Exchange Economies Flashcards
What is a monotone preference?
This is when the consumer’s utility rises as any one of the quantity of consumption bundles increases. It is an ever-increasing preference.
What is a convex preference?
What is defined as a convex set?
This is a preference in which the average consumption between two bundles is preferred, rather than the extreme (which would be equivalent to picking all of either good). This preference exists due to diminishing marginal utility and usually only exists within populations, rather than individuals.
A convex set is one where if you a draw a line between two points, the line drawn would stay in the set. A non-convex set would have some of the line drawn outside of the set (think of the ‘set’ as a badly drawn circle).
What utilities are strictly convex?
Cobb-Douglas utilities (normal indifference curve - curved) are strictly convex.
For a line to be strictly convex, the average bundle has to be preferred to the extreme bundles. That is to say that, for two consumers, they would rather pick the weighted average because picking an extreme would make either one of them have less utility for one good. This is because of diminishing marginal utility.
But, for linear utilities (diagonal downward slope), we can say that the preferences are not strictly convex because the MRS is constant. Regardless of what either consumer picks, they would be indifferent to the weighted average.
What does a Cobb-Douglas utility function look like?
See image.
What is a monotonic transformation?
This is when the utilities derived from each bundle may change but the order of preferences still stay the same. We can do this by :
- Multiplying by a constant (although not a negative or a 1/ fraction)
- Raising by a power
- Taking the natural log.
To understand if a monotonic transformation has taken place, first work out what the change has been (has the utility function been divided by 2). Then write this down as g(u) = 1/2 u. Then differentiate that function with respect to u and you will see that 1/2 > 0, and thus the function is rising and so a monotonic transformation.
What is the MRS of good 1?
How do we calculate it?
This is the amount of good 2 that the consumer is willing to give up in order to get more of good 1 (depending on whichever good is on bottom of the MRS equation). The equation is given below :
What is the Edgeworth box?
The Edgeworth box allows us to graphically represent a market of two commodities between two consumers. Therefore, it also shows us their indifference curves too in which we can analyse their preferences.
Every allocation within and on the Edgeworth box is deemed to be initially feasible (we will use feasible in this course as there is no waste, the total of what was allocated initially will remain the same after looking at preferences).
Think of Agent A starting in the bottom left and agent B starting at the top right. Thus, their consumption also follows the same pattern - i.e agent B consumption is the distance to the endowment from the top right.
What is the endowment allocation in an Edgeworth box?
This is where no trade occurs (initial bundle).
Note: The total quantity of good 1 used is along the bottom and the total quantity of good 2 is shown vertically (remember there are 2 consumers consuming 2 goods).
Are boundary points within the Edgeworth box feasible?
Yes.
- Any point on the bottom would mean that Agent A has none of good 2.
- Any point on the left boundary would mean that Agent A has all of good 2.
- Any point on the top would mean that Agent B has none of good 2.
How can we graphically represent the preferences of both agents in the Edgeworth box?
- Remember that Agent B preferences are shown from the top right-hand side. Thus, Agent B indifference curve is ‘flipped’ in a sense. But, the notion holds true that agent B’s utility would increase the more consumption he has.
What is a pareto-improvement?
This is when the utility of neither consumer decreases but simultaneously the utility of either consumer (or both) will increase from the initial endowment.
Note that the term ‘Pareto improvement’ is with respect to the initial endowment.
Image: In the black area, they are strictly better off.
On the green line, Agent A is indifferent with regards to the initial endowment but agent B is strictly better off.
The reverse is true for any point on the grey line in the set that the indifference curves meet.
Graphically, what would an improvement in trade look like given that we have a coming together of two indifference curves in the Edgeworth box?
When should both agents stop trying to ‘improve their trade’?
When it makes either one of them worse off. There will come a point where the edges of the indifference curve are touching each other - do not mistake this as being tangent. This is when any further movement will make one agent worse off and the other better off (which is not a pareto-improvement).
When no further pareto-improvements can be made, we call it ‘Pareto-optimal’.
What is the ‘contract curve’?
This is the line that joins all the ‘Pareto-optimal’ points.
It is called the contract curve because the agents should come to an agreement on trade at a point that lies on this contract curve.
Note: even though we class all the points on the contract curve as ‘Pareto-optimal’, it does not mean that it is fair. For example, picking a point on the contract curve that lies to the bottom left will result in a greater utility for Agent B as he consumes more of each good.
What are the three conditions required to suggest that ‘Pareto-optimal points occur where the indifference curves are tangent’?
- The preferences need to be convex. (as this is all about trade and so weighted averages are needed). (non-convex also indicates that there is more than one tangency point).
