VII.08 Mathematics and Economic Reasoning
Partha Dasgupta
1 Two Girls
1.1 Becky’s World
Becky, who is ten years old, lives with her parents and an older brother Sam in a suburban town in America’s Midwest. Becky’s father works in a law firm specializing in small business enterprises. Depending on the firm’s profits, his annual income varies somewhat, but it is rarely below $145 000. Becky’s parents met in college. For a few years her mother worked in publishing, but when Sam was born she decided to concentrate on raising a family. Now that both Becky and Sam attend school, she does voluntary work in local education. The family live in a two-story house. It has four bedrooms, two bathrooms upstairs and a toilet downstairs, a large drawing-cum-dining room, a modern kitchen, and a family room in the basement. There is a small plot of land in the rear, which the family use for leisure activities.
Although they have a partial mortgage on their property, Becky’s parents own stocks and bonds and have a savings account in the local branch of a national bank. Becky’s father and his firm jointly contribute to his retirement pension. He also makes monthly payments into a scheme with the bank that will cover college education for Becky and Sam. The family’s assets and their lives are insured. Becky’s parents often remark that, federal taxes being high, they have to be careful with money; and they are. Nevertheless, they own two cars, the children attend camp each summer, and the family take a vacation together once camp is over. Becky’s parents also remark that her generation will be much more prosperous than they. Becky wants to save the environment and insists on biking to school each day. Her ambition is to become a doctor.
1.2 Desta’s World
Desta, who is about ten years old, lives with her parents and five siblings in a village in subtropical, southwest Ethiopia. The family live in a two-room, grass-roofed mud hut. Desta’s father grows maize and tef on half a hectare of land that the government has awarded him. Desta’s older brother helps him to farm the land and care for the household’s livestock: a cow, a goat, and a few chickens. The small quantity of tef produced is sold so as to raise cash income, but the maize is largely consumed by the household as a staple. Desta’s mother works a small plot next to their cottage, growing cabbage, onions, and enset (a year-round root crop that also serves as a staple). In order to supplement household income, she brews a local drink made from maize. As she is also responsible for cooking, cleaning, and minding the infants, her work day usually lasts fourteen hours. Despite the long hours, it would not be possible for her to complete the tasks on her own. (As the ingredients are all raw, cooking alone takes five hours or more.) So Desta and her older sister help their mother with household chores and mind their younger siblings. Although a younger brother attends the local school, neither Desta nor her older sister has ever been enrolled there. Her parents can neither read nor write, but they are numerate.
Desta’s home has no electricity or running water. Around where they live, sources of water, land for grazing cattle, and the woodlands are communal property. They are shared by people in Desta’s village; but the villagers do not allow outsiders to make use of them. Each day Desta’s mother and the girls fetch water, collect fuelwood, and pick berries and herbs from the local commons. Desta’s mother frequently observes that the time and effort needed to collect their daily needs has increased over the years.
There is no financial institution nearby to offer either credit or insurance. As funerals are expensive occasions, Desta’s father long ago joined a community insurance scheme (iddir) to which he contributes monthly. When Desta’s father purchased the cow they now own, he used the entire cash he had accumulated and stored at home, but had to supplement that with funds borrowed from kinfolk, with a promise to repay the debt when he had the ability to do so. In turn, when they are in need, his kinfolk come to him for a loan, which he supplies if he is able to. Desta’s father says that such patterns of reciprocity he and those close to him practice are part of their culture, reflecting their norms of social conduct. He also says that his sons are his main assets, as they are the ones who will look after him and Desta’s mother in their old age.
Economic statisticians estimate that, adjusting for differences in the cost of living between Ethiopia and the United States, Desta’s family income is about $5000 per year, of which $1000 is attributable to the products they draw from the local commons. However, as rainfall varies from year to year, Desta’s family income fluctuates widely. In bad years, the grain they store at home gets depleted well before the next harvest. Food is then so scarce that they all grow weaker, the younger children especially so. It is only after harvest that they regain their weight and strength. Periodic hunger and illnesses have meant that Desta and her siblings are somewhat stunted. Over the years Desta’s parents have lost two children in their infancy, stricken by malaria in one case and diarrhea in the other. There have also been several miscarriages.
Desta knows that she will be married (in all likelihood to a farmer, like her father) when she reaches eighteen and will then live on her husband’s land in a neighboring village. She expects her life to be similar to that of her mother.
2 The Economist’s Agenda
That the lives people are able to construct differ enormously across the globe is a commonplace. In our age of travel, it is even a common sight. That Becky and Desta face widely different futures is also something we have come to expect, perhaps also to accept. Nevertheless, it may not be out of turn to imagine that the two girls are intrinsically very similar: they both enjoy eating, playing, and gossiping; they are close to their families; they like pretty things to wear; and they both have the capacity to be disappointed, get annoyed, be happy. Their parents are also alike. They are knowledgeable about the ways of their worlds. They also care about their families, finding ingenious ways to meet the recurring problem of producing income and allocating resources among family members—over time and allowing for unexpected contingencies. So, a promising route for exploring the underlying causes behind their vastly different conditions of life would be to begin by observing that the constraints the families face are very different: that in some sense Desta’s family are far more restricted in what they are able to be and do than Becky’s.
Economics in large measure tries to uncover the processes that influence how people’s lives come to be what they are. The context may be a household, a village, a district, a state, a country, or the whole world. In its remaining measure, the discipline tries to identify ways to influence those very processes so as to improve the prospects of those who are hugely constrained in what they can be and do. Modern economics, by which I mean the style of economics taught and practiced in today’s graduate schools, does the exercises from the ground up: from individuals, through the household, village, district, state, country, to the whole world. In varying degrees the millions of individual decisions shape the eventualities people all face; as both theory and evidence tell us that there are enormous numbers of unintended consequences of what we all do. But there is also a feedback, in that those consequences go on to shape what people subsequently can do and choose to do. For example, when Becky’s family drive their cars or use electricity, or when Desta’s family create compost or burn wood for cooking, they contribute to global carbon emissions. Their contributions are no doubt negligible, but the millions of such tiny contributions cumulatively sum to a sizable amount, having consequences that people everywhere are likely to experience in different ways.
To understand Becky’s and Desta’s lives, we need first of all to identify the prospects they face for transforming goods and services into further goods and services—now and in the future, under various contingencies. Second, we need to uncover the character of their choices and the pathways by which the choices made by millions of households like Becky’s and Desta’s go to produce the prospects they all face. Third, and relatedly, we need to uncover the pathways by which the families came to inherit their current circumstances.
