be (1 + r)W0. To determine the optimal consumption and borrowing plan, we posit a consumer utility function U(c0, c1) of the form √ c0 + β √ c1. As we shall see, the parameter β serves as a discount factor on future consumption. Smaller values of β imply a larger discount factor on future consumption, which implies that our consumer prefers more consumption today. Formally, the consumer’s optimization problem is given by: MAX {U(c0, c1) : c0 + c1 1 + r = W0}, (4) which may be equivalently expressed as MAX {U(c0, c1(c0)) : 0 ≤ c0 ≤ W0}, (5) where c1(c0) := (1 + r)(W0 − c0). (6) The form of utility implies that consumption in both periods must be positive so that the optimal choice for c0 necessarily lies strictly between 0 and W0. Accordingly, first-order optimality conditions imply that 0 = ∂U ∂c0 + ∂U ∂c1 dc1 dc0 = 1 2 √ c0 − β(1 + r) 2 √ c1 . (7) 1 From (7) the optimal consumption plan necessarily satisfies the condition rc1 c0 = β(1 + r), (8) or, equivalently, c1 = β 2 (1 + r) 2 c0. (9) Substituting (9) into the budget constraint (3) and solving for c0 and c1, the optimal consumption plan is given by: c0 = [ 1 1 + β 2(1 + r) ]W0 := ρ0W0 (10) c1 = [ β 2 (1 + r) 2 1 + β 2(1 + r) ]W0 := ρ1W0. (11) It is important to point out here that the constants ρ0 and ρ1 are independent of present wealth W0; that is, they are known parameters strictly determined by the two discount factors β and r. Note further that the optimal utility is of the form U ∗ (W0) := [√ ρ0 + β √ ρ1] p W0. (12) Example 1. Suppose m0 = 100, m1 = 990, r = 0.10 and β = 0.60. Then c0 = 0.716W0, c1 = 0.312W0 and U ∗ (W0) = 1.182√ W0. Here W0 = 100 + 990/1.1 = 1000. Thus c0 = 716, c1 = 312 and U ∗ (1000) = 37.36. Due to the availability of a capital market at which to borrow or loan, the consumer has increased his utility by almost 30% above the level corresponding to the initial endowment U(100, 990). To obtain the optimal consumption plan, the consumer must borrow b0 = 716 − 100 = 616 today, pay back 678 tomorrow, thereby leaving him with 990 − 678 = 312 to consume in the final period. 2 Consumption and Investment with a Perfect Capital Market We now consider a world in which the consumer now has the opportunity to invest I0 today in production from which he will receive f(I0) tomorrow. The function f(I0) encapsulates the investment opportunities in production available to our consumer. For example, the consumer may wish to obtain an education while he is young, expecting a return on this investment in his working years. It is generally assumed that f(·) is strictly increasing (more investment leads to more return) but exhibits diminishing returns in that the marginal return on an incremental rise in investment declines as the total investment increases. When f(·) is differentiable, these assumptions imply the first derivative is positive and the second derivative is negative. (Such a function is called concave.) To ensure at least some investment would be made (to make our subsequent calculations easier), we also assume that the derivative f 0 (0) is infinite. The equations governing the consumer’s feasible actions today and tomorrow are now: c0 + I0 = m0 + b0