- The preferences need to be smooth. (this is not the case for things like perfect complements - again, cannot determine optimal trade where a consumer is indifferent to consuming one more of a good).
- Tangency points need to occur internally (any point on the boundary is feasible, but not optimal).
Assuming we have smooth and convex preferences that lie inside the Edgeworth box, how can we confirm that a point is Pareto-optimal?
By calculating the MRS of both agents and setting them equal to each other. The point where they are equal is a Pareto-optimal point.
What can be said about Cobb-Douglas utility functions and efficiency (think boundary points)?
Remember that any point on the Edgeworth box boundary is feasible, but not always efficient.
The only points where we can say that it is efficient (i.e. no one is made worse off is the origin of both Agent A and B).
As moving along the indifference curves at any other boundary point could make either agent better off.
How do we find the equation for the contract curve given the following allocations?
- Find MRS of both agents and equate them to each other. (efficient points)
- Total both quantities of goods to determine feasibility.
- Sub these feasible quantities into the efficient points equations to give contract curve equation.
The answer for this one should give us precisely the diagonal for the square of the Edgeworth box - as we have shown that xa2=xa1
What is the core?
With respect to the endowment, it is the set of all Pareto-optimal points (on contract curve) that are Pareto improvements (wrt initial allocation). Thus, in the core, both agents have to be at least as well off as they were wrt initial allocation.
Highlighted in bold to remember.
Note: Even if it is Pareto-optimal, trades can still be blocked by either agent depending on which way go along the contract curve.
This is all the points in the contract curve within an indifference set (where both indifference curves meet) which will not be blocked by either agent.
How do we find the core, given the following worked out contract curve?
Remember, the core is the set of all Pareto optimal points (and thus on contract curve) that are Pareto improvements with respect to the endowment allocation (has to be at least as good as).
Find the contract curve and core of this non-symmetric CD function below :
What are quasi-linear preferences?
These are preferences in which one function can be anything (usually non-linear) and the other function is linear.
Another thing to note is that indifference curves are always vertical/horizontal translates, depending on the arbitrary non-linear function.
Since it is only the non-linear part of the function that can be differentiated (as it is a multiple), we can also say that the MRS is independent of the second variable.
If you are reasonably wealthy, you won’t usually check your preference between buying an extra loaf of bread and saving that money (bread and money being the two goods). Thus, a quasi-linear preference is shown here. Obviously will not be the same preferences if you live in poverty.
What are the key things to note when dealing with quasi-linear preferences (wrt trying to derive a contract curve)?
- Feasibility can only be calculated for good 1 (as it does not matter how much of good 2 you have, it has no effect). Do this after equality of MRS’.
- Thus, when setting the MRS’ equal to each other, you find out the quantity of good 1 that would be pareto-optimal.
Then, depending on what you were given at the start, you can start to calculate how much of good 1 would be pareto optimal for agent B to hold.
Forget about good 2 for quasi-linear preferences.
When is utility maximisation for a consumer achieved?
When his indifference curve is tangent to the budget set.
When does utility maximisation for both consumers occur?
When MRSa = MRSb = p1/p2
Remember, the negative ratio of prices is the slope of the budget set.
The MRS’ of both consumers’ is their indifference curves.
Utility maximisation is achieved when the indifference curve is tangent to the budget set, given that preferences are smooth, convex and internal.
What is the first fundamental welfare theorem?
In any competitive market economy, there will always be a general equilibrium that is Pareto efficient.
- holds true even in cases where there is not a need for convex, smooth preferences (see image), different derivation in this case.
What is the second welfare theorem?
This states that we can gain an equilibrium point (Pareto-optimal) by allocating/re-distributing wealth and then letting the ‘invisible hand’ do its work.
Redistributing wealth/ allocation after transfers can be performed by regulators through the use of taxes, subsidies etc.
Note: Here, preferences HAVE to be convex as we will be unable to gain a Pareto-optimal point once we consider the distribution of wealth (see the difference between slide 82 and slide 89).
Will a consumer want to increase or decrease his consumption of good 1 if p1/p2 < MRS?
Increase.
He is willing to trade 2 units of good 2 to get good 1 but now, he can get good 1 for giving up only one unit of good 2 and so thus his consumption of good 1 would increase.
Work out the following
- Make sure it is in CD form. Apply monotonic transformation if necessary.
- Using demand functions to get xa1 and xa2 (efficient)
- Equate these two to the total income (wa1+wb2).
- You can work out real interest rate from here. Remember that the price ratio is the same as 1+r. (p1/p2=1+r)
- Also write p2/p1 as p for simplicity in calculating feasible demands.
- Sub in p2/p1 into demand functions. If value is less than income (wa1 or wa2), then consumer is lender
