Term 1 week 4 Flashcards
- Draw out a utility maximisation problem for person a and person b.
How do you work out the net demands? - What is net demand denoted by?
- How would the excess / net demand be written?
- Firstly, you look at endowment point then compare it to the optimal point.
If the demanded at optimal is greater than endowment (positive excess demand)
If the demanded optimal is less than endowment it is negative excess demand.
- Net demand is denoted by e
eA1 = (X1A - W1A)
eA2 = (X2A - W2A)
What is the difference between x and w?
X is the demanded amount of a good
W is the endowment of a good.
What is the equation of the budget constraint for an individual in an edgeworth box?
p1A . x1A + p2 . x2A = p1 . w1A + p2 . w2A
What does it mean if you are a net supplier and net demander?
If you are net supplier you sell
If you are net demander you buy.
What it is important about the price line in an edgeworth box?
Price line must pass through endowment point.
What is important to remember when measuring excess demand
person B is flipped so you are measuring excess demand flipped.
What is aggregate excess demand for a good?
What is aggregate excess demand denoted by?
For each person the sum of net demands.
Denoted by z
Z1 = e1A + e1B
Z2 = e2A + e2B
Assume you have found the aggregate excess demand what can you do next?
What is the benefit of the new equilibirum?
If zi> 0 pi increases
if zi<0 pi decreases
This alters the slope
The slope changes until there is tangency
- In the new equilibrium there is no wastage.
What is a walrasian equilibrium?
Also known as a competitive equilibrium.
It is one where the price ratio is at a point they prefer this bundle over any other bundle in the feasible set
Same for person 2.
if you are asked to find a competitive equilibrium what are you looking to find?
Looking to find prices
and equilbrium allocations
Why do you need to be careful between quantity and value of endowment
When working out excess demands you need quantities.
When doing UMP you need value.
What does Walrus Law imply:
If there are n markets and n-1 markets are in equilibrium then the last market must also be in equilibrium.
- What does Walrus Law state?
- What is the Walrus Law mathematically?
- States that the sum of aggregate excess demands is identically zero.
This implies it is possible for all prices not just equilibrium prices. - p1 . z1(p1,p2) + p2 . z2(p1,p2) identically = to 0
What does Walrus Law state if there is excess demand in one market?
Then the other market must have excess supply.
How do you do the proof for the Walrus Law?
-you start with the budget constraints of consumer A and B
-You then add them both together
-Then you subtract to make RHS = 0
-Then you group everything with p1 and p2
-Then you show that p1z1 + p2z2 = 0
What does excess demand is = 0 imply?
-The market is in equilibrium
- Demand is = to supply in that market.
Why can we use p2 as the numeraire when finding an equilibrium?
- This is because of the walrus law.
How do you solve a competitive equilibrium given initial endowments?
What does it give?
Start by making p2 = 1 numeraire to find relative price line.
You then do MRS = price ratio
-this will simplify in terms of one good
-You then sub this back into budget constraint, also putting in the endowment.
-This gives you the demand for one person in one good.
-Then as markets clear in equilbirium
demand for 1 good by two people = endowment of that 1 good by two people.
-Then solve for p1
-This gives Walrasian demands.
What do you do when you can;t calculate price but you still must state competitive equilibrium.
-You sketch it all out on edgeworth box
-CHECK THE SHAPES OF THE UTILITY FUNCTIONS!!!
-You hypothesise a price line then check if it is stable?
-To check stability you see given the price line the optimal choice of both consumers and if this causes excess demand or excess supply this is not stable so it can’t be the line.
Simply what is the first welfare theorem?
What does this assume?
- A competitive equilibrium is always pareto efficient.
-Assumes that competitive equilibrium will always exhaust all gains from trade
-Assumes that people only care about their own consumption.
-Assumes that people behave competitively
What is a big issue with first welfare theorem?
- It is efficient but not equitable, does not allow for equality in distribution.
Why do we say that two people behave competitvely even though they could use market power?
As in the two by two framework we are assuming that it is two different groups with infinite people inside.
How do you prove first welfare theorem?
- You start off by supposing there is a competitive equilibrium x* where everyone is maximising individual utility
-But assume x* is not pareto efficient.
-Then assume there is another allocation x’ that for A is strictly preferred and for B is at least as good.
-As allocations that are more expensive are more preferred then x’ is more expensive than x*
p* x’i > p* xi = p wi
p* x’k> p* xk = p wi
Then sum across both consumers.
As for one person it is strict and for another it is a weak ineqality it becomes strict.
This is now in matrix notation so expand it.
When you do this it shows that for both goods the endowment are less than the allocation.
This means that x’ is not pareto efficient so x* must be pareto efficient.
What does the second welfare theorem say?
Every pareto efficient allocation can be supported as a competitive equilibrium if we get proper redistrubtion of initial endowment.