The last of these is the stuff of economic history. In studying history we could, should we feel bold, take the long view—from about the time agriculture came to be settled practice in the Fertile Crescent (roughly, Anatolia) some eleven thousand years ago—and try to explain why the many innovations and practices that have cumulatively contributed to the making of Becky’s world either did not reach or did not take hold in Desta’s part of the world. (Diamond (1997) is an enquiry into this set of questions.) If we wanted a sharper account, we could study, say, the past six hundred years and ask how it is that, instead of the several regions in Eurasia that were economically promising in about 1400 C.E., it was the unlikely northern Europe that made it and helped to create Becky’s world, even while bypassing Desta’s. (Landes (1998) is an inquiry into that question. Fogel (2004) explores the pathways by which Europe during the past three hundred years has escaped permanent hunger.) As modern economics is largely concerned with the first two sets of enquiries, this article focuses on them. However, the methods that today’s economic historians deploy to answer their questions are not dissimilar to the ones I describe below to study contemporary lives. The methods involve studying individual and collective choices in terms of maximization exercises. The predictions of the theories are then tested by studying data relating to actual behavior. Even the ethical foundations of national economic policies involve maximization exercises: the maximization of social well-being subject to constraints. (The treatise that codified this approach to economic reasoning was Samuelson (1947).)
3 The Household Maximization Problem
Both Becky’s and Desta’s households are microeconomies. Each subscribes to particular arrangements over who does what and when, recognizing that it faces constraints on what its members are capable of doing. We imagine that both sets of parents have their families’ well-being in mind and want to do as well as they can to protect and promote it.1 Of course, both Becky’s and Desta’s parents would have a wider notion of what constitutes their families than I have allowed here. Maintaining ties with kinfolk would be an important aspect of their lives, a matter I return to later. One also imagines that Becky’s and Desta’s parents are interested in their future grandchildren’s well-being. But as they recognize that their children will in turn care about their children, they are right to conclude by recursion that doing the best for their children amounts to doing the best for their grandchildren, for their great grandchildren, and so on, down the generations.
Personal well-being is made up of a variety of constituents: health, relationships, place in society, and satisfaction at work are but four. Economists and psychologists have identified ways to represent well-being as a numerical measure. To say that someone’s wellbeing is greater in situation Y than in situation Z is to say that her well-being measure is numerically higher in Y than in Z. A family’s well-being is an aggregate of its members’ well-beings. As goods and services are among the determinants of well-being (some important examples are food, shelter, clothing, and medical care), the problem that both Becky’s and Desta’s parents face is to determine, from among those allocations of goods and services that are feasible, the ones that are best for their households. However, both pairs of parents care not only about today, but also about the future. Moreover, the future is uncertain. So when the parents think about which goods and services their households should consume, they are concerned not just with the goods and services themselves, but also with when they will be consumed (food today, food tomorrow, and so on) and what will happen in the case of various contingencies (food the day after tomorrow if rainfall turns out to be bad tomorrow, and so forth). Implicitly or explicitly, both sets of parents convert their experience and knowledge into probabilistic judgments. Some of the probabilities they attach to contingencies are no doubt very subjective, but others, such as their predictions about the weather, are arrived at from long experience.
In subsequent sections we shall study the way in which Becky’s and Desta’s parents allocate goods and services across time and contingencies. But here we shall keep the exposition simple and consider a model that is static and deterministic. That is, we shall pretend that the people live in a timeless world, and that they are completely certain about all the information they need in order to make their decisions.
Suppose that a certain household has N members, whom we label 1, 2, . . . , N. Let us think about how we can appropriately model the well-being of household member i. As has already been mentioned, well-being is taken to be a real number that depends in some way on the goods and services consumed and supplied by i. It is traditional to divide goods and services into those consumed and those supplied, and to use positive numbers to represent quantities of the former and negative numbers for the latter. Imagine now that there are M commodities in all. Let Yi (j) represent the quantity of the jth commodity that is consumed or supplied by i. By our convention, Yi (j) > 0 if j is consumed by i (e.g., food eaten or clothing worn) and Yi (j) < 0 if j is supplied by i (e.g., labor). Now consider the vector Yi = (Yi (1), . . . , Yi (M)). It denotes the quantities of all the goods and services consumed or supplied by i. Yi is a point in 
M—the Euclidean space of M dimensions. We now let Ui (Yi) denote i’s well-being. Let us assume that supplying goods and services decreases i’s wellbeing, while consuming them increases it. Because the goods that are supplied by i are measured as negative quantities, we can justifiably assume that Ui (Yi) increases as any of its components Yi increases.
The next step is to generalize the model to one that applies to an entire household. The individual well-beings of the members of the household can be collected together so that they themselves form an N-dimensional vector, (U1 (Y1), . . . , UN (YN)). The household’s well-being is dependent in some way on this vector. That is, we say that the well-being of the household is W (U1(Y1), . . . , UN(YN)), for some function W. (Utilitarian philosophers have argued that W is simply the sum of the Ui.) We also make the natural assumption that W is an increasing function of each Ui (which is certainly the case if W is the sum of the Ui).
Let Y denote the sequence (Y1, . . . , YN). Y is a point in the NM-dimensional Euclidean space NM. It can also be thought of as the matrix you obtain if you make a table of the amounts of each commodity consumed or supplied by each member of the household. Now, it is clear that not every Y in NM can actually occur: after all, the total amount of any given commodity (in the whole world, say) is finite. So we assume that Y belongs to a certain set J, which we regard as the set of all potentially feasible values of Y. Within J we identify a smaller set, F, of “actually feasible” values of Y. This is the set of values of Y from which the household could in principle choose. It is smaller than J because of constraints that the household faces, such as the maximum amount of income it can earn. F is the house-hold’s feasible set.2 The decision faced by a household is to choose Y from the feasible set F so as to maximize its well-being W (U1 (Y1), . . . , UN (YN)). This is called the household maximization problem.
It is reasonable, and mathematically convenient, to assume that the sets J and F are both closed and bounded subsets of NM, and that the well-being function W is continuous. Since every continuous function on a closed bounded set has a maximum, it follows that the household maximization problem has a solution. If, in addition, W is differentiable, the theory of nonlinear programming can be used to identify the optimality conditions the household’s choice must satisfy. If F is a convex set and W is a concave function of Y, those conditions are both necessary and sufficient. The LAGRANGE MULTIPLIERS [III.64] associated with F can be interpreted as notional prices: they reflect the worth to the household of slightly relaxing the constraints.
Let us conduct an exercise to test the power of the modern economist’s way of studying choice. First, let us assume that W is a symmetric and concave function of the individual well-beings Ui (as would be the case if W were the sum of the Ui). The symmetry assumption means that if two individuals exchange their well-beings, then W is unchanged; and concavity means, roughly speaking, that, other things being equal, as a Ui increases, the rate of increase of W does not rise. Let us suppose in addition that the household members are identical: that is, let us set all the functions Ui to be equal to a single function, U, say. Assume also that U is a strictly concave function of the Yi, which means that the rate of increase of well-being declines as consumption increases. Finally, assume that the feasible set F is nonempty, convex, and symmetric. (Symmetry means that if some Y is feasible, and the vector Z is the same as Y except that the consumptions of a pair of individuals in the household have been exchanged, then Z is also feasible.) From these assumptions it can be shown that members of the household would be treated equally: that is, W is maximized when they all receive the same bundle of goods and services.
