Economics Department

MICROECONOMICS II Course Materials – Part I

Application Exercises

with Answer Keys

by Pascal Courty

with assistance of Liliane Karlinger

European University Institute

Florence, Italy October 2004

Application Exercise 1: The Cranberry Market Pascal Courty

EUI, Florence, November 2003 The attached table presents data on surface area harvested, utilized production, and prices for processed cranberries. In contrast with fresh cranberries, processed cranberries are used in drinks, snacks and other food applications. Important features of supply and demand include: Demand Interest in cranberries started to grow in the late 70’s for two main reasons: (1) Some medical research indicates that cranberries can help prevent urinary tract

infections. (2) Consumers started to consume sweetened mixed cranberry drinks (i.e. cranapple). Supply Cranberries are farmed mostly in northern United States and Canada. (1) Growing cranberries typically requires sand peat soil along with low cost water. (2) Land conversion and waiting that bushes reach maturity takes a few years. Based on the available data, try to explain what has happened in the cranberry market. Identify the basic features of demand and supply that can explain the price increase throughout the 80’s and early 90’s and the subsequent price crash. You will be evaluated on your ability to tell a story which uses the concepts learned in class, which is internally consistent, and which is consistent with the evidence. In particular, there is no need to collect additional information and such information if provided will not receive additional credit.

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Utilized Production Price Total acres (10^6 barrels) $ per barrel harvested

10^3 Revenue

1977 2,1 20 missing 421978 2,4 23 missing 55,21979 2,4 28 missing 67,21980 2,6 33 missing 85,81981 2,5 43 missing 107,51982 3 42 missing 1261983 2,9 45 missing 130,51984 3,3 47 missing 155,11985 3,5 46 missing 1611986 3,7 45 missing 166,51987 3,3 45 missing 148,51988 4 46 missing 1841989 3,8 45 missing 1711990 3,3 47 missing 155,11991 4,2 49 28,4 205,81992 4,2 51,6 29,2 216,721993 3,9 50,2 29,4 195,781994 4,7 49,3 31,1 231,711995 4,2 53,4 32,8 224,281996 4,7 65,9 34 309,731997 5,5 63,7 35,7 350,351998 5,4 36,6 36,6 197,641999 6,3 17,2 37,5 108,362000 5,5 17,9 36,6 98,452001 4,8 22,9 34,3 109,92

Answer Key: The Cranberry Market Short-run versus long-run comparative statics

Pascal Courty/Liliane Karlinger EUI, Florence, November 2003

(1) The evolution of demand and price prior to the 1998 crash Demand: Short-run demand for cranberry juice products can be assumed to be fairly inelastic with respect to price: people consume cranberry juice mainly for health/lifestyle reasons (it is not particularly tasty), and so even a sharp drop in price is not likely to induce them to drink a lot more of it. From 1977 to 1997, overall demand increased steadily as more and more people became aware of the health-enhancing effects of cranberries. This can be represented as a gradual rightward shift of the short-run demand curve in the P-Q-plane. Supply: In the long run, there are no sunk costs, and entry is free; farmers can convert land to increase cranberry production, and more and more farmers will find it profitable to enter the industry as demand increases. Yet, the technology may not be exactly CRS, because cranberries will probably grow better in some regions than in others. Thus, one essential input (namely high-quality land) may be in fixed supply in the long run, and so the long-run supply curve will most likely have a positive slope because marginal land is less and less fruitful. (see equation 10.F.1 in MWG, p. 340) Hence, the evolution of price and quantity in the cranberry market from 1977 to 1997 can be understood as equilibrium outcomes due to a steep short-run demand curve shifting along a positively sloped long-run supply curve (see Figure 1).

Quantity

Price

Long-run supply

Short-run demand

19801990

19951997

Figure 1: Evolution of Demand and Price before the Crash of 1998

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(2) The price crash of 1998 Short-run Supply: In any given year, the quantity of cranberries that a farmer can harvest is limited by the amount of land she decided to devote to this crop and the number of bushes she planted several years ago. By the time the bushes mature, the cost incurred in converting the land and growing the bushes (eg fertilizer, pesticides, irrigation) is sunk, i.e. cannot be recovered if the farmer changes her mind and wants to grow a different crop. Thus, the variable cost of production is the cost of harvesting itself (i.e. picking the fruit from the bushes): this cost will probably be low compared to the (sunk) set-up cost, and will be increasing in quantity as more and more harvesting effort is needed to extract an extra unit of cranberries from the bushes. However, once the capacity limit is reached, it will be impossible to increase quantity any further, i.e. the marginal cost will be infinite. (compare with equation 10.F.2 in MWG, p. 340) Figure 2 depicts such a short-run supply function: we see that it is very flat at the beginning, then slightly increasing, and at some point becomes infinitely steep. We will now turn to explaining the price crash of 1998 by short-run comparative statics. Two possible lines of reasoning will be presented here. (a) A negative demand shock (Figure 2) Suppose that, in 1997, the industry was at a long-run competitive equilibrium: a number of J* firms operated in the industry, each at the efficient scale of operation, q*. Then, there might have been a drastic and unexpected drop in demand between 1997 and 1998, represented as a leftward shift of the short-run demand curve in the P-Q-plane. This shift in demand occurred at the “vertical” end of the short-run supply curve, resulting in a sharp price decrease that was not, however, matched by a corresponding reduction in quantity supplied. The years following the 1998 shock can be seen as first steps towards reaching a new long-run equilibrium, with the first farmers reconverting their land or exiting the industry otherwise. This story corresponds to Example 10.F.1 / Figure 10.F.5 in MWG, p. 340 f.; however, the documentation available on the cranberry market does not give us any hints as to whether and why such a demand shock might have occurred. Moreover, the argument relies on a somewhat crude division of dynamic adjustment into two periods (short and long run) which are treated in complete isolation from each other. Hence, let us now turn to an alternative, more sophisticated reasoning that can be very useful in explaining price crashes of the type we observed in the cranberry market for 1998.

