Question

Assume that a consumer lives for T periods, and in each future period, one of the s states may be realized. There are complete markets. Let the utility function of the agent be
U = u(c1) +
T X t=2X s∈S
βt−1πt,su(ct,s)
where the notation is the same as that used in class, with pt,s, denoting the price of Arrow-Debreu security paying off at state s on date t.
a. What do complete markets mean in this context? b. Write down the period and intertemporal budget constraints. Explain the saving/borrowing mechanism in this framework. c. Write down the Lagrangian equation and find the first order conditions. d. Write down the price of the Arrow-Debreu security that pays of in date 4, state 7. e. Assume that the agent has linear utility. How would it effect the price of Arrow-Debreu security?

Answer 1

a) In such a market, the complete set of possible bets on future states of the world can be constructed with existing assets without friction. Here, goods are state-contingent; that is, a good includes the time and state of the world in which it is consumed. For instance, an umbrella tomorrow if it rains is a distinct good from an umbrella tomorrow if it is clear. The study of complete markets is central to state-preference theory. The theory can be traced to the work of Kenneth Arrow (1964), Gérard Debreu (1959), Arrow & Debreu (1954) and Lionel McKenzie (1954). Arrow and Debreu were awarded the Nobel Memorial Prize in Economics (Arrow in 1972, Debreu in 1983), largely for their work in developing the theory of complete markets and applying it to the problem of general equilibrium.

b)

According to Keynes’ absolute income hypothesis current consumption depends only on current income.

But this assumption is not always true. In reality while taking consumption and saving decisions people consider both the present and the future.

The more the people consume in the current period (today or the current year) and the Jess they save, the less they will be able to consume in the next period (tomorrow or next year).

So there is always a choice (trade-off) between current consumption and future consumption. So in making consumption decisions households have to take into consideration their expected future income as also the consumption of goods and services they are likely to be able to afford.

Irving Fisher developed a model to analyse how rational, forward-looking consumers make consumption choices over a period of time.

c)

1. Step 1: Obtain the first-order derivative of f(x).
2. Step 2: Set f'(x) = 0. Solve for x. ...
3. Step 3: Obtain the second-order derivative of f(x).
4. Step 4: Determine the sign of f''(x) at the critical values of x. If f'' < 0, the critical value corresponds to a maximum.