At low levels of consumption, however, the hypothesis that the function U is concave is unreasonable. To see why, we should note that, typically, 60-75% of the daily energy intake of someone in nutritional balance goes toward maintenance, while the remaining 25-40% is expended in discretionary activities (work and leisure). The 60-75% is rather like a fixed cost: over the long run a person needs it as a minimum no matter what he or she does. The simplest way to uncover the implications of such fixed costs is to continue to suppose that F is convex (which is the case, for example, if there is a fixed quantity of food for allocation among members of the household), but that U is a strictly convex function at low intakes of food and a strictly concave function thereafter. It is not hard to show that a poor household in such a world will maximize its well-being by allocating food unequally among its members, while a rich household can afford the luxury of equal treatment and will choose to distribute food equally. Suppose, to take a very stylized example, that energy requirement for daily maintenance is 1500 kcal and that a household of four can obtain at most 5000 kcal for consumption. Then equal sharing would mean that no one would have sufficient energy for any work, so it is better to share the food unequally. On the other hand, if the household is able to obtain more than 6000 kcal, it can share the food equally without jeopardizing its future.
There are empirical correlates of this finding. When food is very scarce, the younger and weaker members of Desta’s household are given less to eat than the others, even after allowance is made for differences in their ages. In good times, though, Desta’s parents can afford to be egalitarian. In contrast, Becky’s household can always afford enough food. Her parents therefore allocate food equally every day.
4 Social Equilibrium
Household transactions in Becky’s world are carried out mostly in markets. The terms of trade are the quoted market prices. In developing a mathematical construction of social outcomes, I continue to imagine, for simplicity, a static, deterministic world. Let P (
0) be the vector of market prices and let M ( 0) be the vector of a household’s endowments of goods and services. (That is, for each commodity j, P (j) is the price of j and M (j) is the amount of j that the household already has.) Recalling our convention that goods consumed are of positive sign and goods supplied are of negative sign, define X = Σ Yi. (Thus, X (j) = Σ Yi (j) is the total amount of commodity j that is consumed by the household.) Then P · X is the total price of goods consumed by the household, minus the total price of goods supplied, and P · M is the total value of its endowments. The feasible set F is the set of household choices Y that satisfy the “budget” constraint P · (X − M) 
0.
The income that Becky’s household earns from the assets it supplies to the market is determined by market prices (Becky’s father’s salary, interest rates on bank deposits, returns on shares owned). Those prices in turn depend on the size and distribution of household endowments of goods and services and on household needs and preferences. They depend too on the ability and willingness of institutions, such as private firms and the government, to make use of the rights they in turn have been awarded. These functional relationships explain why Becky’s father’s skills as a lawyer (itself an asset, termed “human capital” by economists) would not be worth much in Desta’s village, even though they are much valued in the United States. In fact, it was a firm belief that lawyers would continue to prove valuable in the United States that encouraged Becky’s father to be a lawyer.
Although Desta’s household does operate in markets (when her father sells tef or her mother sells the liquor she has brewed), it undertakes many transactions directly with nature; in the local commons and in farming, and in nonmarket relationships with others in the village. Therefore the F that Desta’s household faces is not defined simply by a linear budget inequality, as in the idealized model we have constructed to display Becky’s world, but also reflects the constraints that nature imposes, such as soil productivity and rainfall, the assets it has access to, and the terms and conditions involving transactions with others in the village via nonmarket relationships, a matter I come to later. The constraints imposed by nature are felt by Becky’s household too, but through market prices. For example, should a drought lead to a fall in world cereal production, it would become noticeable to Becky’s household through the high price of cereal. Desta’s household, in contrast, would notice it directly from the reduced harvest from their field.
Desta’s household assets include the family home, livestock, agricultural implements, and their half hectare of land. The skills Desta’s family members have accumulated in farming, managing livestock, and collecting resources from the local commons are part of their human capital. Those skills do not command much return in the global marketplace, but they do shape the household’s feasible set F and are vital to the family’s well-being. Desta’s parents learned those skills from their parents and grandparents, just as Desta and her siblings have learned them from their parents and grandparents. Desta’s family can also be said to own a portion of the local commons: in effect, her household shares its ownership with others in the village. Difficulties in reaching and enforcing agreement with neighbors over the use of the local commons are less severe than they are in the case of global commons, such as the atmosphere as a sink for carbon emissions. This is not only because the required negotiations involve far fewer people when the commons are local, but also because there is likely to be greater congruence of opinions and interests among the users. It helps too that the parties are able to observe whether the agreements they made over the use of local commons are being kept. (See below in our discussion of insurance arrangements in Desta’s world.)
Thus, the choices that are available to individuals are affected by the choices that other people make: this results in feedback. In a market economy, the feedback is in large part transmitted in prices. In nonmarket economies the feedback is transmitted through the terms in which households are able to negotiate with one another.
Let us try to model this situation mathematically. We start by imagining an economy of H households. For ease of exposition, I shall suppose that a household’s well-being can be expressed directly in terms of its aggregate consumption of goods and services, disregarding how this consumption is distributed among the individual members. Let Xh denote the consumption vector in household h (with the usual sign convention), let Jh be the set of potentially feasible vectors Xh, and let Wh (Xh) be h’s well-being.
Within h’s potentially feasible set Jh of consumption vectors lies the actual feasible set Fh. In order to model the feedback we shall explicitly recognize that Fh depends on the consumptions of other households. That is, it is a function of the sequence (X1, . . . , Xh-1, Xh+1, . . . , XH). To save space, we shall denote this sequence, which consists of every household’s consumption vector except h’s, by X_h. Formally, Fh is a function (sometimes called a “correspondence”) that takes objects of the form X_h to subsets of Jh. Household h’s economic problem is to choose its consumption Xh from its feasible set Fh (X_h) in such a way as to maximize its well-being Wh (Xh). The optimum choice depends on h’s beliefs about X-h and the correspondence Fh (X-h).
Meanwhile, all other households are making similar calculations. How can we unravel the feedbacks? One way would be to ask people to disclose their beliefs about the feedbacks. Fortunately, economists avoid that route. So as to anchor their investigation, economists study equilibrium beliefs; that is, beliefs that are self-confirming. The idea is to identify states of affairs where the choices people make on the basis of their beliefs about the feedbacks are precisely those that give rise to those very feedbacks. We call any such state of affairs a social equilibrium. Formally, a sequence 
of household choices is called a social equilibrium if, for every h, the choice 
of household h maximizes the well-being Wh (Xh) over all choices of Xh in its feasible set Fh
.
This raises an obvious question: does a social equilibrium exist? Classic papers by Nash in 1950 and Debreu in 1952 showed that, under a fairly general set of conditions, it always does. Here is a set of conditions that Debreu identified. Assume that each well-being function Wh is continuous and quasi-concave (which means that for any potentially feasible choice in Jh, the set of Xh in Jh for which Wh(Xh) is greater than or equal to Wh() is convex). Assume also that for every household h, the feasible set Fh (recall that this is a subset of Jh) is nonempty, compact, and convex, and depends continuously on the choices X−h made by other households. The proof that under the above conditions a social equilibrium always exists is a relatively straightforward use of THE KAKUTANI FIXED POINT THEOREM [V.11 §2], which is itself a generalization of Brouwer’s fixed point theorem. Alternative sets of sufficient conditions for the existence of social equilibria (which allow the feasible set Fh(X−h) to be nonconvex) have been explored in recent years.