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Quantity

Price

Figure 2: A negative demand shock in 1997

Short-run supply 1998

(marginal cost)

Short-run demand 1997

Short-run demand 1998

Price 1997

Price 1998

Average cost 1998

(b) Expected demand growth did not materialize (Figure 3) As we have argued in section (1), demand grew steadily from 1977 to 1997, and this demand growth was matched by a corresponding supply growth. Now, let us think of this demand growth as a stochastic variable, and argue that farmers hold (correct) expectations about the evolution of this variable. Accordingly, in order to match expected demand in a given year, they have to increase their productive capacity (plant cranberry bushes) well in advance. Now, suppose that in 1998, the actual realization of demand was below its expected value (say, because demand growth is slowing down, and demand is to settle at some long-run equilibrium level; after all, it does not seem reasonable that the cranberry market should grow for ever and ever!). Then, due to entry that occurred years ago (and was fully rational at that time!), aggregate short-run supply kept shifting to the right in 1998 (as it had done over all the previous years), while demand in that same year would not absorb this additional productive capacity, resulting in the observed price crash. Remark: Note that the argument in (a), ie a negative demand shock, implies that total quantity consumed decreases (or at most remains constant) after the shock, while under (b), the quantity consumed still increases. Unfortunately, the data provided in the problem set do not allow us to distinguish between these two predictions, and hence we cannot use the data to decide which of the two explanations is more valid. The data only show capacity utilization and harvested quantities. Note, however, that these figures are not identical with quantity consumed. Cranberry juice is a storable good, and so it might well be that even though quantity harvested still expanded after the price crash, these quantities were not brought to the (final consumer) market, but

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instead stored. In fact, inventories may have started to build up some years before the price crash already, thus aggravating the situation of excess supply experienced in 1998 (and after).

Q uantity

Price

Figure 3: Expected dem and grow th did not m aterialize

Short-run supply 1997

(marginal cost)

D emand 1997 and 1998

Price 1997

Price 1998

Short-run supply 1998

(new entry)

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Application Exercise 2: Rent Pricing in Shopping Malls Pascal Courty

EUI, Florence, November 2003 Try to interpret the attached table on pricing of space in shopping malls assuming the following: a developer typically sets up the shopping mall and rents the space to stores. There is enough competition between developers, so it is reasonable to assume that the rental market is competitive. 1-Use an externality argument to explain why department stores pay less than other stores, and food stores as well as jewelry pay the most. Is the externality you identified “private” (depletable) or “public” (nondepletable)? (for a definition of private/public externalities, see MWG, p. 364 f.) 2-The same stores that are present in a shopping mall are also present in high street locations. Do you think rent prices for high street stores would show the same patterns as the ones identified in shopping malls (i.e. would you expect that department stores would pay less)? Could this be one explanation for the growth of shopping malls? Store Group Median Rent Median Rent per sq.ft/

per square foot ($) Median Sales per sq.ft Department stores (owned) 1.95 0.015 Department stores (unowned) .87 0.005 Clothing and accessories 18.58 0.079 Shoes 22.00 0.085 Food service 32.41 0.095 Gift/specialty 22.00 0.088 Jewelry 42.00 0.076 “owned” department stores: the developer owns the structure, the department store leases space. “unowned” department stores: the department store owns the structure, often does not pay any rent or a much reduced rent.

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Answer Key: Rent Pricing in Shopping Malls Internalizing demand externalities

Pascal Courty/Liliane Karlinger EUI, Florence, November 2003

(1) Demand spillovers and differential rents in shopping malls Consumers are attracted to malls because of the presence of well-known anchor stores – invariably department stores with recognized names. Consumers reduce search costs by shopping at malls. However, the agglomeration of stores inevitably creates free-rider problems, because the success of a mall store depends, in part, on the presence of other stores within the mall, especially on the presence of the mall’s anchor stores. By generating mall traffic, anchors create external economies by indirectly increasing sales and/or reducing promotion and other costs of a host of smaller mall stores. Lesser-known stores can free ride off the reputations of anchors. If these externalities are important, competition among mall developers will internalize these demand spillovers by giving rent subsidies to anchors and by charging higher rents to mall tenants that benefit from spillovers. Note, however, that these rent subsidies are only a very indirect (and, hence, imperfect) remedy for the externality. The “textbook solution” to our problem would be to subsidize the externality-producing activity at the source (see MWG, p. 355 f.), i.e. create a “market for buyers attracted by other stores”: whenever a buyer arrives at the mall with the intention to buy at shop A, but ends up shopping at B as well/instead, then B would pay a “reference fee” to shop A for having generated this buyer’s demand. There are markets which work like this (e.g. among retailers on the internet which refer customers to each other), but in the case of shopping malls, the transaction costs for such a market would probably be too high. Thus, as a “second-best” solution to this missing-market problem, the stores that presumably generate the largest net traffic receive the rent subsidies discussed above. Note that these differential rates are of course independent of advertisement efforts or any other action single shops may take to increase (own and spill-over) demand; thus, they do not fully internalize the demand externality at hand, as they will not in general provide the right incentives to advertise. Now, are the demand externalities discussed above “public” or “private” in nature? Recall that an externality is “private” if the experience of the externality by one agent reduces the amount that will be felt by other agents. Now, say that a consumer was initially attracted to the mall by a big department store, and then decides to have lunch in the mall’s food court. Then, this same consumer might very well decide to buy something at a gift shop or shoe shop afterwards, and so the externality (additional demand) felt by the food court does not reduce possible demand spillovers for third shops. Thus, the demand externality is not perfectly private (as it is in the example of internet retailers mentioned above: if Google refers a customer to Amazon, that same customer will not be referred to, e.g., CDSource). Yet, the externality would only be

