Contingent Claim Economy
() Exchange 1 / 14
Abstract exchange economy
Copyright By PowCoder代写 加微信 powcoder
What is being exchanged?
Commodities or goods. Anything that people want to have but that costs something.
The good is deÖned by its:
physical characteristics.
the geographical place of availability.
the time of availability.
the state (event) where it is available. The umbrella in the rain is di§erent from umbrella in the shine.
() Exchange
The event tree
() Exchange 3 / 14
Two period economy
() Exchange 4 / 14
Abstract exchange economy
Two-period modeló today and tomorrowó with S states tomorrow and M commodities in each state,
The agent lives for two periods, her t = 0 endowment ω0 (i ) .
At date 1 one of the future states is realized, say s with a cyclone it
in, and the agentís endowment is ωs (i).
Tomorrow brings no surprises: once the state is realized everyone observes exactly the same endowments that were expected in that state. (We do not assign probabilities to the states yet)
Together there are (S + 1)M contingent commodities Agent i has endowment of M goods in each state,
2ω01(i) ωS1(i)3 ω ( i ) = 64 . . . . . . . . . 75
ω 0M ( i ) ω SM ( i ) Exchange
Abstract exchange economy
Agent i (planned) consumption bundles are
2×10(i) x1S(i)3 x ( i ) = 64 . . . . . . . . . 75
xM0 (i) xMS (i)
The vector xs (i) is the vector of planned consumptions if state s will
be realized.
Agenti utilityisu(x(i)).
A contingent claim economy is simply a collection of all agents,
f(ui,ω(i)) : i = 1,…,Ig or simply fu,ωg.
() Exchange 6 / 14
Spot market economy
It is not possible to trade the goods until date 1 when the uncertainty is resolved.
Alternatively, it is impossible to trade contingent commodities with di§erent state indices
Commodities available in particular state can be traded, but across states trade is impossible.
Agent then solves
maxui (x (i)) s to p0x0 p0ω0
psxs psωs fors=1,…,S,
where ps are the spot market prices at state s determined as the state
materializes.
She faces S + 1 budget constraints and S + 1 Lagrange multipliers
() Exchange 7 / 14
Contingent claim economy
Suppose we now allow trade in contingent claims.
Contingent claim is a contract to deliver a unit of good m in state s and nothing otherwise.
It is a forward speciÖc to state not only to date.
Let pˆsm be the date 0 price of such contingent claim.
Trade in contingent claims allows the trade across the states and within the states.
Let bsm (i) be the quantity of contingent claims for good m in state s held/issued by consumer i.
If bsm (i) > 0 agent i will be receiving m if s materializes, if bsm (i) < 0 agent i will deliver this contingent commodity.
To achieve xs (i) the agent buys the necessary contingent contracts for state s.
() Exchange 8 / 14
Contingent claim economy
The agents face the prices for contingent claims as determined at date 0.
These prices are allowed to vary across the states. With the matrix of contingent claims prices
2 pˆ 01 pˆ 0M 3 pˆ = 64 . . . . . . . . . 75 ,
pˆS1 pˆSM maxui (x (i)) subject to
pˆ 0 x 0 ( i ) + ∑ pˆ s b s ( i ) pˆ 0 ω 0
s=1 0xs(i)ωs(i)+bs(i) fors=1,...,S,
Agent i solves
Contingent claim economy
BC allows trades between dates 0 and 1 and between di§erent states in date 1.
All exchanges are decided at date 0, including trades in contingent claims.
This means that markets operate at date 0 only. There is a lot of them ó (S +1)M.
The contingent contracts agreed on date 0 are executed on date 1.
() Exchange 10 / 14
Contingent claim economy
By monotonicity of preferences the resource constraint
xs (i) = ωs (i)+bs (i), After rearranging agentís i budget constraint
∑ pˆ s x s ( i ) = ∑ pˆ s ω s ( i )
That is agent i chooses a consumption vector today (x0(i)) and a state-contingent consumption vector for tomorrow (x1(i),...,xS(i)) to maximize her utility subject to the budget constraint.
(S) max ui (x(i)) ∑pˆs (xs (i) ωs (i))=0
holds with equality.
() Exchange 11 / 14
Equilibrium
DeÖnition
A contingent market equilibrium for this contingent claim economy is a pair (x, p) , such that for each agent the chosen x (i ) maximizes her utility subject to her budget constraint given p, and all markets clear:
∑xms (i) = ∑ωsm (i), for s = 0,1,...,S; and m = 1,...,M. i=1 i=1
An equilibrium price is a price vector at which aggregate demand equals aggregate supply for each commodity simultaneously.
Date 0 trading allows us to collapse the sequence of date 1 budget constraints into a single one, thanks to the idea of contingent claims and contingent prices.
Contingent claims construction transforms a complicated intertemporal problem into one of static optimization.
() Exchange 12 / 14
Two state one good example
agent A sells (to B) contingent claims for state 1 agent B sells (to A) contingent claims for state 2
() Exchange 13 / 14
Equilibrium
The beauty of this is that we can use standard tools to make conclusions about the economy.
Does an equilibrium exist? (Yes.)
Is the equilibrium unique? (Usually not.)
If it is not unique, are there at least only a small number of equilibria? (Typically, yes.)
Are the equilibrium allocations e¢ cient? (Yes.) Do Welfare Theorems hold? (Yes)
No need to open the spot markets at date 1 in any state as no new trades will occur.
The arranged prices pˆs for that state clear the markets and each agent maximizes her utility already.
() Exchange 14 / 14
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com