Two-Good Market Analysis
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These slides review the basic analytical tools of output choice that we will need to study trade in a 2-good general equilibrium framework.
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Two-Good Market Analysis
- 2. Goal Develop the tools to analyze trade in a 2-good economy: PPF (and budget constraint, a.k.a. budget line) Preferences (collective and individual) Choice (how the PPF and preferences interact)
- 3. PPF
- 4. PPF: Increasing costs • The slope of the PPF is the opportunity cost of one unit of good 1 in terms of good 2, a.k.a. the marginal rate of transformation of good 1 for good 2 • MRT_1,2 = MC_1/MC_2 • With a concave PPF, the MRT_1,2 is increasing as more of good 1 is produced
- 6. Indifference curves An indifference curve is a set of combinations of good 1 and good 2 that yield the same level of welfare to society We assume society figures out a way to aggregate its individual preferences into a set of collective preferences We make the following assumptions regarding the collective preferences of society More is better than less: A is preferred to B if it has more of at least one of the goods Averages are better than extreme combinations: C is preferred to A and B, if C is a combination that averages A and B Preferences are consistent: if society prefers A to B and B to C, then society also prefers A to C. By the 2nd assumption above, we draw the indifference curves as convex curves.
- 10. Marginal rate of substitution
- 11. Output choice The output bundle (the amounts of good 1 and good 2 produced) will be determined by the interaction between the production possibilities of the economy captured by its PPF and the collective preferences over the two goods captured by its indifference curves. Our focus is on the mechanics of choice. Once we define an equilibrium (or “optimal output choice”), we’ll want to shock the model to see what happens to this optimal bundle if the economic environment changes. Again, if this economy does not trade, the output bundle chosen is also necessarily the consumption bundle.
- 12. Output choice: optimal output bundle
- 13. Output choice: optimal output bundle
- 14. Output choice: optimal output bundle 2
Notes de l'éditeur
- The goal of this presentation is to introduce the tools required to analyze the effect of trade in an economy with two goods. First off, we assume an economy that can only produce two goods.Next, we develop the concepts of (1) production possibilities and (2) preferences (or welfare). We will connect this with the elements of choice theory used in the case of an individual (or of an individual organization, like a household, business, or government).Economists view the choice that a society as a whole (an “economy”) makes – just like the choice that an individual makes – as resulting from the consideration of two elements: (1) what the given society (or individual) can afford and (2) what the given society (or individual) desires or values. Society’s desires or values (or needs or wants) indicate the direction in which it will strive through its choices, what defines its success or well-being. Since we have defined “goods” as whatever increases a society’s well-being, then our society will want more of each of the 2 goods – if it can afford to get them. And this leads me to the second element of choice: society’s possibilities. What a society can consume, depends on the goods available to such society for consumption.Now, if we consider human society as a whole, there’s nobody else to trade with. There’s no other society, some other species from a planet nearby, interested in trading with us. As a result, if we consider the human economy as a whole, we note that it is a closed economy or an autarkic economy, in the sense that we don’t trade with any other non-human society. Consequently, for the human economy as a whole, we can only consume what we ourselves produce.This is, of course, different if we take different individual societies – subsets of the human society, e.g. individual nations or regions or states or – within a given nation-state: individual households, businesses, and governments. An individual nation can trade with another individual nation or with the rest of the world. Similarly, a household or an individual person can trade with the rest of the world.The general principles of trade apply similarly to nations or to smaller individual entities. And we’ll try to make that connection here.