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perfectly public if after arriving at the mall, each consumer would go to all shops in the mall. But if the consumer already bought a sweater at GAP, chances are he will not buy another one at Eddie Bauer’s. So, as we do not know where exactly a customer will end up shopping, i.e. actual sales are unpredictable (for the shops at least), the externality may be considered ex-ante public, even though ex-post it has some features of a private externality. (2) Demand externalities and the decline of high streets Presumably, department stores located in high streets create similar demand externalities as the ones in shopping malls. Some externalities are already internalized within a conventional department store, since a department store resembles a collection of smaller stores offering a variety of goods and services but under single ownership. Nevertheless, some externalities between firms in the high street remained untapped because of diseconomies to department store size. As a consequence, a lesser-known store or the owner of the land benefited by being near a well-known major department store. In a few cities, specialized developers did purchase large blocks of central city land and then offered leases to department stores and other stores at differential rents. More often than not, however, the rents on high streets are flat with respect to store type because ownership of space on high streets is dispersed, and so the major department stores on high streets were NOT compensated for the externalities they created. Hence, it is not surprising that they were eager to leave the high street for malls located elsewhere in the city or in the suburbs. Thus, the inability to internalize externalities may also have contributed to the decline of the central high streets, even though from an overall welfare point of view, high streets may be superior to shopping malls (no car needed, traffic and pollution reduced, ...). For a detailed treatment of the issues discussed above, have a look at: Pashigian, B. Peter / Gould, Eric D. (1998): “Internalizing Externalities: The Pricing of Space in Shopping Malls”, Journal of Law and Economics, vol. XLI, April 1998, p. 115 - 142

Application Exercise 3:IKEA price patterns across countries

Pascal Courty/Liliane KarlingerEUI, Florence, November 2003

IKEA, the Swedish household furniture retailer, produces at total of about12,000 products, and sells them in a total of 140 stores in twenty-nine countries.Product designs are unique to IKEA, thus the goods are di¤erentiated. Yet,IKEA has a number of similarly-sized competitors producing more or less closesubstitutes, and so it seems reasonable to classify the products examined hereas di¤erentiated products sold in an imperfect competition environment.

The following table shows IKEA retail prices for 3 types of mirrors in di¤erentEuropean countries. Alg is a two-pack of a very simple square mirror tile,Guldros is a fairly simple round mirror with beveled glass, and Krabb is a wavymirror tile. While Alg and Guldros have no distinctive features and similarproducts are available in competing stores, Krabb is quite distinct.

Table: Final Retail Selling Prices 1998 (in US$)Alg Guldros Krabb

Austria 24 113 48Belgium 22 111 28Denmark 13 119 34Finland 15 107 21France 21 100 33Germany 22 97 51Italy 23 79 44Netherlands 20 101 20Norway 12 82 27Spain 32 112 34Sweden 15 94 24Switzerland 19 67 27United Kingdom 25 115 30

Note that the products o¤ered in these countries are identical in terms ofboth design and country of origin, and that the …nal retail selling prices listedabove are the local currency catalogue prices, converted into a common currency(US$).

We see that prices for a given type of mirror vary substantially across coun-tries: for instance, prices for the Alg mirror run from the equivalent of 12 US$in Norway to 24 US$ in Austria and 25 US$ in the UK. Conceptually, the …nalretail selling price of a product may be decomposed into the import price of theproduct, the local distribution cost (including di¤erences in VAT rates acrosscountries), and the pro…t mark-up.

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1 - Based on the information given in the table, do you think that the pricedi¤erences we observe can be fully explained by di¤erences in transportationand local distribution cost? Why or why not?

2 - Let us now focus on the Krabb mirror, and assume for simplicity thatIKEA is a monopolist for this speci…c mirror design. Consider two countries,say A(ustria) and B(elgium), where aggregate demand in A is given by xA (p) =a¡ θAp, and aggregate demand in B is given by xB (p) = a¡θBp, with θA < θB .The cost of production is c < a

θBper mirror. This cost already contains all

production, transportation and distribution costs (and VAT), which are assumedto be identical for both countries. Assume also that, for an individual consumer,transportation costs from A to B (and vice versa) are very high, so that the twomarkets are segregated.

2a - Which price will IKEA optimally charge in Austria? Which one inBelgium? Express the monopolist’s relative price-cost margin in each marketin terms of demand elasticities, and relate your …ndings to Exercise 10.E.3 (b)(the ”Ramsey tax problem” of problem set 1).

2b - Suppose now that the European Commission prohibits IKEA from ”dis-criminating” (ie charging di¤erent prices to Austrian and Belgian consumers).What is IKEA’s pro…t-maximizing uniform price? Under which condition willboth countries be served by IKEA? Which pricing regime (uniform or discrimi-natory) will be preferred by Austrian consumers? by Belgian consumers?

2c - Compare the solution obtained under price discrimination with thatof uniform pricing: What happens to IKEA’s pro…ts? What happens to itstotal output? What can you conclude about the change in aggregate welfare(measured as the sum of Marshallian surplus over both markets) if only onecountry is served under the uniform price regime?

2d - (HARDER) Suppose that IKEA has produced some total level Q of themirror. What is the welfare-maximizing way to distribute it between Austriansand Belgians? Use this result to make a statement about the change in aggregatewelfare if both countries are served under the uniform price regime.

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Answer Key: IKEA price patterns across countriesMulti-market price discriminationPascal Courty/Liliane KarlingerEUI, Florence, November 2003

(1) Local distribution cost vs. country-speci…c mark-ups

The price di¤erences observed cannot be explained by di¤erences in trans-portation and distribution costs alone. If this was the case, then we shouldobserve a consistent pricing pattern across countries, ie all types of mirrorsshould be uniformly more expensive in some countries than in others. For in-stance, mirror Alg is much more expensive in Austria than in Denmark (24 vs.13 US$): if this price di¤erence is due to transportation/distribution costs alone,then why is mirror Guldros actually cheaper (!) in Austria than in Denmark(113 vs. 119 US$)?

This suggests that price di¤erences also re‡ect di¤erences in mark-upsacross countries. This evidence supports the following presumptions: (i) in somecountries, IKEA can raise the price above marginal cost (for certain types ofproducts), i.e. IKEA has market power; and (ii) IKEA seems to practicemultimarket (or ”third-degree”) price discrimination: If markets aresegregated (eg by high transportation costs) so that arbitrage across marketscannot take place, then a monopolist can charge higher prices in countries wherethere is more demand for its product.

For IKEA products, it is relatively unlikely that consumer arbitrage takesplace on a substantial scale. Price di¤erences could however be exploited bycompeting retailers to in‡uence their competitive position. Indeed, evidencepresented in Haskel/Wolf (1999) indicates that such arbitrage or competitivepressures helped narrow down the observed price di¤erences over time.

