1 www.lancaster.ac.uk/postgrad/murphys4 / ECON 100 Tutorial: Week 12/office: LUMS C85Note, Questions 1 and 2 are best viewed in “slideshow” mode so that animations are visible. Question 3 has additional notes in the notes section and is best viewed in “normal” mode.
2 Question 1(a)Explain, in detail, what is meant by a Edgeworth box in the context of a 2 good (1,2), 2 person (A,B) model of consumption. (5 marks)Ian’s Answer: An Edgeworth box shows the budget constraints, the consumption patterns of the two consumers for the 2 goods, and their associated (highest) utility levels. The sides of the box is fixed by the levels of production of the two goods.Here is a nice introduction/explanation for the edgeworth box:The point of an edgeworth box is to model a two-agent, two-good exchange economy without production. Each agent is given a utility function and the prices of the goods give a budget constraint. The solution is an example of a competitive equilibrium. It can be used to illustrate the Coase theorem and the general welfare theorems.
3 How to Draw the Edgeworth Box Let’s say we have two people, A and B. A has an initial endowment of 3 units of Good 1 and 2 units of Good 2. B has 2 units of Good 1 and 3 units of Good 2.When we draw the axes for our box, we want to show the total amount (A’s endowment + B’s endowment) of Good 1 on the X axis and the total amount of Good 2 on the Y axis.We graph A’s initial Endowment as a point.Note: Because A & B’s endowments sum to the total amount in the box, when we graph B’s endowment, we’ll find that it will be on the dot that we used to represent A’s endowment.
4 How to Draw the Edgeworth Box Budget Constraint:Our budget constraint is: 𝑃 1 𝑄 1 + 𝑃 2 𝑄 2 =𝑤, where wealth (w) is given by our initial endowment. The slope of our budget constraint is equal to the ratio of the prices of the two goods: 𝑃 1 𝑃 2 .In our case, we don’t have to be quite so technical, and we’re just going to draw in a budget constraint that isn’t quite to scale. The key thing is to make sure it is downward sloping (because both prices are positive) and that it is going through our initial endowment point.Note: Earlier we said that because A & B’s endowments sum to the total amount in the box, when we graph B’s endowment, we’ll find that it will be on the dot that we used to represent A’s endowment.Similarly, If both are facing the same prices (this is generally the case in this type of problem), then the line we use for A’s budget constraint will also represent B’s budget constraint.
5 How to Draw the Edgeworth Box Next, we can draw in the indifference curves.Starting with person A, If the indiff. curve is tangent to the budget constraint at the endowment point, it will look like this:This means that the endowment maximizes A’s utility, subject to the budget constraint. A is in equilibrium.Now let’s suppose that Person B also has an indifference curve that is tangent to the budget constraint at the point of B’s endowment. It would look like this:Because B has an indifference curve that is tangent to the budget constraint at the point of the endowment, B is also in equilibrium.If both agents are in equilibrium, we call this point a competitive equilibrium.We also can call itPareto Efficient.
6 Pareto efficient (PE) allocations I’ve added a few notes to Ian’s slide (Slide 22, lecture 31 to 33):Pareto efficient (PE) allocationsPosition of E depends on incomes of A and BAll ‘s are possible PE allocationsWhile is not PE – both A and B could be better offU*B0BU*AQ1Q20ANote: The blue line here is called the contract curve. It is a set of all possible Pareto Efficient points, it is a collection of points where A’s indifference Curves are perfectly tangent to B’s indifference curves.E
7 Pareto efficient allocations I’ve added a few more notes to Ian’s slide (Slide 22, lecture 31 to 33):Pareto efficient allocationsPosition of E depends on incomes of A and BAll ‘s are possible PE allocationsWhile is not PE – both could be better offWhen two indifference curves cross, as they do here, any point that is within the eye-shaped area (including the “eye-lids”) between those two curves (shaded in gold here) is a Pareto Improvement from the original point where those curves crossed.More specifically, points along the contract curve in this area are both a Pareto Improvement from the red dot and are Pareto Efficient.
