Economics of finance - Week 9 and Week 10 Lec

7/30/2019 Economics of finance - Week 9 and Week 10 Lec

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Economics of Finance

ECON 3107 / 5106

Mariano KulishSchool of Economics

UNSW

Weeks 9 and 10

May 2013



Uncertainty: An Example

• Two periods: today (time 0) and future (time 1)• Today’s state of nature s0 is known. Alternatively, the

probability that s0 occurs π(s0) = 1;

• Set of possible future events - good weather (G) and badweather (B): S = {G, B} .

• G occurs with probability π(s1 = G); B with probabilityπ(s1 = B).

• Notation (state-contingent consumption plans):

• c(s0) - consumption at time 0.• c(s1 = G)−consumption at time 1 if next period’s state is

G;• c(s1 = B)−consumption at time 1 if next period’s state is B;

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Expected Utility: an introduction

• The representative consumer has a time and stateseparable utility function over consumption c(s0) and c(s1).

• Consumers discount future expected utility with timediscount factor β ∈ (0, 1)

• The lower the β, the more impatient are the consumers

• The period utility function u(c) is assumed to be strictlyincreasing and concave;

• The consumer maximises expected utility, U , given by

U = u (c(s0)) + β [π(G) · u (c(G)) + π(B) · u (c(B))] expected discounted future utility

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More on the instantaneous utility:

• u(c) is assumed to be strictly increasing and concave:

u(c) > 0, u(c) ≤ 0

• An example often used in Asset Pricing literature:

• CRRA utility function has marginal utility of the form:

u(c) = c−γ ,

where γ > 0 is the coefficient of relative risk aversion.• The functional forms for CRRA utility function

u(c) =c1−γ

1 − γ , for γ > 0, and γ = 1

u(c) = log c, for γ = 1

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Endowments:

• There is no production;• There is an exogenously given supply of a non-storable

consumption good at each time and state;

• As of date t = 0, the second period endowment is random.

The consumer does not know which state will realize.• Notation (Endowments):

• e (s0) - the initial endowment of consumption good;• e (s1 = G) - the quantity of the consumption good consumer

receives (say apples from a tree) at time 1 if the realized

state is Good Weather;• e (s1 = B) - the endowment available at time 1 in the Bad

Weather state;

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Market structure:

• The consumer can freely borrow or lend in a complete setof basic Arrow securities.

• We assume the existence of two securities: Bad Weathersecurity and Good Weather security.

• One unit of ’G security’ sells at time 0 at a price

q (s0, s1 = G) and pays one unit of consumption at time 1 if state ’G’ occurs and nothing otherwise.

• One unit of ’B security’ sells at time 0 at a priceq (s0, s1 = B) and pays one unit of consumption in state ’B’only.

• In this notation: s0 refers to the state when securities aretraded; s1 = G refers to a particular realization of the states1 when the security pays off.

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Flow budget constraints: Time 0

• In the first period the consumer has initial endowmente(s0). She can consume or buy Arrow securities:

c (s0) + q (s0, s1 = G) · a(s0, s1 = G)

+q (s0, s1 = B) · a(s0, s1 = B) = e (s0)

•

a(s0, s1 = G) - quantity G securities acquired in state s0;• a(s0, s1 = B) - quantity B securities acquired in state s0;

• In our two-period model all trades occur in state s0. Theonly uncertainty is about the realization of the state s1.Therefore, we can use simplified notation:

• for atomic security prices: q (G), q (B)• for quantities of the atomic security purchased (sold):

a(G), a(B)

c0 (s0) + q (G) · a(G) + q (B) · a(B) = e0 (s0)

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Flow budget constraints: Time 1

• If the realized state at time 1 is Good Weather:

•

Each of a(G) G atomic securities pays off 1 unit of consumption;• Bad Weather atomic securities do not pay off at all;• Consumer receives an endowment corresponding to G state:

e(G) and eats every unit of consumption she has got:

c (G) = 1 · a(G) + 0 · a(B) + e(G),

• If the realized state at time 1 is Bad Weather:

• Each of a(B) B atomic securities pays off 1 unit of consumption;

• G atomic securities do not pay off at all;• Consumer receives an endowment corresponding to B state:

e(B) and eats every unit of consumption she has got:

c (B) = 0 · a(G) + 1 · a(B) + e(B).