(2) Third-degree price discrimination

(2a) For each market i = A, B , the monopolist will choose the price pi tosolve

maxX

i=A,B

(pi ¡ c) (a ¡ θipi)

which yields p¤i = 1

2

³aθi

+ c´

for i = A, B . From Exercise 12.B.1, we know thatthe monopolist’s relative price margin must be equal to the inverse of the priceelasticity of demand, ie.

p¤i ¡ cp¤

i=

1εp¤

i

for i = A, B

Hence, optimal pricing implies that the monopolist should charge more in mar-kets with lower elasticity of demand. This rule explains why the prices of goodsin di¤erent countries sometimes do not re‡ect transportation costs and import

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taxes, why students and senior citizens are given discounts by private …rms withno redistribution intention, why legal and medical services are priced accordingto the customer’s income or amount of insurance coverage, and why …rst-timesubscribers to a magazine are given discounts.

We already encountered this concept (the so-called ”Inverse ElasticityRule”) in a di¤erent context, namely in Exercise 10.E.3, where a govern-ment had to choose L ¡ 1 speci…c taxes, one for each commodity traded in the(perfectly competitive) economy. There, we found that the optimal tax rateshould be proportional to the inverse demand elasticity, i.e.

t¤i

p¤i

=p¤

i ¡ ci

p¤i

=µ

1 ¡ µ1

εp¤i

for i = 2, . . . , L

Note the interesting analogy between the two problems: a government rais-ing a certain tax revenue through speci…c commodity taxes while minimizingthe distortive e¤ect of these taxes behaves in some sense like a multi-marketmonopolist who tries to maximize pro…ts.

Recall that the government’s FOC (for a general demand function xl (pl))was

(µ ¡ 1)xl (pl) + µ (pl ¡ cl )x0l (pl) = 0

while the monopolist’s FOC (again for a general demand function xl (pl)) reads

xl (pl) + (pl ¡ cl)x0l (pl) = 0

Thus, the allocation implied by the solution to the tax problem will not ingeneral coincide with the monopoly outcome: after all, the government doesnot seek to maximize its tax revenue, it just wants to meet a certain requiredlevel of revenue (and so, in general, we have µ 6= 1

2). However, the two problemsdo coincide if µ = 1

2 , in which case the government’s tax revenue will be exactlyequal to the monopolist’s pro…ts (and so the latter can be seen as an upperbound on the former).

(2b) If the monopolist will serve both markets, it will choose pu to solve

max (pu ¡ c) [(a ¡ θApu) + (a ¡ θBpu)]

which yields p¤u = 1

2

³2a

θA+θB+ c

´. However, since Austrians are willing to pay

higher prices than Belgians for the same quantities (θA < θB ), it may be betterto charge a price above a

θB(where a

θBis the ”prohibitive price” in the Belgian

market, ie the price at which xB (p) = 0). In fact, if IKEA makes higher pro…tsserving only the Austrian market (at monopoly price p¤

A = 12

¡a

θA + c¢) than

serving both markets at uniform price p¤u < p¤

A , then IKEA will stop servingthe Belgian market altogether. Thus, both market will be served i¤

¦u = (θA + θB )·

12

µ2a

θA + θB¡ c

¶¸2

> ¦A = θA

·12

µaθA

¡ c¶¸2

2

Now, we can conclude that high-valuation consumers (here: Austrians)will always prefer uniform pricing over discriminatory pricing (since p¤

A >p¤

u, they cannot be worse o¤ - but may be better o¤ - if IKEA is forced to chargeuniform prices). However, low-valuation consumers buyers will be preferdiscriminatory pricing: since p¤

u > p¤B , they will either have to pay higher

prices, or they might not even be served anymore under uniform pricing.

(2c) Recall that the monopolist’s price-cost margin should be equal to theinverse of the demand elasticity in each market. Thus, charging the same pricein both markets A and B will in general be suboptimal, ie IKEA is madeworse o¤ if it cannot price-discriminate. As for its output, we have

(i) under price discrimination: QPD = xA (p¤A) + xB (p¤

B) = 12 (a ¡ cθA) +

12 (a ¡ cθB)

(ii) under uniform pricing: if only market A is served, we have QU =xA (p¤

A) = 12 (a ¡ cθA), and if both markets are served, we obtain QU = xA (p¤

u)+xB (p¤

u) = a¡ 12c (θA + θB). But this is exactly the amount we have under price

discrimination, i.e. QPD = QU if both countries are served under uniformpricing!

Now, if uniform pricing implies that market B will not be served, then bothIKEA and consumers in both markets are (weakly) better o¤ if the EuropeanCommission allows for price discrimination, as IKEA would then open theBelgian market without changing the price in the Austrian market.

(2d) Given quantity Q, we maximize consumer surplus by solving

max

qAZ

0

pA (x)dx +

qBZ

0

pB (x)dx

s.t. qA + qB = Q

Let the Lagrangian multiplier be denoted by λ, and conclude from the FOCsthat pA (qA) = pB (qB), ie marginal rates of substitution across markets A andB should be equal at the social optimum. But now, we can draw an interestingconclusion for the welfare e¤ects of the two pricing regimes: we saw in (2c)that if both markets will be served under uniform pricing, then aggregatequantity produced by IKEA will remain unchanged1 . Hence, the pricingregime will only decide about the distribution of this total output among thetwo markets.

Now, under uniform pricing, we have (by de…nition) pu (qA) = pu (qB), whileunder price discrimination, p¤

A (qA) 6= p¤B (qB ). But we just derived that at the

social optimum, pA (qA) = pB (qB ). Thus, if both markets will be served underuniform pricing, then price discrimination is welfare-decreasing.

As a …nal conclusion, we see that price discrimination has ambiguous wel-fare e¤ects: it may either increase welfare (if only one market would be served

1 Note, however, that this result depends crucially on the functional forms chosen for de-mand and production (ie linear demand and CRS technology).

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under uniform pricing) OR decrease welfare (if both markets continue to beserved under uniform pricing).

___________________________________________

For a detailed treatment of the issues discussed above, have a look at:Haskel, Jonathan / Wolf, Holger (1999): "Why Does The ’Law of One Price’

Fail? A Case Study", CEPR Discussion Paper no. 2187, July 1999.

