Questions on General Equilibrium

Due: Monday, April 6, 2020

The following questions are concerned with Overhead Set 8 on General Equilibrium.

1. Say that we have a two-person (Fred and Laurie) exchange economy with two goods (bread and cheese). The Edgeworth box for this economy is as follows:

(a) On this diagram (or your version of it) show how much bread and cheese each person has at the endowment point.

At pt. E: Laurie has CL cheese and BL bread and Fred has CF cheese and BF bread.

(b) Say that at the endowment point Fred has a Marginal Rate of Substitution (MRS) of 3 bread per cheese and Laurie has an MRS of 6 bread per cheese. (i) Illustrate this situation using the Edgeworth box diagram, i.e. add indifference curves to the diagram that go through the endowment point (make sure that their slopes are roughly consistent with the information you are given on the MRSs).

SEE DIAGRAM: Fred is on the difference curve UF1 and Laurie is on the indifference curve UL1. Notice that at the endownment point Laurie’s indifference curve is the steeper of the two since her MRS is 6 bread per cheese and Fred’s is 3 bread per cheese.

(ii) If Fred and Laurie can trade away from the endowment point in (i) which good will each of them want more of and which good will each of them want to give up. Explain. Laurie values cheese more highly than Fred (she will pay up to 6 bread per cheese while Fred will pay at most 3 bread per cheese) so Laurie will ‘buy’ cheese by giving up bread. Specifically, Fred and Laurie will both be made better off by any trade at terms between 6 and 3 bread per cheese (with Laurie trading bread for cheese and Fred trading cheese for bread).

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(iii) Show a Pareto optimal point that the parties might trade to and the amounts of each good traded if they were to move to that point.

Point T is a Pareto optimal point that the parties might trade to. At T Laurie has CLT cheese and BLT bread while Fred has CFT cheese and CFT bread. The arrows show the trades (note Laurie trades bread for cheese and Fred does the reverse).

(c) Say that the prices of bread (PB) and cheese (PC) are determined in competitive markets. (i) Write the equation for the budget constraint and graph it in the Edgeworth box diagram.

The equation and the graph are like those on pp.11-12 of the notes except Cheese takes the place of Clothing and Bread takes the place of Food.

(ii) Why must the budget constraint go through the endowment point? What is its slope? The person buys and sells from their endowment at the market prices.

(d) (i) Draw a diagram with an endowment point and initial indifference curves like those in part (b) but now add a budget constraint where at the current prices of bread and cheese both Fred and Laurie want to buy bread and sell cheese.

The budget line will have to be quite steep in order to get Laurie to sell cheese (her MRS is 6 bread per cheese so the Pcheese/PBread>6 for her to be willing to sell it (in other words for her to sell the market must pay her more than her MRS per cheese). The broken blue-purple line is such a budget line – Laurie wants to move from the endowment (E) to point g where the blue-purple budget line is tangent to her red indifference curve and Fred wants to trade from E to f (where the budget line is tangent to the higher blue indifference curve). The arrows on the axes are the desired trades at the going market prices. In this case the trades can’t be made since no one wants to sell bread and no one wants to buy cheese.

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(ii) What will happen to prices in (d) (i)? Say that prices now change so as to give general equilibrium. Illustrate this outcome.

Both are selling cheese so its price will fall due to excess supply. Both are trying to buy Bread so the excess demand raises its price. The slope of the budget line –(Pcheese/Pbread) will become flatter than that of the budge line in (d)(i). A possible outcome is like the following (where point T is the equilibrium outcome):

At T indifference curves are tangent to the budget line at a common point. So both Fred and Laurie are maximizing utility (tangency) and the desired trades are feasible since they wish to be at the same point in the box.

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2. The Second Fundamental Theorem of Welfare Economics says that any point on the contract curve can be supported by a competitive equilibrium. This question has you illustrate and explain this argument. Start with an Edgeworth box diagram for a two-person, two good economy. In your diagram assume that there is a total of 100 units of food and 100 units of clothing which will be split between two people (Person A and Person B).(a) Let’s say that the initial endowment is quite unequal giving Person A 90 units of both goods and Person B 10 units of each good. Illustrate this endowment point in the Edgeworth box.

(b) Say that if A and B trade from this endowment point they would trade to an efficient point where A has 95 units of clothing and 86 units of food while B has the remaining units of each good. Add indifference curves to your diagram in (a) to show this situation. SEE BELOW

(c) Add a contract curve to your picture (note that it must go through the point Person A and Person B trade to in part (b)). Draw the contract curve so that it just happens to also go through the point where A has 50 of each good and B has 50 of each good. SEE BELOW

(d) According to the Second Fundamental Theorem of Welfare Economics a government could use redistribution and markets to attain the outcome on the contract curve where each person gets 50 of each good. Explain how this could be done and illustrate.

A budget line can be drawn through the desired efficient point (where each person gets 50 of each good) where the slope of this budget line is the same as the slope of A and B’s indifference curves at that point. This budget line is the solid black line above. The slope of this budget line gives the relative price (Pclothing/Pfood) that could give this outcome as a competitive equilibrium. Redistribution of the initial endowments to any point on this budget line will give the desired result, i.e. once at a post-redistribution allocation on the budget line markets will determine the desired prices and A and B will trade to the midpoint of the box.