In Becky’s world, a social equilibrium is called a market equilibrium. A market equilibrium is a price vector P* (≥ 0) and a consumption vector 
for each household h, such that maximizes Wh(Xh) subject to the budget constraint P* · (Xh − Mh) ≤ 0, and such that the demands for goods and services across households are feasible (i.e., Σ(Xh − Mh) ≤ 0). That market equilibria are social equilibria, in the sense in which we have defined the latter term here, was demonstrated by Arrow and Debreu in 1954. Debreu (1959) is the definitive treatise on market equilibria. In that book, Debreu followed the leads of Erik Lindahl and Kenneth J. Arrow, by distinguishing goods and services not only in terms of their physical characteristics, but also in terms of the date and contingency in which they appear. Later in this article we shall expand the commodity space in that way to study savings and insurance decisions in both Becky’s and Desta’s worlds.
One cannot automatically assume that a social equilibrium is just or collectively good. Moreover, except for the most artificial examples, social equilibrium is not unique—which means that a study of equilibria per se leaves open the question of which social equilibrium we should expect to observe. In order to probe that question, economists study disequilibrium behavior and analyze the stability properties of the resulting dynamic processes. The basic idea is to hypothesize about the way people form beliefs about the way the world works, track the consequences of those patterns of learning, and check them against data. It is reasonable to limit such a study by considering only those learning processes that converge to a social equilibrium in stationary environments. Initial beliefs would then dictate which equilibrium is reached in the long run (see, for example, Evans and Honkapohja 2000). Since the study of disequilibria would lengthen this article greatly, we shall continue to study social equilibria here.
5 Public Policy
Economists distinguish between what they call private goods and public goods. For many goods, consumption is rivalrous: if you consume a bit more from a given supply of such a good (e.g., food), others have that much less to consume. These are private goods. The way to assess their consumption throughout the economy is to add up the amounts consumed by all individual households; which is what we did in the previous section when arriving at the notion of a social equilibrium. Not all goods are like that, however. For example, the extent of national security on offer to you is the same as that on offer to all households in your country. In a just society the law has that same property, as has the state: not only is consumption not rivalrous, but in addition, no one can be prevented from availing himself or herself of the entire amount available in the economy. Public goods are goods of this second kind. One models the quantity of a public good as a number G, and the quantity Gh consumed by each household h is deemed to equal G. An example of a public good that has a global coverage is the Earth’s atmosphere: the whole world benefits from it jointly.
If the supply of public goods is left to private individuals, then problems arise. For example, even though everyone in a city would benefit from a cleaner, healthier environment, individuals have a strong incentive to free-ride on others when it comes to paying for that cleaner environment. Samuelson showed in 1954 that such a situation resembles the prisoner’s dilemma: each party has a strategy that is best for him/her, regardless of what strategies the other parties choose, even though there is another set of strategies, one per party, that is better for everybody. Under such circumstances, one usually needs public measures, such as taxes and subsidies, in order for it to be in the interest of private individuals to act in a way that implements the collectively preferred outcomes. In other words, the dilemma can be expected to be resolved effectively not by markets but by politics. It is widely accepted in political theory that government should be charged with imposing taxes, subsidies, and transfers, and should be engaged in supplying public goods. The government is also the natural agency to supply infrastructure, such as roads, ports, and electrical cables, requiring as they do investments that are huge in comparison with individual incomes. We shall now extend our earlier model to include public goods and infrastructure, so that we can study the government’s economic task.
Let us assume that social well-being is a numerical aggregate of household well-beings. Thus, if V is social well-being, we write it as V(W1, . . . , WH). It is natural to postulate that V increases as any Wh increases. (One example of such a function V is the one prescribed by utilitarian philosophy, namely, W1 + · · · + WH.) The government chooses what quantities to supply of the various public goods and infrastructure commodities. These numbers can be modeled by two vectors, which we will call G and I, respectively. The government also chooses to impose on each household h certain transfers Th of goods and services (for example, providing health care and charging income tax). Let us write T for the sequence (T1, . . . , TH). Whether or not a particular choice of vectors G and I is actually feasible for the government will depend on T, so we define KT to be the set of feasible pairs of vectors (G, I), given the choice of T.
Because we have introduced a new set of goods, we shall have to modify the household well-being functions by enlarging their domains. The obvious notation to express this extra dependence is to write Wh(Xh, G, I, Th) for the well-being of household h. Moreover, h’s feasible set Fh now also depends on G, I, and Th; so we write the set of feasible household choices as Fh(G, I, Th, X−h).
To try to determine the optimum public policy, imagine a two-stage game. The government has the first move, choosing T and then G and I from KT . Households go second, reacting to decisions made by the government. Imagine that a social equilibrium X* = (
, . . . , ) is reached and that the equilibrium is unique. (We assume that if there are multiple equilibria, the government can select among them by resorting to public signals.) Clearly, this equilibrium X* is a function of G, I, and T. An intelligent and benevolent government will anticipate it and choose T, G, and I from KT in such a way as to maximize the resulting social well-being V(W(), . . . , W()).
The public policy problem we have just designed, involving as it does a double optimization, is technically very difficult. It transpires, for example, that even in some of the simplest model economies one can imagine, Fh(G, I, Th, X−h) is not convex. This means that the social equilibrium cannot be guaranteed to depend continuously on G, I, and T, as was shown by Mirrlees in 1984. This in turn means that standard techniques are not suitable for the government’s optimization problem. In fact, of course, even “double optimization” is a huge simplification. The government chooses; people respond by trading, producing, consuming; the government chooses again; people respond once again—and so forth in an unending series of moves and countermoves. Identifying the optimum public policy involves severe computational difficulties.
6 Matters of Trust: Laws and Norms
The previous examples demonstrate that a fundamental problem facing people who would like to transact with one another concerns trust. For example, the extent to which parties trust one another shapes the sets Fh and KT . If the parties do not trust one another, what could have been mutually beneficial transactions will not take place. But what grounds does a person have for trusting someone to do what he promises to do under the terms of an agreement? Such grounds can exist if promises can be made credible. Societies everywhere have constructed mechanisms to create credibility of this kind, but in different ways. What the mechanisms have in common, however, is that individuals who fail to comply with agreements without a good reason are punished.
How does that common feature work?
In Becky’s world the rules governing transactions are embodied in the law. The markets Becky’s family enters are supported by an elaborate legal structure (a public good). Becky’s father’s firm, for example, is a legal entity; as are the financial institutions he deals with in order to accumulate his retirement pension, to save for Becky’s and Sam’s education, and so on. Even when someone in the family goes to the grocery store, the purchases (paid for with cash or by card) involve the law, which provides protection for both parties (the grocer, in case the cash is counterfeit or the card is void; the purchaser, in case the product turns out on inspection to be substandard). The law is enforced by the coercive power of the state. Transactions involve legal contracts backed by an external enforcer, namely, the state. It is because Becky’s family and the grocery store’s owner are confident that the government has the ability and willingness to enforce contracts (i.e., to continue to supply the public good in question) that they are willing to make transactions.