8 Question 1(b)Show what the allocation of resources would be at some non-equilibrium prices – for example where the price of good 1 is too high and the price of good 2 is too low so that there is excess demand for good 2 and excess supply of good 1. (5 marks)Ian’s Answer: The dashed line shows a budget constraint that results in A and B’s demand for good 2 being greater than supply (the height of the box), and the overall demand for good 1 being less than available supply (the width of the box). There is excess demand for 2 of ED2 and excess supply of 1 of ES1.
9 Question 1(b) This graph shows a situation where the My notes:This graph shows a situation where theendowment is in disequilibrium. This isevident because A’s indifference curvethat is tangent to the budget constraint isNOT at the point of A’s endowment.Instead, it is at some other point. (we can only have one indifference curve that is tangent to the budget constraint for each person, because indifference curves cannot cross each other (for a given agent).)I have drawn in hypothetical indifference curve for A and B that actually do go through the endowment point. You can see that they are not tangent to the budget constraint.This means that each individual would increase utility by moving to a higher (further from their origin) indifference curve, until they are at one that is just tangent to their budget constraint.To get there, A would need to sell some of G1 and buy some more G2.
10 Question 1(b)My notes, ctd.:Likewise, B can sell some good 1 and buy good 2 and move down and to the right, to reach a place where he maximizes utility (where indiff. curve is tangent to budget constraint).The problem is they can’t both do that if there are no other agents in the economy to trade with.So at these prices the demand for good 2 is greater than the total level of good 2 in the economy, so there is excess demand for good 2. For a similar reason there is excess supply of good 1.
11 To take this a bit further, If only A and B are in this economy how can they deal with this disequlibrium and get to a place where both can maximize their utility?To get an idea of how this can happen, let’s temporarily ignore prices and take a look at A’s indifference curve that goes through the endowment point.A will increase utility if he can move to an indifference curve further away from the origin (shown by the arrows).The same is true for B.Both can increase their utility by consuming at any point in the gold region.There is a line which represent points where A and B’s indiff. curves are tangent to each other. This line is called the contract curve and is represented in grey.If the prices shift so that the price ratio intersects the contract curve within the gold region, then they can trade with each other, which increases both utilities compared to their utilities at E.This slide and the next one are my explanation of how we go from Ian’s diagram in Question 1(b) to Ian’s diagram in Question 1(c)
12 To look at this with more detail, Here is an example of two utility isoquants (indifference curves) which are tangent, so we can see that they define a point on the contract curve:This slide and the previous one are my explanation of how we go from Ian’s diagram in Question 1(b) to Ian’s diagram in Question 1(c)How can A and B get here from the original graph in 1(b)? They can negotiate prices so that their budget constraint (the ratio of prices) that goes through the endowment also goes through the gold-colored region, specifically, also through the contract curve.If we clean this graph up, we will have the same (not exactly to scale) graph as we see in the next slide for part 1(c).
13 To look at this with more detail, Here is an example of two utility isoquants (indifference curves) which are tangent, so we can see that they define a point on the contract curve:This slide and the previous one are my explanation of how we go from Ian’s diagram in Question 1(b) to Ian’s diagram in Question 1(c)How can A and B get here from the original graph in 1(b)? They can negotiate prices so that their budget constraint (the ratio of prices) that goes through the endowment also goes through the gold-colored region, specifically, also through the contract curve.If we clean this graph up, we will have the same (not exactly to scale) graph as we see in the next slide for part 1(c).
14 Question 1(c)Argue what would then happen to these prices in a competitive model and show (and explain) what the resulting competitive equilibrium looks like in this box. (5 marks)Ian’s Answer: In the face of ED2 and ES1 we would, in a competitive market system, expect the price of 2 to rise and the price of 1 to fall until the markets were in equilibrium and the Edgewoth box looked like:The two agents would negotiate new prices so that they can exchange and end up consuming in the gold region (from the previous two slides), so their utilities are tangent to the new budget constraint, so their utilities are tangent to each other, and so their consumption levels are on the contract curve.
