Weight lifter

Question 1

Offer curve

Answer 1

Demand curve

Question 2

utility function assumptions

Answer 2

1. continuous
2. strictly increasing
3. strictly quasi-concave

Question 3

Properties of demand

Answer 3

1. Unique x(p,w) which solves consumer problem
2. x is continuous
3. x is hod0
4. exhausts budget constraint
5. boundary condition

Question 4

Boundary condition

Answer 4

if own price is zero, demand is infinite for that product

Question 5

Excess demand

Answer 5

z = demand - endowment
Satisfies same properties as demand
Even in aggregate with Walras Law for budget exhaustion

Question 6

Walras Law in z

Answer 6

pz(p)=0

Question 7

markets clear in equil if z(p*)=…

Answer 7

0

Question 8

Fixed point theorems

Answer 8

Brower’s - single-valued, continuous function has fixed pt

Kakutani’s - upper hemi-continuous stuff

Question 9

Equilibrium existence proof

Answer 9

Probably worth looking into this

for 2 goods can use hod0 with boundary condition, fixed point theorems and Walras Law

Question 10

First Welfare Theorem

Answer 10

If u increasing, every Walrasian equilibrium is PO

Question 11

Second welfare theorem

Answer 11

any PO allocation can be supported as an equilibrium

requires equilibrium existence conditions: U is continuous, increasing, quasiconcave

Question 12

WARP

Answer 12

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)

Question 13

Law of demand in aggregation if holds for individuals:

Answer 13

Preserved by aggregation

Question 14

1st line of proof that strict LoD implies Warp in z(p)

Answer 14

let p.w=(Ap’).w

Question 15

Gross subs implies unique equilibrium

proof involves

Answer 15

increasing the price of one good increases demand for another forcing it out of equilibrium

Question 16

If z(p) obeys WARP, equilibrium prices are a convex set proof

Answer 16

consider p’=tp+(1-t)p**
where ps are equils but p’ assumed not
applying warp leads to a contradiction of W.L.

Question 17

Price tatonnement

Answer 17

rate of change of price pk proportional to aggregate excess demand for k

Question 18

GARP

Answer 18

Transitivity

p1.x2

Question 19

Afriat’s Theorem

Answer 19

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

Question 20

Brown-Matzkin Theorem

Answer 20

Can find out if observed wealths and prices are walrasian rationalisable (can be explained by increasing U)

Question 21

D

Answer 21

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

Question 22

Dz

Answer 22

payoff in each state from holding portfolio of assets, z (containing multiple assets)

Question 23

Incomplete markets

Answer 23

Span(D) (rank) less than number of states

there are payoff vectors which cannot be achieved by any portfolio choice

Question 24

Budget constraints in financial asset world

Answer 24

x0=w0-q.z
x-0=w-0+Dz
2nd one is a vector

Question 25

Proof that in financial market equil sum of consumption = sum of endowments in each period

Answer 25

equil prices set sum of each asset = 0

Question 26

Constrained feasible x-0

Answer 26

if x-0=w-0+Dz

Question 27

constrained P.O. if

1WT says

Answer 27

there does not exist constrained feasible pareto improvement

1WT says every equil allocation is CPO

Question 28

If markets are complete, CPO allocations are also

Answer 28

PO

Question 29

Invariance theorem

Answer 29

Changes in assets which do not change the span do not change consumption bundles or anything in the ‘real’ sector, only financial variables

Question 30

If D’, D have the same span, there exists a matrix K s.t. D’=DK where K has what property

Answer 30

invertible

Question 31

Proof of invatiance

Answer 31

budget set with q,w,D is same as with qK,w,D’ by using K^-1z

Question 32

Arbitrage if

Answer 32

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

Question 33

Fundamental Theorem

Answer 33

q=pD
equilibrium price q admits no arbitrage if this holds
p can be interpreted as probabilities or prices

Question 34

if q=pD then p* is

Answer 34

a no-arbitrage equil price

works other way round if p* is equil then so is q*

Question 35

There always exists a no arbitrage euqil

Answer 35

this proof won’t be asked