In most of the prior examples, production was very simple. The economy consisted of one or more apple producers, each of whom grew trees with the same output (63 apples if the weather was good, 48 apples if the weather was bad).
Consider now a slightly more complex firm: one that will produce 20 apples today, 63 apples in the future if the weather is good and 48 apples in the future if the weather is bad. This production plan can be plotted as a point in a three-dimensional diagram in which the axes plot apples today, apples next year if the weather is good, and apples next year if the weather is bad, as shown below.
As before, assume that it is possible to swap any one of the three time-state claims for one or more of the other two. Any two swap ratios can be utilized to summarize all possibilities. As before we use the prices (present values) of the atomic securities, each of which represents a payment for one and only one future state of the world.
While a firm's production plan will plot as one point in this type of diagram, by utilizing swaps (including traditional market exchanges), the firm can achieve any point on a plane that goes through that point. The plane is easily constructed, once the production plan and market prices (swap terms) are known. If the firm issues more than one class of security, each class will plot at a particular point on a lower plane, but the total value of the various classes will be the same. The set of decisions taken by the firm to move to different points on one or more planes can be considered its financing plan.
Of course, the shareholders need not consume the mix corresponding to the point selected by the firm via the combination of its production plan and its financing plan. They, too, can utilize swaps (market exchanges) to collectively achieve any desired points on any desired planes of their own that collectively have the same value as the firm's production plan. Typically, each shareholder will obtain one point on his or her own plane, and utilize market trades to transform it into another point. The total of the consumption-investment combinations chosen by the shareholders will, of course, lie on the plane going through the firm's production point and have the same value as the production plan.
In a world of full information and zero transactions costs, it is the distance from the origin of the plane on which a firm's production plan plots that matters. The particular location chosen on the plane (or sub-planes) is, in such a setting, irrelevant to shareholders. If the firm does not choose a point or points that they collectively prefer, they can utilize market exchanges to move to such a point or points themselves.
In our simple world, the measure of the value of a production plan is the distance from the origin of the plane that represents combinations of time-state claims that can be obtained from it. Given market prices, all such planes are parallel to one another, so one can measure the desirability of any such plane using virtually any chosen dimension. It is conventional to do this using the "now" axis. In other words, the desirability of a production plan is measured in terms of the number of present goods for which it could be traded. Equivalently, we measure the present value of the plan.
The term wealth is often used to indicate a person's present value. For a firm financed entirely through equity, the goal of management can be stated as the maximization of shareholder wealth. The analogous criterion for a firm with more complex financing is the maximization of the wealth of those holding claims on the firm's production (the firm's stakeholders).
Given the ability of a firm's stakeholders to trade time-state claims on their own, the financing plan chosen by a firm should be irrelevant. So should any swaps made by the firm, once its production plan is in place. The key issues concern the "trades with nature" that result in the production plan. In the type of world we have described, management should concentrate on choosing a plan that will produce the maximum possible present value for required investment. Aspects of financing via issuance of different types of securities, risk-reduction via swaps, etc. are of no importance.
More generally, a firm should undertake new projects if and only if their value exceeds the cost of marketable alternatives with the same time-state payments. The manner in which the needed resources are obtained is (at best) of secondary importance.
While such statements are correct in the world under discussion, they are hardly likely to apply without qualification in the world as it is. Transactions costs, asymmetric information, concentration of managers' human and other capital in the firms for which they work, and a host of other issues make some corporate financial decisions more advantageous than others. However, such aspects cannot be adequately analyzed until the essential framework for understanding economies without such features has been built. Hence we continue to deal with our simple frictionless world in which information concerning time-state payoffs is known to all.
Consider a firm with given resources. It will typically have to choose among a number of alternative production plans. Some will be inefficient. In this context, a plan is inefficient if there is an alternative plan that provides more of at least one time-state claim and no less of any time-state claims. Inefficient plans can be rejected out of hand, since each is dominated by at least one other alternative. But this will typically still leave for consideration a number of efficient plans -- i.e. plans that are not inefficient.