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3. Assume two industries (housing and wheat) and two inputs (labour and land). Say that the supply of labour (L) and land (T) is fixed at 10,000 workers and 8,000 acres respectively. At the initial allocation of inputs the housing industry has 5000 workers and 2000 acres of land while the wheat industry has 5000 workers and 6000 acres of land. Say that housing output is 4000 houses and wheat output is 20,000 bushels at this initial allocation. whic

(a) Let the wheat industry’s marginal rate of technical substitution (MRTS) at the initial allocation be 4 acres of land for 1 worker. What does this MRTS mean in words?

At the current allocation of inputs the wheat industry can keep output at its current level by hiring 1 more worker and reducing land input by 4 acres.

(b) Say that the MRTS in the housing industry is 2 acres of land for 1 worker at the initial allocation. Given the information in (a) and (b) is the initial allocation efficient? If not, how should labour and land be reallocated between the industries? Give a numerical example of such a reallocation and make sure that your example shows that the reallocation raises output in both industries.

It is not efficient since the MRTS’s differ between housing and wheat producers. The MRTS tells your that labour must be twice as productive as land in the housing industry (1 worker produces as much as 2 acres of land) while in the wheat industry labour is 4 times as productive as land (it takes 4 acres to replace 1 worker). Given that labour is so much more productive in the wheat industry, labour should be shifted from the housing to the wheat industry, e.g. shifting 1 worker from housing to wheat and 3 acres of land from wheat to housing will raise output in both industries. This reallocation gives Wheat 1 extra worker at a cost of 3 acres of land --- the wheat industry MRTS is 4 acres per worker so it could have given up 4 acres and kept output constant since it gave up only 3 acres wheat output must have risen. The housing industry MRTS was 2 acres per worker yet it gave up 1 worker and received 3 acres of land in return (2 extra acres would have kept housing output constant – getting 3 extra acres means output of houses has risen).

Note that it is the difference in MRTSs that implies inefficiency not the ratios of output to input as some people suggested.

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(c) Draw an Edgeworth production box showing the initial allocation, your proposed reallocation and the pre- and post- reallocation isoquants for each industry.

Start with 5000 labour in each industry and 2000 acres of land in housing and 6000 acres in wheat. The picture is drawn so that the slope of the Wheat isoquant is steeper than the slope of the Housing isoquants at this starting allocation (recall MRTS is higher in Wheat than Housing). The arrows on the horizontal axes show the reallocation of labour from the Housing to the Wheat industry and the arrows on the vertical axes show the reallocation of land from Wheat to Housing. This reallocation moves the economy into the area between the two original isoquants and so raises output in both industries (i.e. both industries are on higher isoquants relative to their origins after the shift of inputs between industries). Note that though my reallocation has raised output in both industries I haven’t drawn the new allocation as efficient (more reallocation could still raise output further).

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5. Say that an economy produces Food and Cars. The Production Possibility Frontier (PPF) for this economy is given in the following diagram:

Recall from the notes that a PPF can be derived from an Edgeworth Production Box. Say that the Food and Car industries use only two inputs capital (K) and labour (L) to produce their output (so K and L are on the axes in the Edgeworth Production Box). (a) Why might an economy end up at a point like ‘b’, i.e. under the PPF? Explain. In an Edgeworth Production Box show an allocation of K and L between the two industries that could give an outcome like point b.

Point b could occur if at the associated allocation of inputs between the two industries MRTSs were not equal across industries (i.e. a picture like the initial allocation of inputs back in question 3). Another possible cause of a point like b is that some inputs are unemployed.

(b) What is happening at a point like point ‘a’ in the diagram? Explain. In an Edgeworth Production Box show an allocation of K and L between the two industries that could give an outcome like point a.

Pt. a is an efficient allocation. Inputs have been allocated between industries so that MRTSs are equal (isoquants are tangent) – see pt. a in the production box diagram below.

(c) What is going on at point c? Pt. c is not feasible given current quantities of inputs and current production technologies.

(d) Say that the economy starts at pt. ‘a’ and that the Marginal Rate of Transformation at that point is 0.5 cars for 1 unit of food. If the typical consumer has a marginal rate of substitution of 1 car per unit of food how should inputs be reallocated between the car and food industry in order to achieve output mix efficiency? Explain why this change is necessary to achieve output mix efficiency and show in a diagram roughly where on the PPF the economy will locate after reallocation.

The MRS tells you that consumers value food and cars equally while the MRT tells you that it costs the economy 0.5 cars per unit of food, i.e. the cost of food is low compared to how consumers value it. So this economy can raise food output by 1 and lower car output by 0.5. If it then takes 0.5 cars away from the consumer and gives the consumer the extra unit of food the consumer will be better off (their MRS tells you that they could have given up 1 car for an extra food and still been just as well off as before the change but the consumer only had to give up 0.5 food – so the consumer must be better off).

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As the shift in output occurs along the PPF the economy will be shifting inputs from the car to the wheat industry.

So the economy is moving down the PPF (producing more food and fewer cars), e.g. from pt. a to pt h. In terms of the production box the economy would be moving inputs from the car industry to the food industry (see the production box panel, I’ve assumed that the inputs are capital (K) and labour (L)). The move from a to h on the PPF corresponds to the move from a to h in the production box while the arrows show the change in allocation of inputs.

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