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Application Exercise 4: The 2x2 Production Model with international trade

Pascal Courty EUI, Florence, November 2003

The goal of this exercise is: (1) to extend the 2x2 Production Model presented in MWG, Section 15.D, by adding

a second country to it (the 2x2x2 model), thus allowing us to use the results of MWG, 15.D, to derive predictions on international trade patterns. (Note: Assume that all results presented in MWG, 15.D., are correct; when using them here, you don’t need to prove them again!)

(2) to take these predictions to the data, along the lines of Leontief (1953). Consider two economies, call them “home” and “abroad”. Initially, there is no trade between these two countries, i.e. they are both at an “autarky” equilibrium. In particular, each country is endowed with two factors (call them L for home labor endowment, K for home capital endowment, and L*, K* for the corresponding abroad endowments). These inputs are used to produce two types of goods, call them good 1 and good 2. The price for one unit of labor is denoted by w (wage), and the price for one unit of capital is r (rental rate). Production of good 1 is labor intensive, and production of good 2 is capital intensive (in the sense of Def. 15.D.1, i.e. aL1(w,r)/aK1(w,r) > aL2(w,r)/aK2(w,r) at all factor prices (w, r)). We also assume that factors are perfectly mobile across industries, and that both factor markets and product markets are perfectly competitive. The two countries have identical CRS technologies of producing these two goods, and the representative consumers (one in each country) have identical homothetic preferences over goods 1 and 2 (labor and capital do not have any consumption value to consumers). These preferences can be translated into demand functions Zi(.) (and Z*

i(.)) for home (and foreign) demand for good i=1,2. However, countries do differ in their factor endowments: the home country is labor abundant relative to the foreign country, so that L/K > L*/K*. (For simplicity, assume L = L* and K* > K). The autarky equilibrium in each country is interior, i.e. both countries produce a strictly positive amount of both goods (call these outputs Yi(.) (and Y*

i(.)) for home (and foreign) output of good i=1,2). Let us now introduce free international trade in the two goods, while the factors continue to be immobile across countries. Trade is balanced (i.e. imports of one country equal exports of the other), and the new (free-trade) equilibrium will be an interior equilibrium (i.e. each country continues to produce both goods).

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2

(1) Theoretical Predictions (1a) Denote by pa = p1

a/p2a the equilibrium relative price of good 1 and 2 under

autarky at home (and pa* the one abroad), and call p the (common) equilibrium price under free trade. How can they be ordered, ie which one is the highest/second highest/lowest? (Hint: first, compare pa and pa*; assume they are equal, and show that this leads to a contradiction; develop your arguments from this result) (1b) Given free trade, characterize excess demands for good 1 and 2, both at home and abroad, ie determine the sign of zi(p) = Yi(p) – Zi(p) and zi

*(p) = Y*i(p) – Z*

i(p) for good i=1,2. Use this result to make your predictions for the following: What is the pattern of trade in goods between the two countries? In particular, which good will the home country export, and which one will it import? (1c) Compare equilibrium factor prices under autarky ((wa,ra) and (wa*,ra*)) and free trade ((w,r) and (w*,r*)). Will home (foreign) wages increase or decrease after allowing for free trade? What happens to home and foreign rental rates under free trade, compared to their autarky values? (1d) Under a free-trade equilibrium, denote by good i the export good of country 1, and by good j the import good of country 1. How does the relative factor intensity of country 1’s export good compare to that of its import good, ie is aLi(.)/aKi(.) larger or smaller than aLj(.)/aKj(.)? (2) Taking the model to the data The following table shows the amounts of labor and capital used in the production of $1 million of US exports and imports for the year 1947. Take for granted that the US was capital-abundant (relative to the rest of the world) in 1947.

Exports Imports Capital ($ million) $ 2.5 $ 3.1 Labor (person-years) 182 170

(2a) Which one of the predictions you derived in (1a) to (1d) can be tested using the data contained in this table? (2b) Do the numbers in the table corroborate or contradict this prediction? (2c) If the data contradict your prediction, what are possible explanations for this contradiction? Discuss briefly. (2d) Is there another prediction (from the list (1a) to (1d)) that would be conducive to testing? What kind of data would you need to have to perform such a test? Describe briefly how you would set up this test.

Answer Key: The 2x2 Production Model with international trade

The Heckscher-Ohlin Theorem Pascal Courty/Liliane Karlinger EUI, Florence, November 2003

(1) Theoretical Predictions: the Heckscher-Ohlin Theorem In order to derive the pattern of trade between the two countries, we proceed by first establishing what the relative product price is in each country under autarky. As we shall see, the pattern of autarky prices can then be used to predict the pattern of trade: a country will export the good whose free trade price is higher than its autarky price, and import the other. (1a) Claim: pa* > p > pa Proof: The autarky home equilibrium is established where an indifference curve of the representative consumer is tangent to the home PPF. Thus, the equilibrium autarky relative price will be some pa = p1

a/p2a. Now, suppose pa = pa*, and let’s see whether

this assumption leads to a contradiction. (i) assumed identical technologies across countries + interiority of autarky

equilibria use factor price equalization theorem: (wa,ra) = (wa*,ra*) (ii) under pa = pa* and (wa,ra) = (wa*,ra*), with L = L* and K* > K by assumption

use Rybcszynski theorem: Y*2(pa*) > Y2(pa*) and Y*

1(pa*) < Y1(pa*) (iii) by identical homothetic consumer perferences: foreign consumers will demand

goods 1 and 2 in the same proportions as home consumers, ie Z*2(pa*) =

kZ2(pa*) and Z*1(pa*) = kZ1(pa*) where k>1 is some constant

(iv) If pa clears home markets, then pa* = pa cannot clear foreign markets! We have: z1

*(pa) = Y*1(pa) – Z*

1(pa) < 0 and z2*(pa) = Y*

2(pa) – Z*2(pa) > 0. By

convexity of the PPF and preferences, z1*(.) is increasing in relative price, and z2

*(.) is decreasing in relative price foreign markets will clear for some pa* > pa. Under free trade, we have common relative price, called p. We just argued: z1(pa) = 0 and z1

*(pa) < 0. Thus, world excess demand for good 1 at pa is z1(pa) + z1*(pa) < 0.

