Nivel Individual:

Previous analyses in this book have worked with the representative-agent assumption.

This means that we neglected all potential di¤erences across individuals and assumed that they are all the same (especially identical preferences, labour income and initial wealth). We now extend the analysis of the last section by (i) allowing individuals to give loans to each other. These loans are given at a riskless endogenous interest rate r: As before, there is a “normal”asset which pays an uncertain return rt+1:We also (ii) assume there are two types of individuals, the risk-neutral ones and the risk-averse, denoted by i = a; n. The second assumption is crucial: if individuals were identical, i.e. if they had identical preferences and experienced the same income streams, no loans would be given in equilibrium. In this world with heterogeneous agents, we want to understand who owns which assets. We keep the analysis simple by analyzing a partial equilibrium setup.

Households

The budget constraints (8.1.6) and (8.1.7) of all individuals i now read

w_t= cⁱ_t+ sⁱ_t; (8.2.1)

cⁱ_t+1 = sⁱ_t 1 + rⁱ : (8.2.2)

In the …rst period, there is the classic consumption-savings choice. In addition, there is an investment problem as savings need to be allocated to the two types of assets.

Consumption in the second period is paid for entirely by capital income. Interests paid on the portfolio amount to

rⁱ = ⁱr_t+1+ 1 ⁱ r: (8.2.3)

Here and in subsequent chapters, ⁱ denotes the share of wealth held by individual i in the risky asset.

We solve this problem by the substitution method which gives us an unconstrained maximization problem. A household i with time preference rate and therefore discount factor = 1= (1 + ) maximizes

U_tⁱ = u w_t sⁱ_t + E_tu 1 + rⁱ sⁱ_t ! max

sⁱ_t; ⁱ

(8.2.4)

by now choosing two control variables: The amount of resources not used in the …rst period for consumption, i.e. savings sⁱ_t;and the share ⁱ of savings held in the risky asset.

First-order conditions for sⁱ_t and ⁱ, respectively, are

u⁰ cⁱ_t = E_t u⁰ cⁱ_t+1 1 + rⁱ ; (8.2.5)

Eu⁰ cⁱ_t+1 [r_t+1 r] = 0: (8.2.6)

Note that the …rst-order condition for consumption (8.2.5) has the same interpretation, once slightly rewritten, as the interpretation in deterministic two-period or in…nite horizon models (see (2.2.6) in ch.2.2.2 and (3.1.6) in ch. 3.1.2). When rewriting it as

E_t

we see that optimal behaviour again requires us to equate marginal utilities today and tomorrow (the latter in its present value) with relative prices today and tomorrow (the latter also in its present value). Of course, in this stochastic environment, we need to express everything in expected terms. As the interest rates and consumption tomorrow are jointly uncertain, we can not bring it exactly in the form as known from above in (2.2.6) and (3.1.6). However, this will be possible, further below in (9.1.10) in ch. 9.1.3.

The …rst-order condition for says that expected returns from giving a loan and holding the risky asset must be identical. Returns consist of the interest rate times marginal utility. This condition can best be understood when …rst thinking of a certain environment. In this case, (8.2.6) would read rt+1 = r : agents would be indi¤erent when holding two assets only if they receive the same interest rate on both assets. Under uncertainty and with risk-neutrality of agents, i.e. u⁰ = const:, we get Etr_t+1 = r :Agents hold both assets only if the expected return from the risky assets equals the certain return from the riskless asset.

Under risk-aversion, we can write this condition as Etu⁰ cⁱ_t+1 r_t+1 = E_tu⁰ cⁱ_t+1 r(or, given that r is non-stochastic, as Etu⁰ cⁱ_t+1 r_t+1 = rE_tu⁰ cⁱ_t+1 ). This says that agents do not value the interest rate per se but rather the extra utility gained from holding an asset: An asset provides interest of rt+1 which increases utility by u⁰ cⁱ_t+1 r_t+1:The share of wealth held in the risky asset then depends on the increase in utility from realizations of rt+1 across various states and the expectations operator computes a weighted sum of theses utility increases: Etu⁰ cⁱ_t+1 r_t+1 ⁿ_j=1u⁰ cⁱ_jt+1 r_{jt+1 j};where j is the state, rjt+1

the interest rate in this state and j the probability for state j to occur. An agent is then indi¤erent between holding two assets when expected utility-weighted returns are identical.