15 Question 1(d)Explain, with reference to the diagram from part (c), why the competitive equilibrium is Pareto efficient (5 marks)Ian’s Answer: At the equilibrium (the black dot) the MRS for A = MRS for B because both consumers face the same relative prices (as in this diagram).The equilibrium satisfies Pareto efficiency because the economy cannot move from this point without making one or both individuals worse off.That is, no Pareto improvements over this equilibrium are possible – therefore, this equilibrium is Pareto efficient.Note also: The set of all such Pareto efficient points is called the contract curve.Ian’s Answer:
16 Question 2(a)Explain the Pareto criterion and why this may not, in practice, be very useful in evaluating the social welfare consequences of a policy change (2 marks)Ian’s answer: A “Pareto improvement” is a change that made at least one person better off and no-one worse off.“Pareto efficiency” is a situation where no Pareto improvement is possible.My examples:In practice most policy changes will involve some loss in utility for some group: For example it may be impossible to give to the poor without taking something from the rich. It may be impossible to improve the NHS without putting some people out of jobs, raising taxes, increasing wait time or making it harder to get certain prescriptions, etc. Even though these policies are not Pareto improvements, they can still provide more utility to society as a whole.Early childhood education is the surest way to lower crime, lower poverty, and raise the GDP for at risk groups and for the country as a whole… 20 years from now. So if someone dies before the children affected by the policy are grown, they miss the benefit but do have to help pay for the policy.
17 Question 2(b)Suppose a policy change involves a large number of winners and a small number of losers such that the financial gains outweighed the financial losses. Could you argue that the winners could then compensate the losers and still be better off themselves? If not, why not? (3 marks)Ian’s Answer: Just because the financial gains outweigh the financial losses does NOT mean that the gainers can compensate the losers.For example, suppose the winners were all rich and the losers were all poor then diminishing marginal utility of income might imply that the UTILITY gains to the financial winners might be small relative to the large UTILITY losses of the poor. In which case, the winners could NOT compensate the losers.Only if winners and losers were randomly selected (so, they were equally rich, on average) could you argue that compensation would work. Alternatively, if utility were a linear junction of income this could be assured.”My answer/example:In the example of early childhood education, we cannot compensate the losers because we do not know who might not live to see the benefits down the road.You could come up with an insurance market or loan program so the costs today could be paid by exactly those who benefit tomorrow. This would increase the demand for loans, increasing the cost of borrowing money, so even then some costs would be born by people today.This type of argument could be made for most any policy.Ian’s answer is more general: “Just because the financial gains outweigh the financial losses does NOT mean that the gainers can compensate the losers.For example, suppose the winners were all rich and the losers were all poor then diminishing marginal utility of income might imply that the UTILITY gains to the financial winners might be small relative to the large UTILITY losses of the poor. In which case, the winners could NOT compensate the losers.Only if winners and losers were randomly selected (so, they were equally rich, on average) could you argue that compensation would work. Alternatively, if utility were a linear junction of income this could be assured.”
18 Question 2(c) Two out of three prefer A to B. Suppose there are three individuals in a society(1, 2, 3). And there are three possible outcomesthat could be attained (A,B,C). Each individual’srankings of these three alternatives is given here:Explain why majority voting would giverise to a Condorcet Paradox. (5 marks)Ian’s Answer:Two out of three prefer A to B.Two out of three prefer B to C, and two out of three prefer C to A.Thus voting results in a non-transitive SOCIAL ranking of the three alternatives.A consequence of this Condorcet Paradox is that the order in which pairwise votes takes place effects the final outcome (show this).Thus, the outcome depends on the agenda – what’s decided first.Additional notes that I’ve added:The Condorcet Paradox is when collective preferences can be non-transitive), even if the preferences of individual voters are transitive.123FirstABCSecondThirdThe voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e., not transitive), even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.What does transitive mean? The transitive property is that if A>B, and B>C, then A must be > C.
19 Question 2(c)Ian’s Answer stated: A consequence of this Condorcet Paradox is that the order in which pairwise votes takes place effects the final outcome.123FirstABCSecondThirdLet’s look at what that means:What would happen if votes were taken on A vs B and then on A vs C?What is selected as the first choice/what is the overall winner?What if, on the other hand, votes were taken on A vs C and then on B vs C?What is selected as the first choice in this case?Do you see how the outcome depends on which vote is taken first? This is the Condorcet Paradox.The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e., not transitive), even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
20 Question 2(d)Suppose the rankings in part (c ) where replaced by the following:Explain what the social choice would be under majority voting. (5 marks)Ian’s Answer: B beats C, B beats A, and C beats A. So this is consistent:B wins, then C, then A.My note:In this scenario, each individual’s preferences are transitive and the preferences determined by social choice/the majority vote are also transitive. So, no Condorcet Paradox here.123FirstACBSecondThirdNote, in Ian’s solution, there was a typo, not sure if it was fixed before posting to moodle: He wrote, “Answer: C beats B, B beats A, and C beats A.”