d) This section explains Arrow-Debreu prices in an equilibrium context, where they originated, see Arrow (1953) and Debreu (1959). We consider at first a singleperiod model with uncertain states, that will be extended to multiple periods later. For this exposition, we restrict ourself to a single consumption good only, and consider a pure exchange economy without production. Let (Ω, F) be a measurable space of finitely many outcomes ω ∈ Ω = {1, 2, . . . , m}, where the σ-field F = 2Ω is the power set of all events A ⊂ Ω. There is a finite set of agents, each seeking to maximize the utility u a (c a ) from his consumption c a = (c a 0 , ca 1 (ω)ω∈Ω) at present and future dates 0 and 1, given some endowment that is denoted by a vector (e a 0 , ea 1 (ω)) ∈ R 1+m ++ . For simplicity, let consumption preferences of agent a be of the expected utility form u a (c a ) = U a 0 (c0) + Xm ω=1 P a (ω)U a ω (c1(ω)), where P a (ω) > 0 are subjective probability weights, and the direct utility functions U a ω and U a 0 are, for present purposes, taken to be of the form U a i (c) = d a i c γ/γ with relative risk aversion coefficient γ = γ a ∈ (0, 1) and discount factors 1 d a i > 0. This example for preferences satisfies the general requirements (insaturation, continuity and convexity) on preferences for state contingent consumption in Debreu (1959), which need not be of the separable subjective expected utility form above. The only way for agents to allocate their consumption is by exchanging state contingent claims, for delivery of some units of the (perishable) consumption good at a specific future state. Let qω denote the price at time 0 for the state contingent claim that pays q0 > 0 units if and only if state ω ∈ Ω is realized. Given the endowments and utility preferences of the agents, an equilibrium is given by consumption allocations c a∗ and a linear price system (qω)ω∈Ω ∈ R m + such that, a) for any agent a, his consumption c a∗ maximizes u a (c a ) over all c a subject to budget constraint (c a 0 − e a 0 )q0 + P ω (c a 1 − e a 1 )(ω)qω ≤ 0, and b) markets clear, i.e. P a (c a t − e a t )(ω) = 0 for all dates t = 0, 1 and states ω . An equilibrium exists and yields a Pareto optimal allocation; see Debreu (1959), Chapter 7, or the references below. Relative equilibrium prices qω/q0 of the Arrow securities are determined by first order conditions from the ratio of marginal utilities evaluated at optimal consumption: For any a, qω q0 = P a (ω) ∂ ∂ca 1 U a ω (c a∗ 1 (ω)). ∂ ∂ca 0 U a 0 (c a∗ 0 ). To demonstrate existence of equilibrium, the classical approach is to show that excess demand vanishes, i.e. markets clear, by using a fixed point argument, see Chapter 17 in Mas-Colell et al. (1995). To this end, it is convenient to consider c a , e a and q = (q0, q1, . . . , qm) as vectors in R 1+m. Since only relative prices matter, we may and shall suppose that prices are normalized so that Pm 0 qi = 1, i.e. the vector q lies in the unit simplex ∆ = {q ∈ R 1+m + | Pm 0 qi = 1}. The budget condition a) then reads compactly as (c a −e a )q ≤ 0, where the left-hand side is the inner product in R 1+m. For given prices q, the optimal consumption of agent a is given by the inverse of the marginal utility, evaluated at a multiple of the state price density (see (6) for the general definition in the multi-period case), as c a∗ 0 = c a∗ 0 (q) = (U a 0 ′ ) −1 (λ a q0) and c a∗ 1,ω = c a∗ 1,ω(q) = (U a ω ′ ) −1 (λ a qω/Pa (ω)), ω ∈ Ω , where λ a = λ a (q) > 0 is determined by the budget constraint (c a∗ −e a )q = 0 as the Lagrange multiplier associated to the constrained optimization problem a). Equilibrium is attained at prices q ∗ where the aggregate excess demand z(q) := X a (c a∗ (q) − e a ) vanishes, i.e. z(q ∗ ) = 0. One can check that z : ∆ → R 1+m is continuous in the (relative) interior ∆int := ∆ ∩ R 1+m ++ of the simplex, and that |z(q n)| goes to ∞ when q n tends to a point on the boundary of ∆. Since each agent exhausts his budget constraint a) with equality, Walras’ law z(q)q = 0 holds for any 2 q ∈ ∆int. Let ∆n be an increasing sequence of compact sets exhausting the simplex interior: ∆int = ∪n∆n. Set ν n(z) := {q ∈ ∆n | zq ≥ zp ∀p ∈ ∆n}, and consider the correspondence (a multi-valued mapping) Π n : (q, z) 7→ (ν n (z), z(q)), that can be shown to be convex, non-empty valued, and maps the compact convex set ∆n × z(∆n) into itself. Hence, by Kakutani’s fixed point theorem, it has a fixed point (q n∗ , zn∗ ) ∈ Πn(q n∗ , zn∗ ). This implies that (1) z(q n∗ )q ≤ z(q n∗ )q n∗ = 0 for all q ∈ ∆n , using Walras’ law. A subsequence of q n∗ converges to a limit q ∗ ∈ ∆. Provided one can show that q ∗ is in the interior simplex ∆int, existence of equilibrium follows. Indeed, it follows that z(q ∗ )q ≤ 0 for all q ∈ ∆int, implying that z(q ∗ ) = 0 since z(q ∗ )q ∗ = 0 by Walras’ law. To show that any limit point of q n∗ is indeed in ∆int, it suffices to show that |z(q n∗ )| is bounded in n, recalling that z explodes at the simplex boundary. Indeed, z = P a z a is bounded from below since each agent’s excess demand satisfies z a = c a−e a ≥ −e a . This lower bound implies also an upper bound, by using (1) applied with some q ∈ ∆1 ⊂ ∆n, since 0 < ǫ ≤ qi ≤ 1 uniformly in i. This establishes existence of equilibrium. To ensure uniqueness of equilibrium, a sufficient condition is that all agents’ risk aversions are less or equal to one, that is γ a ∈ (0, 1] for all a, see Dana (1993). For multiple consumption goods, above ideas generalize if one considers consumption bundles and state contingent claims of every good. Arrow (1953) showed that in the case of multiple consumption goods, all possible consumption allocations are spanned if agents could trade as securities solely state contingent claims on the unit of account (so called Arrow securities), provided that spot markets with anticipated prices exists for all other goods exists in all future states. In the sequel, we only deal with Arrow securities in financial models with a single numeraire good that serves as unit of account, and could for simplicity be considered as money (‘Euro’). If the set of outcomes Ω were (uncountably) infinite, the natural notion of atomic securities is lost, although a state price density (stochastic discount factor, deflator) may still exist, which could be interpreted intuitively as an Arrow-Debreu state price per unit probability.

e) In financial economics, a state-price security, also called an Arrow-Debreu security (from its origins in the Arrow-Debreu model), a pure security, or a primitive security is a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state occurs at a particular time in the future and pays zero numeraire in all the other states. The price of this security is the state price of this particular state of the world. The state price vector is the vector of state prices for all states.[1][2][3] As such, any derivatives contract whose settlement value is a function of an underlying asset whose value is uncertain at contract date can be decomposed as a linear combination of its Arrow-Debreu securities, and thus as a weighted sum of its state prices.

example:-

Imagine a world where two states are possible tomorrow: peace (P) and war (W). Denote the random variable which represents the state as ω; denote tomorrow's random variable as ω1. Thus, ω1 can take two values: ω1=P and ω1=W.

Let's imagine that:

• There is a security that pays off £1 if tomorrow's state is "P" and nothing if the state is "W". The price of this security is qP
• There is a security that pays off £1 if tomorrow's state is "W" and nothing if the state is "P". The price of this security is qW

The prices qP and qW are the state prices.

The factors that affect these state prices are:

• "Time preferences for consumption and the productivity of capital"[4]. That is to say that the time value of money affects the state prices.
• The probabilities of ω1=P and ω1=W. The more likely a move to W is, the higher the price qW gets, since qW insures the agent against the occurrence of state W. The seller of this insurance would demand a higher premium (if the economy is efficient).
• The preferences of the agent. Suppose the agent has a standard concave utility function which depends on the state of the world. Assume that the agent loses an equal amount if the state is "W" as he would gain if the state was "P". Now, even if you assume that the above-mentioned probabilities ω1=P and ω1=W are equal, the changes in utility for the agent are not: Due to his decreasing marginal utility, the utility gain from a "peace dividend" tomorrow would be lower than the utility lost from the "war" state. If our agent were rational, he would pay more to insure against the down state than his net gain from the up state would be.

that the right answer all the best.