Weight lifter Flashcards
Offer curve
Demand curve
utility function assumptions
- continuous
- strictly increasing
- strictly quasi-concave
Properties of demand
- Unique x(p,w) which solves consumer problem
- x is continuous
- x is hod0
- exhausts budget constraint
- boundary condition
Boundary condition
if own price is zero, demand is infinite for that product
Excess demand
z = demand - endowment
Satisfies same properties as demand
Even in aggregate with Walras Law for budget exhaustion
Walras Law in z
pz(p)=0
markets clear in equil if z(p*)=…
0
Fixed point theorems
Brower’s - single-valued, continuous function has fixed pt
Kakutani’s - upper hemi-continuous stuff
Equilibrium existence proof
Probably worth looking into this
for 2 goods can use hod0 with boundary condition, fixed point theorems and Walras Law
First Welfare Theorem
If u increasing, every Walrasian equilibrium is PO
Second welfare theorem
any PO allocation can be supported as an equilibrium
requires equilibrium existence conditions: U is continuous, increasing, quasiconcave
WARP
p’.x(p,w)<=p’.x(p’,w’)=w’ (x(p,w) affordable at p’, but obviously x(p’,w’) chosen instead)
implies p.x(p’,w’)>p.x(p,w)=w (x(p’,w’) must be unaffordable at p)
Law of demand in aggregation if holds for individuals:
Preserved by aggregation
1st line of proof that strict LoD implies Warp in z(p)
let p.w=(Ap’).w
Gross subs implies unique equilibrium
proof involves
increasing the price of one good increases demand for another forcing it out of equilibrium
If z(p) obeys WARP, equilibrium prices are a convex set proof
consider p’=tp+(1-t)p**
where ps are equils but p’ assumed not
applying warp leads to a contradiction of W.L.
Price tatonnement
rate of change of price pk proportional to aggregate excess demand for k
GARP
Transitivity
p1.x2
Afriat’s Theorem
for observed prices and quantities (p,x) the following statements are equivalent
1. the consumer is maximising a strictly increasing and concave utility function
2. GARP is obeyed
3. the following inequality holds for some As, Bs:
As
Brown-Matzkin Theorem
Can find out if observed wealths and prices are walrasian rationalisable (can be explained by increasing U)
D
payoff matrix with states on the vertical axis (rows) and securities on the horizontal (columns)
ie if one asset: D=(1,2,3)’
paying 1 in state one, 3 in state 3
Dz
payoff in each state from holding portfolio of assets, z (containing multiple assets)
Incomplete markets
Span(D) (rank) less than number of states
there are payoff vectors which cannot be achieved by any portfolio choice
Budget constraints in financial asset world
x0=w0-q.z
x-0=w-0+Dz
2nd one is a vector
Proof that in financial market equil sum of consumption = sum of endowments in each period
equil prices set sum of each asset = 0
Constrained feasible x-0
if x-0=w-0+Dz
constrained P.O. if
1WT says
there does not exist constrained feasible pareto improvement
1WT says every equil allocation is CPO
If markets are complete, CPO allocations are also
PO
Invariance theorem
Changes in assets which do not change the span do not change consumption bundles or anything in the ‘real’ sector, only financial variables
If D’, D have the same span, there exists a matrix K s.t. D’=DK where K has what property
invertible
Proof of invatiance
budget set with q,w,D is same as with qK,w,D’ by using K^-1z
Arbitrage if
qz<0 or DZ>0 with the other holding with equality
ie get positive price now for 0 cost later or positive payoff later for 0 price now
Fundamental Theorem
q=pD
equilibrium price q admits no arbitrage if this holds
p can be interpreted as probabilities or prices
if q=pD then p* is
a no-arbitrage equil price
works other way round if p* is equil then so is q*
There always exists a no arbitrage euqil
this proof won’t be asked