What is the basis of that confidence? After all, the contemporary world has shown that there are states and there are states. Why should Becky’s family trust the government to carry out its tasks in an honest manner? A possible answer is that the government in her country worries about its reputation: a free and inquisitive press in a democracy helps to sober the government into believing that incompetence or malfeasance would mean an end to its rule come the next election. Notice how the argument involves a system of interlocking beliefs about the abilities and intentions of others. The millions of households in Becky’s country trust their government (more or less!) to enforce contracts, because they know that government leaders know that not to enforce contracts efficiently would mean being thrown out of office. In their turn, each party to a contract trusts the other to refrain from reneging (again, more or less!), because each knows that the other knows that the government can be trusted to enforce contracts. And so on. Trust is maintained by the threat of punishment (a fine, a jail term, dismissal, or whatever) for anyone who breaks a contract. Once again, we are in the realm of equilibrium beliefs, held together by their own bootstraps. Mutual trust encourages people to seek out mutually beneficial transactions and engage in them. As the formal argument that supports the above claim is very similar to the one showing that social norms contain mechanisms for enforcing agreements, we turn to the place of social norms in people’s lives.
Although the law of contracts exists also in Desta’s country, her family cannot depend on it because the nearest courts are far from their village. Moreover, there are no lawyers in sight. As transport is enormously costly, economic life is shaped outside a formal legal system. In short, crucial public goods and infrastructure are either unavailable, or, at best, in short supply. But even though there is no external enforcer, Desta’s parents do make transactions with others. Credit (not dissimilar to insurance in her village) involves saying, “I will lend to you now if you promise to repay me when you can.” Saving for funerals involves saying, “I agree to abide by the terms and conditions of the iddir.” And so on. But why should the parties have any confidence that the agreements will not be broken?
Such confidence can be justified if agreements are mutually enforced. The basic idea is this: a credible threat by members of a community that stiff sanctions will be imposed on anyone who breaks an agreement can deter everyone from breaking it. The problem is then to make the threat credible. In Desta’s world credibility is achieved by recourse to social norms of behavior.
By a social norm we mean a rule of behavior followed by members of a community. A rule of behavior (or “strategy” in economic parlance) reads like, “I will do X if you do Y,” “I will do P if Q happens,” and so forth. For a rule of behavior to be a social norm, it must be in the interest of everyone to act in accordance with the rule if all others act in accordance with it. Social norms are equilibrium rules of behavior. We will now see how social norms work and how transactions based on them compare with market-based transactions. To do this we will study insurance as a commodity.
7 Insurance
To insure oneself against a risk is to act in ways to reduce that risk. (Formally, a RANDOM VARIABLE [III.71 §4] 
is said to be riskier than a random variable 
if there is a random variable 
with zero mean such that has the same distribution as + . In this case, and have the same mean but is more “spread out.”) As long as it does not cost too much, riskaverse households will want to reduce risk by purchasing insurance: in fact, avoiding risk would seem to be a universal urge. To formalize these notions, consider an isolated village, such as Desta’s. Suppose for simplicity that it contains H identical households. If household h’s food consumption is Xh (represented by a single real number), let us say that its well-being is W(Xh). We shall assume that W′(Xh) > 0 (that is, more food leads to greater well-being) and that W″(Xh) < 0 (the more food you already have, the less you benefit from yet more). We shall confirm below that the second property of W, its strict concavity, implies, and is implied by, risk aversion; but the basic reason is simple: if W is strictly concave, then you gain less when you are lucky than you lose when you are unlucky.
For simplicity, let us suppose that the production of food by a household h, which is subject to chance factors such as the weather, involves no effort. Since the output is uncertain, we represent it by a random variable h, with expected value μ, which is assumed to be positive. We shall denote expectations by 
.
If a household h is completely self-sufficient, then its expected well-being is simply (W(h)). However, the strict concavity of W implies that W(μ) > (W (h)). To put this in words: h’s well-being at the average level of production is greater than the expectation of h’s well-being if the production is random. This means that h will prefer a sure level of consumption to a risky one with mean equal to that sure level. In short, h is risk averse. Define a number 
by W() = (W(h)). So is the level of production that achieves the expected well-being. This will be less than μ, and so μ − is a measure of the cost of the risk that a self-sufficient household bears. Notice that the greater the “curvature” of W is, the greater the cost is of the risk associated with h. (A useful measure of curvature turns out to be −XW″(X)/W′(X). We will make use of this measure when discussing intertemporal choices.) To see how households could gain by pooling their risks, let us write h = μ + 
h, where h is a random variable with mean zero, variance σ2, and finite support. Suppose for simplicity that the random variables h are identical (i.e., they do not depend on h). Let the correlation coefficient of any two of these distributions be ρ. It turns out that, as long as ρ < 1, households can reduce their risks by agreeing to share their outputs. Suppose that households are able to observe one another’s outputs. Given that the random variables h are identical, the obvious insurance scheme is to share out the outputs equally. Under this scheme, h’s uncertain food consumption becomes the average of 1 , . . . , H, which is an improvement on self-sufficiency because (W(Σh′/H)) > (W(h)). The problem is that, without an enforcement mechanism, the agreement to share will not stick, because once each household knows how much food every household has produced, all but the unluckiest households will wish to renege. To see why, notice first that the luckiest households will renege because their outputs are above the average; but this means that the next luckiest set of households will renege because their outputs are above the reduced average; and so on, down to the unluckiest households. Since households know in advance that this will happen if there are no enforcement mechanisms, they will not enter the scheme in the first place: the only social equilibrium is pure self-sufficiency and there is no pooling of risk.
Let us call the insurance game just described the stage game. Although pure self-sufficiency is the only social equilibrium for the stage game, we shall now see that the situation changes if the game is played repeatedly. To model this, let us use the letter t to denote time, and let us take time to be a nonnegative integer. (The game might, for instance, take place every year, with 0 standing for the current year.) Let us assume that the villagers face the same set of risks in each time period, and that the risk in one year is independent of the risks in all other years. Also assume that, in each period, once food outputs are realized, households decide independently of one another whether they will abide by the agreement to share their produce equally or whether they will renege on it.
Although future well-being is important to a household, it will typically be less important than present well-being. To model this we introduce a positive parameter δ, which measures how much a household discounts its future well-being. The assumption is that, when making calculations at t = 0, a household divides its well-being at time t by a factor (1 + δ)t: that is, the importance decays by a certain fixed percentage at each time period. We shall now show that, provided δ is sufficiently small (i.e., provided that households care enough about their future well-being), there is a social equilibrium in which households abide by the agreement to share their aggregate output equally.