Conversely, for pa*: z1(pa*) + z1*( pa*) > 0. By continuity of excess demand functions,

there must be a price p, with pa* > p > pa, such that z1(p) + z1*(p) = 0. This is the

equilibrium price with free trade. By Walras’ Law, p will clear world market for good 2 as well.

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(1b) Claim: z1(p) > 0, z2(p) < 0 and z1*(p) < 0, z2

*(p) > 0 Proof: This follows from pa* > p > pa and continuity of excess demand functions. Thus, country 1 (the labor-abundant country) will export good 1 (the labor-intensive good) and import good 2 (the capital-intensive good), and vice versa for country 2. This result is known as Heckscher-Ohlin Theorem: Each country will export the good that uses its abundant factor intensively. (see also (1d)) (1c) Claim: (w,r) = (w*,r*), and wa* > w > wa, while ra* < r < ra Proof: Identical technologies + common relative price under free trade apply factor equalization theorem: (w,r) = (w*,r*) For results regarding autarky factor prices: under factor intensity assumption (ie aL1(w,r)/aK1(w,r) > aL2(w,r)/aK2(w,r) at all factor prices (w, r)) + interiority of autarky and free trade equilibria + p > pa apply Stolper-Samuelson theorem: w > wa and r < ra. The reverse argument applies for (wa*,ra*). Thus, in each country, the price of the factor used intensively in its export good will rise, the other factor price will fall, after we allow for free trade. (1d) Claim: aL1(.)/aK1(.) > aL2(.)/aK2(.) Proof: We showed in (1b) that country 1 will export good 1 and import good 2. By the factor intensity assumption, we have aL1(w,r)/aK1(w,r) > aL2(w,r)/aK2(w,r) at all factor prices, and hence in particular at free-trade equilibrium factor prices. Hence, country 1’s export good will be labor-intensive, while its import good will be capital-intensive. To summarize: the Heckscher-Ohlin model tries to explain trade patterns as the result of differences in factor endowments, while assuming that all countries have access to the same technologies. As it turns out, this model performs rather poorly as a predictor of actual trade patterns, and we will now turn to one classical example of such a failure of the HO model, which is known in the trade literature as “Leontief’s Paradox”. (2) Confronting the Heckscher-Ohlin model with data (2a) The numbers presented in the table can be found in Leontief (1953), who was the first to confront the Heckscher-Ohlin model with data. The table says that to produce 1 “unit” of exports (here: $1 million of exports), the US used $2.5 million of capital and 182 person-years of labor. Thus, we can compute the ratio of these two numbers to obtain the capital-labor ratio for US exports: in 1947, each person employed in producing exports was working with $ 13,700 worth of capital, while the capital/labor ratio in imports was $ 18,200. These two numbers correspond to the relative factor intensities of US exports and imports, ie we are testing hypothesis (1d).

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(2b) In terms of our model above, the US corresponds to country 2 (the capital-abundant country), while the “rest of the world” corresponds to country 1 (the labor-abundant country), ie Kus/Lus > Krow/Lrow. Thus, adapting hypothesis (1d) to the situation of the US, we would expect its exports to be more capital-intensive than its imports, ie aK,US-ex(.)/aL,US-ex(.) > aK,US-im(.)/aL,US-im(.). However, the data contradict this prediction: aK,ex(.)/aL,exi(.) = 13,700, while aK,im(.)/aL,im(.) = 18,200. Remarkably, this is in fact higher than the capital/labor ratio found for US exports. This finding came to be called “Leontief’s Paradox”. (2c) A wide range of explanations have been offered for this paradox, either criticizing the assumptions of the HO-model or the measurement and testing strategies used by Leontief: ♦ US and foreign technologies are probably not the same; ♦ By focusing only on labor and capital, Leontief ignored land; ♦ Labor should have been disaggregated by skill; ♦ The data for 1947 may be unusual, since World War II had just ended; ♦ The US was not engaged in free trade, as the HO model assumes; ♦ factors are not perfectly immobile, as the HO model assumes (capital can be

exported and imported, e.g. through foreign direct investments, and workers can immigrate and emigrate)

♦ The implications of the HO model in a world of (strictly) more than 2 factors/countries/goods are different from the predictions of the 2x2x2 model (see Leamer, 1980, for the “factor content” version of the HO model).

(2d) Testing Prediction (1d) does not seem to be the most straight-forward test of the Heckscher-Ohlin model; Leontief’s choice was probably determined by the (limited) data material available to him. Now, as an alternative to (1d), Prediction (1c), ie the claim that factor prices should equalize across countries engaged in free trade, could lend itself very well to testing the HO model. We would need observations on wages (possibly disaggregated by skills) and rates of return on physical capital across various countries (that are trading partners) over a certain time horizon. We will probably find that this prediction will be contradicted by the data as well.

Application Exercise 5:Risk-sharing in medieval villagesPascal Courty/Liliane KarlingerEUI, Florence, November 2003

The typical English village in the 14th century can be thought of as a closedeconomy, with little or no grain exports or imports and little or no labor mi-gration. Farmers grew three types of grain, namely wheat, barley, and oats.Besides, most households had a small garden where they would grow some veg-etables. The most important consumption good was food, and a household’sfood consumption would consist of the harvest from their …elds and the yield oftheir garden.

Crop harvests ‡uctuated considerably both across time and space, and weredetermined by unpredictable events like droughts, hailstorms, fungoid diseases,or depredations by birds and insects.

Let us …rst look at variations across time (see Table 1): with a coe¢cientof variation of .35, available output would fall below half of mean value everytwelve years or so. This shortfall is something historians associate with disaster.In fact, English villages su¤ered from famine at roughly this frequency.

Table 1: Average Coe¢cient of Variation for Three Cropson the Winchester Demesnes, 1335-49

Wheat Barley OatsAverage Coe¢cient of Variation .42 .35 .55Standard Deviation of the Average .09 .12 .15Standard Error of the Average .02 .02 .03

As far as spatial ‡uctuations are concerned, it is interesting to note thateven very nearby places would experience sizable variations in their harvest, asis shown in Table 2 below.