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Sequential Market Equilibrium: An Example

• A Sequential Market Equilibrium in this economy is definedas an allocation c (s0) , c (G) , c (B) , a (G) , a (B) , and prices

q (G) , q (B) such that:• Given the prices, the allocation solves the consumer’s

problem of maximizing expected utility

u (c(s0)) + β [π(G) · u (c(G)) + π(B) · u (c(B))]

subject to a sequence of budget constraints

c (s0) + q (G) · a(G) + q (B) · a(B) = e (s0) ,

c (G) = a(G) + e(G),c (B) = a(B) + e(B).

• Prices are such that markets clear in every period and state:

c (s0) = e (s0) ; c (G) = e(G); c (B) = e(B),

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household discountfactor ( the level ofpatience



Constrained Optimization: A refresher

• To deal with the consumer’s problem we have to maximizea function subject to several equality constraints.

• Consider a problem of choosing x,y,z to maximize theobjective function f subject to equality constraints g1 andg2 :

maxx,y,z

f (x,y,z)

s.t. g1 (x,y,z) = b1,

g2 (x,y,z) = b2,

where b1 and b2 are constants.

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Constrained Optimization: A refresher (cont’d)

Under certain technical conditions, the following algorithm willobtain the necessary conditions for the maximum:

• Form a Lagrangian as

• the objective function f • minus Lagrange multiplier for constraint 1, λ1, times the

difference between LHS and RHS of the constraint 1•

minus Lagrange multiplier for constraint 2, λ2, times thedifference between LHS and RHS of the constraint 2

L (x,y,z ,λ1, λ2) = f (x,y,z) − λ1 [g1 (x,y,z) − b1]

− λ2 [g2 (x,y,z) − b2]

• Take partial derivatives of the Largangian with respect toits every argument and equate them to zero;

• The values of x,y,z that maximize f must satisfy theresulting 5 equations for some λ1, λ2.

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Solving the Consumer’s Problem:

• The problem:

• Choose c(G), c(B), c(s0), a(G), a(B)

• to maximizeu (c(s0)) + β [π(G) · u (c(G)) + π(B) · u (c(B))]

• subject to

c (s0) + q (G) · a(G) + q (B) · a(B) = e (s0) ,

c (G) = a(G) + e(G),

c (B) = a(B) + e(B).

• Form the Lagrangian

L = u (c(s0)) + β [π(G) · u (c(G)) + π(B) · u (c(B))]

− λ0 [c (s0) + q (G) · a(G) + q (B) · a(B) − e (s0)]

− λ1 [c (G) − a(G) − e(G)]

− λ2 [c (B) − a(B) − e(B)]

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Solving the Consumer’s Problem: (cont’d)

• Equate the partial derivatives of the Lagrangian to zero:

• Partials w.r.t. c(G), c(B), c(s0), a(G), a(B)

∂L/∂c(s0) = u (c(s0)) − λ0 = 0

∂L/∂c(G) = βπ(G) · u (c(G)) − λ1 = 0

∂L/∂c(B) = βπ(B) · u (c(B)) − λ2 = 0

∂L/∂a(G) = −λ0 · q (G) + λ1 = 0

∂L/∂a(B) = −λ0 · q (B) + λ2 = 0

• Partial w.r.t. the multipliers λ0, λ1, λ2 are just theconstrains:

c (s0) + q (G) · a(G) + q (B) · a(B) − e (s0) = 0

c (G) − a(G) − e(G) = 0

c (B) − a(B) − e(B) = 0

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Solving the Consumer’s Problem: (cont’d)

• Expressing atomic prices as functions of consumption

allocations:

• The values of the multipliers:

λ0 = u (c(s0))

λ1 = βπ(G) · u (c(G))

λ2 = βπ(B) · u (c(B))

• The prices of the basic Arrow securities:

q (G) =λ1

λ0= βπ(G)

u (c(G))

u (c(s0))

q (B) =λ2

λ0= βπ(B)

u (c(B))

u (c(s0))

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The Prices of Basic Arrow Securities

• Combine the solution to the consumer’s problem with themarket clearing conditions:

• Atomic prices as functions of consumption allocations:

q (G) = βπ(G)u (c(G))

u (c(s0))

q (B) = βπ(B)u (c(B))

u

(c(s0))

• Market clearing conditions:

c (s0) = e (s0) ; c (G) = e(G); c (B) = e(B),

•

The Prices of Basic Arrow Securities:

q (G) = βπ(G)u (e(G))

u (e(s0)),

q (B) = βπ(B)u (e(B))

u (e(s0))

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A More General Environment: Many Future States

• Still only two periods: today (time 0) and future (time 1)• Today’s state of nature s0 is known. Alternatively, the

probability that s0 occurs π(s0) = 1;

• A future event s1 ∈ S from the set of possible events S

occurs with probability π(s1);• Notation (state-contingent endowments and consumption

plans):

• c0(s0)− consumption at time 0;• c1(s1)− consumption at time 1 if next period’s state is s1;• e0 (s0)− the initial endowment;• e1(s1)− the endowment at time 1 if next period’s state is s1;

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Consumer’s Problem• The consumer takes the prices of Arrow securities{q (s0, s1)}s1∈S as given.