Let h(t) be the uncertain amount of food available to household h at time t. If all households are participating in the agreement, then h(t) will be μ + (Σ h′)/H, and if there is no agreement, then it will be μ + h. At time t = 0 the total expected well-being of household h, present and future, is 
(W(h (t)))/(1 +δ)t. (To calculate this we took, for each t ≥ 0, the expected well-being of h at time t and divided it by (1 + δ)t. Then we added these numbers up.)
Now consider the following simple strategy that h might adopt: it begins by participating in the insurance scheme and continues to participate so long as no household has reneged on the agreement; but it withdraws from the scheme from the date following the first violation of the agreement by some household. Game theorists have christened this the “grim strategy,” or simply grim, because of its unforgiving nature. Let us see how grim could support the original agreement to share aggregate output equally at every date. (For a general account of repeated games and the variety of social norms that can sustain agreements, see Fudenberg and Maskin (1986).)
Suppose that household h believes that all other households have chosen grim. Then h knows that none of the other households will be the first to defect. What should h do then? We will show that if δ is small enough, h can do no better than play grim. As the same reasoning would be applicable to all other households, we should conclude that, for small enough values of δ, grim is an equilibrium strategy in the repeated game. But if all households play grim, then no household will ever defect. Grim can therefore function as a social norm for sustaining cooperation. Let us see how the argument works.
The basic idea is simple. As all other households are assumed to be playing grim, household h would enjoy a one-period gain by defecting if its own output exceeded the average output of all households. But if h defects in any period, all other households will defect in all following periods (they are assumed to be playing grim, remember). Therefore, h’s own best option in all following periods will be to defect also, which means that subsequent to a single deviation by h, the outcome can be predicted to be pure self-sufficiency. So, set against a one-period gain that household h would enjoy if its output exceeded the average output of all households is the loss it would suffer from the following date because of the breakdown of cooperation. That loss exceeds the one-period gain if δ is small enough. So, if δ is sufficiently small, household h will not defect, but will adopt grim; implying that grim is an equilibrium strategy and equal sharing among households in every period is a social equilibrium.
To formalize the above argument, we consider the situation in which h’s incentive to defect is greatest. Let A and B be the minimum and maximum possible outputs of any household. Then the maximum gain that household h could possibly enjoy from defecting at t = 0 arises if h happens to produce B and all other households happen to produce A. Since the average output in this eventuality is (B + (H − 1)A)/H, the one-period gain that household h would enjoy from defecting is
But h knows that if it defects, the expected loss in each subsequent period (i.e., from t = 1 onward) will be (W(Σ ,h′/H)) − (W(h)). In order to simplify the notation, let us write (W(Σ h′ /H)) − (W(h)) as L. Household h can then calculate that the expected total loss it will suffer from defecting at t = 0 is L 
(1 + δ)−t, which equals L/δ. If this future loss exceeds the present gain from defecting, then household h will not want to defect. In other words, h will not want to defect if
or
But if h does not find it in its interest to defect when the one-period gain from defection is the largest possible, it will certainly not want to defect in any other situation. We conclude that if inequality (1) holds, then grim is an equilibrium strategy and equal sharing among households in every period is a resulting social equilibrium. Notice that, as we said, this will happen if δ is sufficiently small.
We usually reserve the term “society” to denote a collective that has managed to find a mutually beneficial equilibrium. Notice, however, that another social equilibrium of the repeated game is each household for itself. If everyone believed that all others would break the agreement from the start, then everyone would break the agreement from the start. Noncooperation would involve each household selecting the strategy: renege on the agreement. Failure to cooperate could be due simply to a collection of unfortunate, self-confirming beliefs, and nothing else. It is also easy to show that noncooperation is the only social equilibrium of the repeated game if
We now have in hand a tool for understanding how a community can slide from cooperative to noncooperative behavior. For example, political instability (in the extreme, civil war) can mean that households are increasingly concerned that they will be forced to disperse from their village. This translates into an increase in δ. Similarly, if households fear that their government is now bent on destroying communal institutions in order to strengthen its own authority, δ will increase. But from (1) and (2) we know that if δ increases sufficiently, then cooperation ceases. The model therefore offers an explanation for why, in recent decades, cooperation at the local level has declined in the unsettled regions of sub-Saharan Africa. Social norms work only when people have reasons to value the future benefits of cooperation.
In the above analysis, we allowed for the possibility that, in each period, household risks were positively correlated. Moreover, the number of households in any village is typically not large. These are two reasons why Desta’s household is unable to attain anything like full insurance against the risk they face. Becky’s parents, in contrast, have access to an elaborate set of insurance markets that pool the risks of hundreds of thousands of households across the country (even the world, if the insurance company is a multinational). This helps to reduce individual risk more than Desta’s parents can, because, first, spatially distant risks are more likely to be uncorrelated, and, second, Becky’s parents can pool their risk with many more households. With enough households and enough independence of their risk, THE LAW OF LARGE NUMBERS [III.71 §4] practically guarantees that equal sharing among those households will provide each one with the average μ. This is an advantage of markets, backed by the coercive power of the state as an external enforcer: in a competitive market, insurance contracts are available, enabling people who do not know one another to do business through third parties, in this case the insurance companies.
Many of the risks that Desta’s parents face, such as low rainfall, will in fact be very similar for all households in their village. Since the insurance they are able to obtain within their village is therefore very limited, they adopt additional risk-reducing strategies, such as diversifying their crops. Desta’s parents plant maize, tef, and enset (an inferior crop), with the hope that even if maize were to fail one year, enset would not let them down. That the local resource base in Desta’s village is communally owned probably also has something to do with a mutual desire to pool risks. Woodlands are spatially nonhomogeneous ecosystems. In one year one group of plants bears fruit, in another year some other group does. If the woodland were divided into private parcels, each household would face a greater risk than it would under communal ownership. The reduction in individual household risks owing to communal ownership may be small, but as average incomes are very low, household benefits from communal ownership are large. (For a fuller account of the management of local commons in poor countries, see Dasgupta (1993).)
8 The Reach of Transactions and the Division of Labor
Payments in Becky’s world are made in money, expressed in U.S. dollars. Money would not be required in a world where everyone was known to be utterly trustworthy, people did not incur computational costs, and transactions were costless: simple IOUs, stipulating repayment in terms of specific good and services, would suffice in that world. However, we do not live in that world. A debt in Becky’s world involves a contract specifying that the borrower is to receive a certain number of dollars and that he promises to repay the lender dollars in accordance with an agreed schedule. When signing the contract the relevant parties entertain certain beliefs about the dollar’s future value in terms of goods and services. Those beliefs are in part based on their confidence in the U.S. government to manage the value of the dollar. Of course, the beliefs are based on many other things as well; but the important point remains that money’s value is maintained only because people believe it will be maintained (the classic reference on this is Samuelson (1958)). Similarly, if, for whatever reason, people feared that the value would not be maintained, then it would not be maintained. Currency crashes, such as the one that occurred in Weimar Germany in 1922–23, are an illustration of how a loss in confidence can be self-fulfilling. Bank runs share that feature, as do stock market bubbles and crashes. To put it formally, there are multiple social equilibria, each supported by a set of self-fulfilling beliefs.