Table 2: Correlations between Yields in Close and Far VillagesWinchester Demesnes, 1335-49

Wheat Barley Oats AverageAverage R (distant villages) .55 .15 .38 .38Standard Deviation .18 .23 .90 .09Average R (near villages) .68 .57 .66 .66Standard Deviation .15 .15 .22 .09Signif. Level of Di¤erence .074 .001 .01 .0001

There are several explanations for this phenomenon:

² Soil and topography can vary considerably over a two- to three-squaremile typical village. Relatively clay soils are retentive of rain and do wellin a dry year, but fare poorly in a wet one.

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² late frost in England is frequently spotty in its incidence, particularlyacross slightly di¤erent altitudes.

² hailstorms damage crops in an area a mile or two long and only a fewhundred yards wide.

² fungoid diseases are local in e¤ect as well, with spores spread in a randomway by wind and rain.

² lay of the land and its exposure to winds also determine sensitivity tovarying weather patterns.

Considering this high exposure to risk, it may seem surprising that there waslittle or no storage of grain (apart from what was meant to be used as seed):estate accounts show that grain still in storage at the time of harvest (termed”carryover”) were often zero, suggesting that large-scale grain storage was nottechnically feasible.

The same records also contain no evidence whatsoever for borrowing andlending between lord and villagers, though uses of the lord’s grain for seed, live-stock, household sta¤, and feudal obligations, were all recorded. This absenceof a …nancial institution seems surprising as well.

Finally, Figure 1 (enclosed) reveals that landholdings in the typical medievalvillage were characterized by a striking geometric pattern of long narrow strips.The holdings of family 1 in the graph are shaded black, showing how fragmentedtheir land was, with many separated holdings throughout this three-square milevillage. Their holdings are typical of all other families in that village.

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(1a) Based on the information above, how was risk accommodated for inthe medieval English village?

(1b) Name at least one more possible way of spreading risk, and makeconjectures about why this mechanism was not used in the village economy.

(2) Let us now consider a one-period pure exchange economy subject touncertainty of the following type: the village consists of just two farmers, j =1, 2. There are two types of land, e.g clay land and sandy land, denoted byk = 1, 2. Household 1 owns all the land of type 1, and household 2 owns all theland of type 2. Under autarky, each farmer eats the produce from her own land.Let cj denote the units of food consumed by household j, and let j ’s utilityfunction over cj be de…ned by

Uj (cj ) =(cj)

d

(1 ¡ d)d

where parameter d satis…es d 6= 0 and d 6= 1.Denote by ej (ε) household j’s endowment (ie harvests from j’s land) as

a function of some publicly observed vector of shocks, where ε = (ε1, ε2, ε3)represents all possible states of the world (e.g. state 1 is ”fungoid disease”, state2 is ”hailstorm”, state 3 is ”insect depredation”). Denote a typical element ofε by εs , where s = 1, 2,3, and let the probability of event εs be denoted byπs .Table 3 reports the event-contingent harvests from the two land types.

Table 3: Event-Contingent Payo¤s (Harvests) from Two Land Typesclay land sandy land total harvest

e1 (ε) e2 (ε) e (ε) = e1 (ε) + e2 (ε)Event ε1 10 2 12Event ε2 6 4 10Event ε3 2 5 7

(2a) Compute the coe¢cients of absolute and relative risk aversion for thisutility function. Comment brie‡y. (For a de…nition of the coe¢cients of absoluteand relative risk aversion, see MWG, Chapter 6, p. 190 and 194)

(2b) Suppose that before the state of the world is revealed, there is aplanning period t0. Set up the social planner’s problem as one of maximizingboth agent’s expected utility (as a function of state-contingent consumption),where each household enters the social welfare function with a weight λj, andλ1 + λ2 = 1.

(2c) Derive Pareto-optimal consumption for household j as a function of to-tal state-dependent endowment (denoted e (ε)) and parameters only. Commenton the properties of this optimal consumption rule. Does the autarky solutionsatisfy the conditions for Pareto-optimal consumption?

(2d) Now, suppose that instead of allocating consumption directly, the socialplanner just redistributes the land such that each household j holds a fractionαj of both types of land (with α1+α2 = 1). After ”slicing up” and redistributing

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the land in this fashion, each farmer will farm her share of each of the two typesof land, and consume whatever she harvests (there will be no ex-post transfersamong farmers once the landholdings were reassigned). How should the socialplanner set the shares (α1, α2)?

(2e) Could the result obtained in (2d) help us explain the fragmented pat-tern of landholdings in medieval English villages as a solution to a portfoliodiversi…cation problem? Comment brie‡y.

(2f) Assume that all three states of the world are equally likely, ie. πs = 13

for all s = 1, 2, 3. Now, suppose that apart from the two land plots that arebeing farmed already, ie k = 1, 2, there are two plots of wasteland available tothe village (ie potentially arable land that has not been cultivated yet). Denotethe two plots by k = 3, 4. Villagers can only farm one of the two additionalplots, but not both of them. Given the harvests predicted for each of the twoplots (see table 4 below), which of the two should the village cultivate in additionto k = 1, 2? Justify your answer.

Table 4: Predicted Harvests for land types 3 and 4plot 3: e3 (ε) plot 4: e4 (ε)

Event ε1 1 6Event ε2 3 3Event ε3 6 1

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Answer Key: Risk-sharing in medieval villagesPortfolio Diversi…cation through Fragmented landholdings

Pascal Courty/Liliane KarlingerEUI, Florence, November 2003

(1a) Villagers were presumably risk averse and disliked variability, but hadvery bad storage facilities. Since a household’s crop harvests were determinedby both aggregate and idiosyncratic shocks (the general weather plus hailstormsand crop disease), this would deliver land plots as diverse ”endowments”, withthe speci…ed variance-covariance statistics given in Tables 1 and 2. Then, plotholdings seem to represent the solution to a portfolio diversi…cation prob-lem, with divisions or fragmentation achieving risk-spreading across the villageterritory.

(1b) As discussed in the text, individual consumption smoothing acrosstime through storage was apparently not feasible. Likewise, no …nancialinstitutions had evolved, so borrowing and lending was not an option either.Finally, villagers could have teamed to set up an insurance (or risk-sharing)scheme whereby village output would be pooled and redistributed amongeach other in a way that ”lucky” farmers would give part of their output to”unlucky” ones.

However, a typical villager might have had less of an incentive to work hard ifs/he was assured consumption by the pooling of village output. This incentiveproblem might have prevented e¢cient insurance from being implemented inthis way.