Risk-neutral and risk-averse behaviour

For risk-neutral individuals (i.e. the utility function is linear in consumption), the

…rst-order conditions become

1 = E [1 + ⁿr_t+1+ (1 ⁿ) r] ; (8.2.8)

Ert+1= r: (8.2.9)

The …rst-order condition for how to invest implies, together with (8.2.8), that the endoge-nous interest rate for loans is pinned down by the time preference rate,

1 = [1 + r], r = (8.2.10)

as the discount factor is given by = (1 + ) ¹ as stated before in (8.2.4). Reinserting this result into (8.2.9) shows that we need to assume that an interior solution for Ert+1 = exists. This is not obvious for an exogenously given distribution for the interest rate rt+1

but it is more plausible for a situation where rt is stochastic but endogenous as in the analysis of the OLG model in the previous chapter.

For risk-averse individuals with u (c^a_t) = ln c^a_t;the …rst-order condition for consumption reads with i in (8.2.3) being replaced by a

1

c^a_t = E 1

c^a_t+1[1 + ^ar_t+1+ (1 ^a) r] = E 1

s^a_t = 1

s^a_t (8.2.11) where we used (8.2.2) for the second equality and the fact that st as a control variable is deterministic for the third. Hence, as in the last section, we can derive explicit expressions.

Use (8.2.11) and (8.2.1) and …nd

s^a_t = c^a_t , w^t c^a_t = c^a_t , c^at = 1 1 + w_t: This gives with (8.2.1) again and with (8.2.2)

s^a_t =

1 + w_t; c^a_t+1=

1 + [1 + ^ar_t+1+ (1 ^a) r] w_t: (8.2.12) This is our closed-form solution for risk-averse individuals in our heterogeneous-agent economy.

Let us now look at the investment problem of risk-averse households. The derivative of their objective function is given by the left-hand side of the …rst-order condition (8.2.6) times . Expressed for logarithmic utility function, and inserting the optimal consumption result (8.2.12) yields

It can now easily be shown that (8.2.13) implies that risk-averse individuals will not allocate any of their savings to the risky asset, i.e. ^a = 0. First, observe that the derivative of the expression EX= ( + ^aX) from (8.2.13) with respect to ^a is negative

d

d ^aE X

+ ^aX = E X²

( + ^aX)² < 0 8 ^a:

The sign of this derivative can also easily be seen from (8.2.13) as an increase in ^aimplies a larger denominator. Hence, when plotted, the …rst-order condition is downward sloping in ^a: Second, by guessing, we …nd that, with (8.2.9), ^a = 0 satis…es the …rst-order condition for investment,

a= 0) E X

+ ^aX = EX = 0:

Hence, the …rst-order condition is zero for ^a = 0: Finally, as the …rst-order condition is monotonically decreasing, ^a = 0 is the only value for which it is zero,

E X

+ ^aX = 0 , ^a = 0:

This is illustrated in the following …gure.

N

N

0 ^a

dU_t^a

d = (1 + )E ₊^XaX

Figure 8.2.1 The …rst-order condition (8.2.13) for the share ^a of savings held in the risky asset

The …gure shows that expected utility of a risk-averse individual increases for negative

a and falls for positive ^a. Risk-averse individuals will hence allocate all of their savings to loans, i.e. ^a = 0. They give loans to risk-neutral individuals who in turn pay a certain interest rate equal to the expected interest rate. All risk is born by risk-neutral individuals.