The use of money enables transactions to be anonymous. Becky frequently does not know the salespeople in the department stores of her town’s shopping mall, nor do they know Becky. When Becky’s parents borrow from their bank, the funds made available to them come from unknown depositors. Literally millions of transactions occur each day between people who have never met and will never meet in the future. The problem of creating trust is solved in Becky’s world by building confidence in the medium of exchange: money. The value of money is maintained by the state, which has an incentive to maintain it because, as we saw earlier, it wishes not to destroy its reputation and be thrown out of office.
In the absence of infrastructure, markets are unable to penetrate Desta’s village. Becky’s suburban town, by contrast, is embedded in a gigantic world economy. Becky’s father is able to specialize as a lawyer only because he is assured that his income can be used to purchase food in the supermarket, water from the tap, and heat from cooking ovens and radiators. Specialization enables people to produce more in total than they would be able to if they were each required to diversify their activities. Adam Smith famously remarked that the division of labor is limited by the size of the market. Earlier we noted that Desta’s household does not specialize, but produces pretty much all of its daily requirements from a raw state. Moreover, the many transactions it enters into with others, being supported by social norms, are of necessity personalized, thus limited. There is a world of a difference between laws and social norms as the basis of economic activities.
9 Borrowing, Saving, and Reproducing
If you do not have insurance, then your consumption will depend heavily on various contingencies. Purchasing insurance helps to smooth out this dependence. We shall see presently that the human desire to smooth out the dependence on contingencies is related to the equally common desire to smooth out consumption across time: they are both a reflection of the strict concavity of the well-being function W. The flow of income over a person’s lifetime tends not to be smooth, so people look for mechanisms, such as mortgages and pensions, that enable them to transfer consumption across time. For instance, Becky’s parents took out a mortgage on their house because at the time of purchase they did not have sufficient funds to finance it. The resulting debt decreased their future consumption, but it enabled them to buy the house at the time they did and thereby raise current consumption. Becky’s parents also pay into a pension fund, which transfers present consumption to their retired future. Borrowing for current consumption transfers future consumption to the present; saving achieves the reverse. Since capital assets are productive, they can earn positive returns if they are put to good use. This is one reason why, in Becky’s world, borrowing involves having to pay interest, while saving and investing earn positive returns.
Becky’s parents also make a considerable investment in their children’s education, but they do not expect to be repaid for this. In Becky’s world, resources are transferred from parents to children. Children are a direct source of parental well-being; they are not regarded as investment goods.
A simple way to formulate the problem Becky’s parents face when they arrange transfers of resources across time is to imagine that they view themselves as part of a dynasty. This means that, in reaching their consumption and saving decisions, they take explicit note not only of their own well-being and the well-being of Becky and Sam, but also of the well-being of their potential grandchildren, great grandchildren, and so on, down the generations.
To analyze the problem, it is notationally tidiest to assume that time is a continuous variable. At time t (which we take to be greater than or equal to 0), let K(t) denote household wealth and X(t) the consumption rate, which is some aggregate based on the market prices of what they consume. In practice, a household will want to smooth its consumption across both time and contingencies, but in order to concentrate on time we shall consider a deterministic model. Suppose that the market rate of return on investment is a positive constant r. This means that if household wealth at time t is K(t), then the income it earns from that wealth at t is rK(t). The dynamical equation describing the dynasty’s consumption options over time is then
The right-hand side of the equation is the difference between the dynasty’s investment income at time t (which is r times its wealth at t) and its consumption at t. This amount is saved and invested, so it gives the rate of increase of the dynasty’s wealth at t. The present time is t = 0 and K(0) is the wealth that Becky’s parents have inherited from the past. Earlier, we assumed that the household allocates its consumption across contingencies by maximizing its expected well-being. The corresponding quantity for allocating consumption across time is
where, as before, we assume that W satisfies the conditions W′(X) > 0 and W″(X) < 0. The parameter δ is once again a measure of the rate at which future well-being is discounted—owing to shortsightedness, the possibility of dynastic extinction, and so on. The difference between this and the previous δ is that now we are considering a continuous model rather than a discrete one, but the decay is still assumed to be exponential. In Becky’s world the rate of return on investment is large; that is, investment is very productive. So it makes empirical sense to suppose that r > δ. We will see presently that this condition provides Becky’s parents with the incentive to accumulate wealth and pass it on to Becky and Sam, who in turn will accumulate their wealth and pass that on, and so on. For simplicity, let us suppose that the “curvature” of W, which is −XW″(X)/W′(X), is equal to a parameter α, whose value exceeds 1.3 As we saw earlier, strict concavity of W means that you gain less from increasing consumption than you lose from decreasing it by the same amount. The strength of this effect is measured by α: the larger it is, the greater the benefit of any smoothing you are able to do.
Becky’s parents’ problem at t = 0 is to maximize the quantity in (4) by making a suitable choice of the rate at which they consume their wealth (namely, X(t)), subject to the condition (3), together with the conditions that K(t) and X(t) should not be negative.4 This is a problem in the CALCULUS OF VARIATIONS [III.94]. But it is of a somewhat unusual form, in that the horizon is infinite and there is no boundary condition at infinity. The reason for the latter is that Becky’s parents would ideally like to determine the level of assets that the dynasty ought to aim at in the long run; they do not think it is appropriate to specify it in advance. If we assume for the moment that a solution to the optimization problem exists, then it turns out that it must satisfy the Euler–Lagrange equation:
This equation is easily solved, and gives
However, for this problem we are free to choose X(0). Koopmans showed in 1965 that X(t) in (6) is optimal if W′(X(t))K(t)e−δt → 0 as t → ∞. It transpires that, for the model in hand, there is a value of X(0), which we shall write as X*(0), such that the condition (3) and Koopmans’s asymptotic condition are satisfied by the function X(t) given in (6). This implies that X*(0)e(r−δ)t/α is the unique optimum. Consumption grows at the percentage rate (r − δ)/α and dynastic wealth accumulates continually in order to make that rising consumption level possible. All other things being equal, the larger the productivity of investment r, the higher the optimum rate of growth of consumption. By contrast, the larger the value of α, the lower the rate of growth of consumption, since there is a greater wish to spread it out among the generations.
Let us conduct a simple exercise with our finding. Suppose the annual market rate of return is 4% (i.e., r = 0.04 per year)—a reasonable figure for the United States—that δ is small, and that α = 2. Then we can conclude from (6) that optimum consumption will grow at an annual rate of 2%; meaning that it will double every thirty-five years—roughly, every generation. The figure is close to the postwar growth experience in the United States.
For Desta’s parents the calculations are very different, since they are heavily constrained in their ability to transfer consumption across time. For example, they have no access to capital markets from which they can earn a positive return. Admittedly, they invest in their land (clearing weeds, leaving portions fallow, and so forth), but that is to prevent the productivity of the land from declining. Moreover, the only way they are able to draw on the maize crop following each harvest is to store it. Let us see how Desta’s household would ideally wish to consume that harvest over the annual cycle.