Efficient plans will generally plot on a "hill" in a diagram with time-state payments on the axes -- a hill that "bulges out" from the origin of the diagram. As long as production technologies may be undertaken at any desired scale with proportionate results, this surface is guaranteed never to "cave in". Consider two plans, X and Y. Point X might represent planting an entire orchard with one type of apple tree, and point Y planting it with another type. Now consider a plan in which half the orchard is planted with one type of tree and the other half with the other. If each tree has the same characteristics, no matter how many are planted, the result will lie half-way between points X and Y in every dimension -- i.e. on the straight line connecting the points in the diagram. If no better alternative can be found, at least this linear set of combinations will be available. Thus the surface will not cave in. If a better alternative is in fact available, the surface will bulge out.
The set of efficient production possibilities can be termed the production possibility frontier. If it bulges out, the technology evidences decreasing returns to scale -- a widely-observed phenomenon. A rather crude analogy holds that it looks something like an upside-down mixing bowl (although perhaps a somewhat irregular one).
Insuring that production is efficient is, at base, a technical issue. This criterion can be met without reference to market prices. However, the choice of the best (optimal) plan from among the efficient plans requires the use of market prices.
The rule is simple: for given resources, among such plans select the one with the largest present value. Graphically, this involves selecting the point on the production possibility frontier at which a value plane (every point on which has the same market value) is tangent to (touches but does not intersect) the frontier. This point represents the wealth-maximizing production plan. Its selection is the only important decision made by management in our setting. In the real world, it is still likely to be by far the most important decision for a firm -- considerably more important than financing decisions.
Again, a crude analogy: think of the valuation plane as a cookie sheet. Then the optimal production plan lies at the point at which the cookie sheet touches a point on the upside-down mixing bowl.
Firms (producers) play a key role in an economy. Individuals (consumers) are the other key players. Ultimately, of course, all resources (including both physical and human capital) are owned by individuals, so people function in both roles.
Given his or her resources, an individual will "own" a particular combination of time-state claims. For example, a worker may expect to receive a salary of 100 apples today and a salary of 110 apples in the future if the weather is good or 90 apples if the weather is bad. Such an endowment will plot as a point in the kind of diagram we have been using. However, given the ability to trade in markets, the individual can choose to consume any point lying on the value plane that includes his or her endowment point. This can be done by lending, borrowing, or any of a host of investment transactions.
An individual's wealth can be measured by the amount of present consumption that he or she could obtain by exchanging all future prospects for present values, plus the value of present prospects already attained. Acting in one's role as producer, it is desirable to choose a career, location, education, etc. to maximize this present value. We ignore here the importance of intangibles that are not fully tradable, such as peace of mind, integrity, etc., (but do not wish to leave the impression that they are unimportant). In any event, given such decisions, the individual as consumer faces a value plane representing the available combinations of time-state claims. This is sometimes termed a budget plane or consumption opportunity set.
It is trivial, but nonetheless correct, to suggest that a consumer should select from among available consumption combinations the one that he or she likes best. Ultimately, this depends on individual preferences. However, it is possible to argue that for most people such preferences are likely to have certain characteristics.
It is useful to consider sets of time-state claims among which a consumer exhibits indifference. For example, if a given consumer considers combinations X, Y and Z equally desirable, we say that he or she is indifferent among them. Graphically, they lie on the same indifference surface. A given consumers' preferences can then be represented by a set of such surfaces. Such surfaces will not intersect, since this would involve a contradiction. Assuming non-satiation of preferences, i.e. that each time-state claim is a good (more is better), such surfaces will not have the appearance of the production possibility surface. Rather, they are likely to "curve away" from the origin in a diagram with time-state payments on the axes. In most cases, they will have the appearance of mixing bowls that are "right-side up". Of course, there will be many of them, each representing a set of alternatives preferred to that on the surface below (closer to the origin). The appearance will thus be something like that a set of nested mixing bowls.