In other words, even if a farmer’s e¤ort was observable to all other farmers(a reasonable assumption in a small village where everybody interacts frequentlywith everyone else), his/her e¤ort may not have been enforceable. In economiclingo, the problem they faced was probably not a moral hazard (or ”hiddenactions”) problem, but rather a 1/N (or team) problem. Note that thefragmentation-of-landholdings mechanism does not entail incentive problems, aseach farmer’s consumption will still fully depend on his/her individual farminge¤orts.

(2a) The coe¢cient of absolute risk aversion (CARA) is de…ned as

rA (x) = ¡u00(x)u0(x)

Thus, for Bernoulli utility de…ned as Uj (cj ) = (cj)d

(1¡d)d , we obtain

rA (cj ) = ¡U 00

j (cj )U 0

j (cj )=

1 ¡ dcj

The coe¢cient of relative risk aversion (CRRA) is de…ned as

rR (x) = ¡xu00(x)u0(x)

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and so, in terms of our utility function, we have

rR (cj ) = cjrA (cj ) = 1 ¡ d

Thus, our Bernoulli utility function is of the constant-relative-risk-aversiontype.

(2b) The social planner’s ex ante maximization problem is:

maxfc1(εs),c2(εs)g,s=1,2,3

2X

j=1

λjEεUj (cj (εs))

s.t.2X

j=1

cj (εs) ·2X

j=1

ej (εs) = e (εs) for all s = 1, 2, 3

where λj is household j’s weight in the social welfare function, Eε is the expec-tation of Uj (cj (εs)) with respect to the random variable ε, and cj (εs) is j ’sconsumption in state s.

(2c) Denote by µs the Lagrangian multiplier for each of the s = 1, 2, 3 con-straints of the above problem. Then, the FOC for the social planner’s problemwith respect to cj (εs) is

λjπsU 0j (cj (εs)) = µs

for all j = 1, 2 and s = 1, 2, 3. Thus, for any state s = 1, 2, 3 we must have:

λ1πsU 01 (c1 (εs)) = λ2πsU 0

2 (c2 (εs))

ie in each state s, weighted marginal utilities are equated across agents (recallthis optimality principle from the IKEA exercise of problem set 3!). Now, insertU 0

1 (c1 (εs)) = 11¡d (c1 (εs))

d¡1 and c2 (εs) = e (εs) ¡ c1 (εs) (by the economy’sresource constraint) to obtain

c1 (εs) =λ

1d¡12

λ1

d¡11 + λ

1d¡12

e (εs)

We see that Pareto-optimal state-contingent consumption by each household isa constant fraction of total endowment in state s! In other words, the Pareto-optimal sharing rule we obtain is linear in the total endowment.

Now, under autarky, we have the following consumption shares for consumer1:

c1 (ε1) = 1012 e (ε1) ' 0.83e (ε1)

c1 (ε2) = 610 e (ε2) = 0.6e (ε2)

c1 (ε3) = 27 e (ε3) ' 0.29e (ε3)

We see that under autarky, farmer 1’s consumption as a share of total en-dowment is varying across states! Thus, autarky cannot be Pareto-optimal.

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(2d) Under this land-redistribution scheme, household j’s state-contingentconsumption will be

c1 (εs) = αj [e1 (εs) + e2 (εs)]

for all s = 1,2, 3. Thus, set household 1’s optimal share in each type of land to

α¤1 =

λ1

d¡12

λ1

d¡11 + λ

1d¡12

(and α¤2 = 1 ¡ α¤

1) to fully replicate the Pareto-optimal outcome.

(2e) We saw that by appropriately redistributing land plots (which areinterpreted as endowments here), the social planner can perfectly replicate aPareto-optimal risk-sharing outcome that would otherwise require direct trans-fer payments among agents. Note, however, that our …ndings depend criticallyon the functional form chosen for our Bernoulli utility, which yields optimalstate-contingent consumption as a constant share of total endowment. If op-timal consumption were instead changing non-linearly with total endowment,our land-fragmentation mechanism would only provide partial risk-sharing andtherefore would not be Pareto-optimal anymore.

(2f) Since πs = 13 for all s = 1,2, 3, the expected additional harvests from

plot 3 and plot 4 are exactly the same (Eεe3 (ε) = Eεe4 (ε) = 13 (1 + 3 + 6)).

Thus, in terms of expected payo¤, the two plots are identical, so the choicebetween plot 3 and 4 cannot be based on expected payo¤.

However, given that the village will continue to farm plots 1 and 2 as well,we may think of the choice between plot 3 and 4 as one of optimal portfoliodiversi…cation. Compare the total state-dependent endowments for the twoadditional plots k = 3, 4:

Table 5: Total Event-Contingent Payo¤s (Harvests) from k = 3, 4farming plot 3 farming plot 4

e1 (ε) + e2 (ε) + e3 (ε) e1 (ε) + e2 (ε) + e4 (ε)Event ε1 12 + 1 = 13 12 + 6 = 18Event ε2 10 + 3 = 13 10 + 3 = 13Event ε3 7 + 6 = 13 7 + 1 = 8

Now, note that if plot 3 is farmed, the total village harvest will be the sameacross all states of the world, ie there would be no aggregate uncertaintyat all. Instead, if plot 4 is farmed, then the expected total harvest is exactlythe same (1

3 (18 + 13 + 8) = 13), but aggregate endowment will ‡uctuate acrossstates. In other words, the portfolio (k = 1, 2, 4) is a mean-preserving spreadof (k = 1, 2, 3) (ie (k = 1, 2, 3) second-order stochastically dominatesthe portfolio (k = 1, 2, 4)), so that risk-averse agents will always prefer (k =1, 2,3). In particular, farming plot 3 will allow for complete insurance ofall households (ie farmer j’s consumption will be the same across all states, so

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that risk can be fully avoided), and is therefore Pareto-optimal for any choiceof weights (λ1, λ2).

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For a detailed treatment of the issues discussed above, have a look at:Townsend, Robert M. (1993): "The medieval village economy: a study of

the Pareto-mapping in general equilibrium models", Princeton, NJ: PrincetonUniversity Press, 1993.

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