Let K(0) be the harvest, measured, say, in kilocalories. As rats and moisture are a potent combination, stocks depreciate. If X(t) is the planned rate of consumption and γ the rate of depreciation of the maize stock, then the stock at t satisfies the equation
Here, γ is assumed to be positive and both X(t) and K(t) nonnegative. Let us imagine that Desta’s parents regard their household’s well-being over the year to be 
W(X(t)) dt. As with Becky’s household, let −XW″(X)/W′(X) be equal to a number α > 1. Desta’s parents’ optimization exercise is to maximize W(X(t)) dt, subject to (7) and the condition that K(1) ≥ 0.
This is a straightforward problem in the calculus of variations. It can be shown that the optimum maize consumption declines over time at the rate γ/α. This explains why Desta’s family consume less and become physically weaker as the next harvest grows nearer. But Desta’s parents have realized that the human body is a more productive bank. So the family consumes a good deal of maize during the months following each harvest so as to accumulate body mass, but they draw on that reserve during the weeks before the next harvest, when maize reserves have been depleted. Across the years maize consumption assumes a sawtooth pattern. (Readers may wish to construct the model that incorporates the body as a store of energy: see Dasgupta (1993) for details.)
As Desta and her siblings contribute to daily household production, they are economically valuable assets. Her male siblings, however, offer a higher return to their parents, because the custom (itself a social equilibrium!) is for girls to leave home on marriage and for boys to inherit the family property and offer security to their parents in old age. Because of an absence of capital markets and state pensions, male children are an essential form of investment. The transfer of resources in Desta’s household, in contrast to Becky’s, will be from the children to their parents.
The under-five mortality rate in Ethiopia was, until relatively recently, in excess of 300 per 1000 births. So, parents had to aim at large families if they were to have a reasonable chance of being looked after by a male child in their old age. But fertility is not entirely a private matter, since people are influenced by the choices of others. This gives rise to a certain inertia in household behavior even under changing circumstances, which is why even though the under-five mortality rate has fallen in Ethiopia in recent decades, Desta has five siblings.5 High population growth has placed additional pressure on the local ecosystem, meaning that the local commons that used to be managed in a sustainable manner no longer are. That they are not is reflected in Desta’s mother’s complaint that the daily time and effort required to collect from the local commons has increased in recent years.
10 Differences in Economic Life among Similar People
In this article, I have used Becky’s and Desta’s experiences to show how it can be that the lives of essentially very similar people can become so different (for further elaboration, see Dasgupta (2004)). Desta’s life is one of poverty. In her world people do not enjoy food security, do not own many assets, are stunted and wasted, do not live long (life expectancy at birth in Ethiopia is under fifty years), cannot read or write, are not empowered, cannot insure themselves well against crop failure or household calamity, do not have control over their own lives, and live in unhealthy surroundings. The deprivations reinforce one another, so that the productivity of labor effort, ideas, physical capital, and of land and natural resources are all very low and remain low. The rate of return on investment is zero, perhaps even negative (as it is with the storage of maize). Desta’s life is filled with problems each day.
Becky suffers from no such deprivation (for example, life expectancy at birth in the United States is nearly eighty years). She faces what her society calls challenges. In her world, the productivity of labor effort, ideas, physical capital, and of land and natural resources are all very high and continually increasing; success in meeting each challenge reinforces the prospects of success in meeting further challenges.
We have seen, however, that, despite the enormous differences between Becky’s and Desta’s lives, there is a unified way to view them, and that mathematics is an essential language for analyzing them. It is tempting to pronounce that life’s essentials cannot be reduced to mere mathematics; but in fact mathematics is essential to economic reasoning. It is essential because in economics we deal with quantifiable objects of vital interest to people.
Acknowledgments. In describing Desta’s life, I have received much guidance from my colleague Pramila Krishnan.
Further Reading
Dasgupta, P. 1993. An Inquiry into Well-Being and Destitution. Oxford: Clarendon Press.
——. 2004. World poverty: causes and pathways. In Annual World Bank Conference on Development Economics 2003: Accelerating Development, edited by F. Bourguignon and B. Pleskovic, pp. 159–96. New York: World Bank and Oxford University Press.
Debreu, G. 1959. Theory of Value. New York: John Wiley.
Diamond, J. 1997. Guns, Germs and Steel: A Short History of Everybody for the Last 13,000 Years. London: Chatto & Windus.
Durlauf, S. N., and H. Peyton Young, eds. 2001. Social Dynamics. Cambridge, MA: MIT Press.
Evans, G., and S. Honkapohja. 2001. Learning and Expectations in Macroeconomics. Princeton, NJ: Princeton University Press.
Fogel, R. W. 2004. The Escape from Hunger and Premature Death, 1700–2100: Europe, America, and the Third World. Cambridge: Cambridge University Press.
Fudenberg, D., and E. Maskin. 1986. The folk theorem in repeated games with discounting or with incomplete information. Econometrica 54(3):533–54.
Landes, D. 1998. The Wealth and Poverty of Nations. New York: W. W. Norton.
Ramsey, F. P. 1928. A mathematical theory of saving. Economic Journal 38:543–49.
Samuelson, P. A. 1947. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press.
——. 1958. An exact consumption loan model with or without the social contrivance of money. Journal of Political Economy 66:1002–11.
1. As suggested by McElroy and Homey in 1981, a realistic alternative would be to suppose that household decisions are reached by negotiation between the various parries (see Dasgupta 1993, chapter 11). Qualitatively, nothing much is lost in my assuming optimizing households here.
2. Presently we will see why we need to distinguish J from F, rather than looking just at F.
3. This means that W has the form B − AX−(α−1), where A (which is a positive number) and B (which can be of either sign) are the two arbitrary constants that arise when we integrate the curvature of W to arrive at W itself. We will see presently that the values that are adopted for A and B have no bearing on the decisions that Becky’s parents will want to make; that is, Becky’s parents’ optimum decision is independent of A and B. Notice that, as α > 1, W(X) is bounded above. The above form is particularly useful in applied work, because in order to estimate W(X) from data on household consumption, one has to estimate only one parameter, α. Empirical studies of saving behavior in the United States have revealed that α is in the range 2–4.
4. This problem originated in a classic paper by Ramsey (1928). Ramsey insisted that δ = 0 and devised an ingenious argument to show that an optimum function X(t) exists despite the fact that the integral in (4) does not converge. For simplicity, I am assuming δ > 0. As W(X) is bounded above and r > 0 (meaning that it is feasible for X(t) to grow indefinitely), we should expect (4) to converge if X(t) is allowed to rise fast enough.
5. See Dasgupta (1993) for the use of interdependent preferences to explain fertility behavior. In the notation of the section on social equilibria, we are to suppose that household h’s well-being has the form Wh(Xh, X−h), where one of the components of Xh is the number of births in the household, and that the higher the fertility rate is among other households in the village, the larger the desired number of children in h. The theory based on interdependent preferences interprets transitions from high to low fertility rates as bifurcations. Fertility rates are expected to decline even in Ethiopia. Interdependent preferences are currently being much studied by economists (see Durlauf and Young 2001).