The figure below shows one "cut" of this relationship. The horizontal axis plots an amount consumed in the present, while the vertical axis plots an amount consumed for certain in the future. In such a trade-off, only the timing of consumption is of relevance, since there is no uncertainty.
Each curve in the figure shows combinations of these two consumption items among which the investor is indifferent. Only a few such curves are shown, of the very many that could be drawn. The key assumption is that each curve becomes flatter as one goes from the upper left portion to the lower right portion. Equivalently, the added amount of the good on the X axis that the consumer will require to give up a unit of the good on the Y axis will be larger, the larger the amount of X and smaller the amount of Y.
The figure below shows another type of trade-off. Here the amount consumed in the present is assumed to be fixed, with only the amounts to be consumed under each of the two possible future states to be determined. The vertical axis plots the amount consumed if the weather is good, while the horizontal axis plots the amount consumed if the weather is bad. Needless to say, the diagram reflects preferences when these are still contingent claims -- i.e. before the actual weather pattern is known.
Here, too the consumer is assumed to have indifference curves that get flatter as one moves from the upper-left to the lower-right. In a sense, this is no different than in the first figure. However, given the nature of the goods in question, there are further implications.
Consider combinations X and Y. Since they are on the same indifference curve, the consumer considers one as desirable as the other. For convenience, we denote payments B and G in bad and good weather, respectively, as [B G]. In the figure, X is thus [20 80] and Y is [50 40]. Now consider combination Z, which pays [35 60] and plots midway between X and Y. Clearly, the individual would prefer it, since it lies on a higher (better) indifference curve.
Now assume that there are two consumers, each with the preferences shown in the figure. Assume, moreover, that one holds securities providing combination X, while the other holds securities providing combination Y. Together, their portfolios will pay [70 120].
Imagine that a clever entrepreneur sets up a mutual fund, suggesting that both consumers "invest" their shares in return for half the payments received by the fund. Under this arrangement, each will obtain [35 60]. But this is combination Z. The first consumer has traded X for Z, and is happier. The second consumer has traded Y for Z and is also happier. In fact, the entrepreneur could take a bit of the action and still make both consumers (who are now investors) happier.
Needless to say, the increased happiness is an ex ante construct in this case. After the fact, one of the two investors will be better off, and the other worse off, than had the change not been made. But this involves hindsight, which always is characterized by 20/20 vision.
As the example illustrates, consumers with preferences of the assumed sort will find it desirable to diversify. In an important sense, they are risk-averse.
Each producer in an economy of the sort we have analyzed will choose a production plan based on available resources, technological possibilities and current market prices. Similarly, each consumer will choose a consumption plan based on wealth, preferences and market prices. But what will assure that the total amount of each time-state claim provided by producers will equal the total amount that consumers collectively wish to consume?
The answer, of course, lies in the role of prices in a market economy.
Given a set of prices, if consumers wish to consume more of one time-state claim than producers wish to produce, there will be "buying pressure" (the quantity demanded will exceed the quantity supplied), and the price will rise. Conversely, if consumers wish to consume less of a time-state claim than producers wish to produce, there will be "selling pressure" (the quantity supplied will exceed the quantity demanded) and the price will fall. Such changes will continue until equilibrium is achieved -- i.e. the market for each time-state claim clears (quantity demanded equals quantity supplied). The prices that accomplish this are termed equilibrium prices.
There is no point debating the "causes" of such prices. Both supply (production opportunities) and demand (consumer preferences) determine them. Equilibrium prices result from the interaction of both forces, as well as the initial distribution of wealth (including human abilities).
When an equilibrium is achieved, the set of the amounts of time-state claims produced and the set of the amounts consumed will be the same. This combination can be termed the societal aggregate product.
Assume that in a society the aggregate product in dollars or apples is qtotal:
Present 100 Future if Bad Weather 50 Future if Good Weather 150
Denote the three alternatives as N (now), B (future if weather is bad) and G (future if weather is good). Assume that, as before, the equilibrium price of 1B is 0.665N, and the equilibrium price of 1G is 0.285N. Thus the price vector p is:
Future Future Present if Bad if Good 1.000 0.665 0.285
The value of the aggregate product, p*qtotal is $176.00.