• At time 0, she must choose:• the quantities of the securities to hold {a (s0, s1)}s1∈S ,• time-0 consumption, c0(s0),• state-contingent plan for the next period consumption{c1 (s1)}s1∈S .

• The consumer’s objective is• to maximize expected utility

u (c0 (s0)) + β s1∈S

π (s1) u (c1 (s1)) ,

• subject to period-0 constraint

c0 (s0) +s1∈S

q (s0, s1) · a (s0, s1) = e0 (s0) ,

• and a series of period-1 constrains for every possible state:

c1 (s1) = a (s0, s1) + e1 (s1) , for all s1 ∈ S.

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S



Sequential Market Equilibrium :A Sequential Market Equilibrium in this economy is defined asan allocation c0 (s0) , {c1 (s1)}s1∈S , {a (s0, s1)}s1∈S and prices

{q (s0, s1)}s1∈S such that:• given the prices {q (s0, s1)}s1∈S the allocation solves

max u (c0 (s0)) + β s1∈S

π (s1) u (c1 (s1)) ,

subject to

c0 (s0) +s1∈S

q (s0, s1) · a (s0, s1) = e0 (s0) ,

c1 (s1) = a (s0, s1) + e1 (s1) , for all s1 ∈ S 1;• Markets clear in every period and state:

c0 (s0) = e0 (s0) ,

c1 (s1) = e1 (s1) , for all s1 ∈ S 1.

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S l h C ’ P bl



Solving the Consumer’s Problem:

• The Lagrangian for the consumer’s problem can written as

L = u (c0 (s0)) + β s1∈S

π (s1) u (c1 (s1))

− λ0 (s0)c0 (s0) + s1∈S

q (s0, s1) · a (s0, s1) − e0 (s0)−s1∈S

λ1 (s1)

c1 (s1) − a (s0, s1) − e1 (s1)

.

• To obtain the necessary conditions we must differentiatethe Lagrangian with respect to c0 (s0) , {c1 (s1)}s1∈S 1

, and{a (s0, s1)}s1∈S 1

and equate them to zero.

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C ’ O i li C di i



Consumer’s Optimality Conditions:

• The corresponding optimality conditions are

u (c0 (s0)) − λ0 (s0) = 0,

βπ (s1) u (c1 (s1)) − λ1 (s1) = 0, for s1 ∈ S,

−λ0 (s0) q (s0, s1) + λ1 (s1) = 0, for s1 ∈ S.

• The prices of basic Arrow securities as functions of consumption:

q (s0, s1) =λ1 (s1)

λ0 (s0)

, for s1 ∈ S.

q (s0, s1) = βπ (s1)u (c1 (s1))

u (c0 (s0)), for s1 ∈ S.

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P i f B i A S iti



Prices of Basic Arrow Securities

• The prices of basic Arrow securities from consumer’sproblem:

q (s0, s1) = βπ (s1)u (c1 (s1))

u (c0 (s0)), for s1 ∈ S.

• Market clearing requires that ct (st) = et (st) , for t = 1, 2,the prices of the Arrow securities must satisfy

q (s0

, s1

) = βπ (s1

)u (e1 (s1))

u (e0 (s0)), for all s

1 ∈S.

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St h ti Di t F t



Stochastic Discount Factor

• The prices of the Arrow securities are given by

q (s0, s1) = βπ (s1)

u (e1 (s1))

u (e0 (s0)) , for all s1 ∈ S.• The discount factor is the sum of all atomic security prices

df (1) =

s1∈S

q (s0, s1) =

s1∈S

βπ (s1)u (e1 (s1))

u (e0 (s0))

• The stochastic discount factor, m1, is a random variable

• the value of which is unknown at t = 0;

• the value of which at time 1 is m1 (s1) = β u(e1(s1))

u(e0(s0))if state

s1 is realized;

• Then the discount factor is

df (1) =s1∈S

π (s1) β u (e1 (s1))

u (e0 (s0))=s1∈S

π (s1) m1 (s1) = E [m1]

where E [·] is the expectation operator.

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Economics of finance - Week 9 and Week 10 Lec

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