21 Question 2(e)Explain what is meant by “single peaked preferences”. Are the preferences in parts (c ) and (d) single peaked? (5 marks)Answer: See MT pMy answer: Individual’s preferences for all three individuals in (d) are single peaked and for individuals 1 and 2 in (c). But 3 in (c) is double peaked.We can do social ordering for (d) but not for (c). The social ordering for (d) is single peaked.For an example of what Ian means by “single peaked preferences,” see:
22 Question 3(a)Suppose an economy was characterised by a monopoly in one industry. Suppose that industry exhibited constant returns so that MC was constant, and suppose the demand curve was linear. Show, using an appropriate diagram, what the equilibrium price and output would be. Show the profits of the firm and the deadweight loss (relative to the competitive price) (7 marks)
23 Question 3(b) Ian’s Answer: What you may not remember is Explain why a policy of regulating this firm toforce it to set p=MC would pass a cost-benefitcriterion. Indeed, show that the benefits ofsuch a policy would be exactly 50% greaterthan the costs of doing so. (7 marks)Ian’s Answer:What you may not remember isthat the MR curve has the sameintercept as the Demand curve,but double the slope.This implies that the monopoly output in this picture is exactly half of the competitive output. That means that the DWL triangle (which has the same height and base as the profit box) must have half the area of the profit box.
24 Question 3(b) Profit, =(Pm-Pc)*Qm DWL = ½ (Pm-Pc)*(Qc-Qm) Here’s the math/explanation behind Ian’s answer:Profit, =(Pm-Pc)*QmDWL = ½ (Pm-Pc)*(Qc-Qm)Since the slope of MR is twice the slope of D,the competitive quantity is exactly twice thatof the monopoly quantity: Qc = 2Qm. So:= ½ (Pm-Pc)*(2Qm-Qm)= ½ (Pm-Pc)*(Qm)= ½ The benefits of making this firm behave competitively is: + DWL = + ½ = 3/2 = 1.5 The costs are just , So the benefits are 50% greater than the costs!
25 Question 3(c)Since the gains of making this industry competitive are large relative to the costs, the gains from doing so outweigh the losses. It would pass a cost-benefit criterion. Indeed, consumers could collectively compensate the losers of such a policy and still be better off. Explain why it may be difficult to rely on such policy to make the economy efficient? (6 marks)The consumers would be better off by 3 DWL, and the firm would be worse off by 2 DWL.So the consumers could say to the monopolist:If you set p=MC we’ll give you 2 DWL so you’ll be no worse off, and we will end up being better off by 1 DWL. The gainers can compensate the losers!BUT, there may be costs associated with coordinating all of the consumers to pay this bribe.There might be a “free rider” problem – each consumer would be better off if she didn't contribute to the bribe, so the dominant strategy is to not cooperate. And the bribe doesn't get paid and everyone ends up worse off!This is similar to what Coase said about how establishing property rights and allowing a market in those rights can lead to pareto improvements. You should look up and think about the Coase theorem. One way to state such a mindset is that if legally binding property rights can be established and a market can be applied to a situation, a pareto improving equilibrium can be found. Note: there is no guarantee that the equilibrium will be truly optimal if agents are not fully rational and do not have full information.
26 Notes about the ExamFriday the 24th – Check your Timetable for time/locationBring a pencil, eraser, calculator and your library card #Please write neatly! If the graders can not read your answer you will not get points for it!50 minutes: 5 questions (each with parts a, b, … etc.)Exam material could be from weeks 6 – 12 (including 12)Questions will likely be similar to those in tutorials, particularly the last few weeks.(note: this is an educated guess on my part, not a guarantee!)It will likely take a few weeks for marks to come out.Good luck!