Assume there are two people in this society, each with the same initial wealth ($88.00). If their preferences were the same, each could select [50 ; 25; 75 ] of [N ; B ; G ], and the markets would clear (equivalently, their aggregate preferred holdings would equal the aggregate product). If their preferences and/or wealths differed, they would generally choose different mixes. However, for the markets to clear, their aggregate preferred holdings must equal the aggregate product.
Equivalently, we might represent each individual's consumption choices in terms of the relative values of the holdings. Assume that individual X's wealth is $59.00 and that she chooses [40; 20; 20]. In this context it seems appropriate to call her investor X. Her consumption proportions will then be those shown below.
Holding Price Value Proportion N 40 1.000 40.00 0.6780 B 20 0.665 13.30 0.2254 G 20 0.285 5.70 0.0966 ------ ------ 59.00 1.0000
Thus Investor X has chosen to consume 67.8% of her wealth now, and to invest the remaining 32.2%. Of the amount invested, 70% (0.2254/0.3220) will be used to purchase claims that pay off in state B and 30% (0.0966/0.3220) to purchase claims that pay off in state G (or some combination of other securities that gives the same overall set of exposures).
In this situation, Investor Y will have a wealth of $117.00 and choose the following portfolio:
Holding Price Value Proportion N 60 1.000 60.00 0.5128 B 30 0.665 19.95 0.1705 G 130 0.285 37.05 0.3167 ------ ------ 117.00 1.0000
We can also characterize the aggregate product in proportional terms; thus:
Holding Price Value Proportion N 100 1.000 100.00 0.5682 B 50 0.665 33.25 0.1889 G 150 0.285 42.75 0.2429 ------ ------ 176.00 1.0000
In this world, Investor X has 33.52% of total wealth (59.00/176.00) while Investor Y has the remaining 66.48%. If we were to compute a wealth-weighted average of their two consumption mixes (expressed in proportions), we would obtain the aggregate mix (also expressed in proportions). In this sense, consumers collectively consume the aggregate product of current and contingent goods.
Now consider a related set of calculations in which only future claims are included. The resulting three investment portfolios would then be characterized as follows:
Investor X ------------ Holding Price Value Proportion B 20 0.665 13.30 0.70 G 20 0.285 5.70 0.30 ------ ----- 19.00 1.00 Investor Y ----------- Holding Price Value Proportion B 30 0.665 19.95 0.35 G 130 0.285 37.05 0.65 ------ ----- 57.00 1.00 Society ------- Holding Price Value Proportion B 50 0.665 33.25 0.4375 G 150 0.285 42.75 0.5625 ------ ------- 76.00 1.0000
The aggregate portfolio is often termed the market portfolio.
In terms of invested wealth, Investor X has 25% (19.00/76.00) while Investor Y has 75% (57.00/76.00) of the total. Relative to the market, Investor X is overweighted in B (70.0%, compared with 43.75%) and underweighted in G (30.0%, compared with 56.25%), while Investor Y is underweighted in B (35.0%, compared with 43.75%) and overweighted in G (65.0%, compared with 56.25%). However, their weighted average holdings will equal those of "the market" precisely. Not surprisingly, if one or more investors chooses to hold less than market proportions of a security, other investors must choose to hold more than market proportions, and the total value of the first group's underweighting must equal that of the second group's overweighting.
In one sense, this is simply an accounting identity: that which is, must be held. But in a world in which prices have adjusted to achieve equilibrium, no one holds more or less than desired. Thus investors who underweight or overweight relative to the market portfolio do so voluntarily ( for what must seem to them at the time good reasons).
An investor can choose to hold securities in market proportions. On average, taking wealth into account, investors must do so. In this sense, investing in the market portfolio represents a default position. The Analyst must thus address two key questions:
Should the overall portfolio diverge from the market? If so, which types of securities should be underweighted and which ones should be overweighted?
Practical and theoretical problems abound, but it is important to ask the right questions.