- We begin by introducing the concept of the production possibilities curve or production possibilities frontier. For simplicity, we will call it the PPF of an economy.You have probably encountered this concept before, but it is particularly useful when studying trade. That is why we will expand in our understanding of this concept.The PPF is plotted in a 2-good space: The horizontal axis indicates quantities of good 1 and the vertical axis indicates quantities of good 2. Each point in the 2-good space (in the nonnegative quadrant) indicates a given combination of good 1 and good 1.The PPF is defined as the set of the possible combinations of good 1 and good 2 that this economy can produce in a given period of time, given also the productive resources (natural resources, means of production, and labor power) available to it during the period of production. We will assume that, at a given point in time, say, at the beginning of the production period (and also during the production period) the amount of productive resources available is fixed. We also assume that the technology embedded or embodied in those resources is fixed: the stock of knowledge, scientific and empirical, all the know-how, the workers’ skills, etc. are fixed. Finally, we assume that – to pin down the PPF – this economy employs fully all of its available resources and given technology. In other words, there is full employment.The area below and to the left of the PPF is said to be the feasibility or feasible set – it is the set of all the bundles of good 1 and good 2 that the economy can produce with the resources and technology available. Note that this set includes the boundary: the PPF. However, the feasibility set includes all the combinations of good 1 and good 2 that the economy can produce with the available resources and technology without necessarily employing them fully. Clearly, since we assume that this economy will seek to get the most of each good that it can afford, there is no reason why this society (if it can control the economic process) would want to accept anything less than full employment. Since there are only two goods that can be produced in this period, and it is not possible to produce anything else or leave anything for the future, then it would be unwise for this economy to produce at less than full employment. Producing at less than full employment entails unrecoverable waste. Note that, as you probably learned in macroeconomics, full employment does not mean that the economy is producing the most it can produce with no regard for the future, but rather the level of employment of its resources that would ensure its continuity or sustainability into the indefinite future. So, people are not overworked, working two shifts, etc., the same with factories and machines, which require repair and maintenance, but just working normally. Similarly, it means that there’s no slack, stopped factories and unemployed workers, but a normal level of employment. This is a notion similar to the Aristotelian “just mean.”Now, everything above and beyond (outside of) the PPF is unfeasible. They are combinations of the 2 goods that cannot be produced sustainably with the resources and technology available. Note that the PPF indicates various *possible* combinations of the 2 goods that can be produced by this economy. Of course, only one of them *will* actually be produced. If with full employment, then the combination will be a point on the PPF. If there’s unemployment, then the output point or bundle will be inside of the feasibility set. If the economy chooses to produce some combination of the 2 goods, we will call that the *output* choice, bundle, point, or combination.What the PPF intends to highlight is, precisely, what it entails for an economy to choose one output bundle in comparison to other possible ones. Economists say that, since the resources and technology available are given and finite, in other words, since human productivity is finite, then the production of more of one good requires that we sacrifice the production of the alternative good to a certain extent. That extent is indicated by the PPF.One other thing here is that all points on the PPF are all, from the viewpoint of the production possibilities of society, efficient. We will say that they are productively efficient. This does not mean that they are optimal or efficient in general, because we haven’t yet taken into consideration the preferences of society. The point here is that we won’t expect society to pick bundles that do not belong on the PPF, since points outside of the PPF are either unfeasible or productively inefficient or wasteful of the available resources.It may be convenient at this point to imagine this as a closed economy – an economy in isolation, which has consequently to produce all it wishes to consume. In this case, the output bundle will also be the same as the *consumption* bundle.You will note that I have drawn the PPF as a curve concave to the origin. Economists tend to believe that this type of PPF is the paradigmatic one, and we will study it next.
- A concave PPF describes the case when the sacrifice an economy must incur to produce one additional unit of a given good is increasing. And this sacrifice of a certain amount of good for the sake of one additional unit of the other good is what we call the “cost” of producing one unit of the latter good.Suppose for example that the economy has chosen to produce 4 units of good 1 and 20.1 units of good 2, a point along the PPF and therefore feasible. Now suppose that the economy ponders what it would take to produce ten additional units of good 1. If the economy is to remain on the PPF and it decides to produce 14 units of good 1, then the economy will only be able to produce 3 units of good 2. In other words, the 10 additional units of good 1 will cost this economy 3 - 20.1 = - 17.1, that is, 17.1 units of the good. Now, let’s determine the cost of *one* single unit of good 1 if it decides to make this move. Since it takes a sacrifice of 17.1 units of good 2 to produce 10 extra units of good 1, then the economy must sacrifice 17.1/10 = 1.71 units of good 2 per unit of good 1 if it is to make such a move. Note that, geometrically or graphically, the cost of one additional unit of good 1 is the *slope* of a line joining the points representing these two output bundles – the initial and the final one. The slope = change in good 2/change in good 1 = (3-21.1)/(14-4) = -17.1/10 = -1.71. Note that the slope is negative, since one of the goods (good 2) must decrease for the other good (good 1) to increase. What the slope does is show how much of good 2 decreases to produce 1 additional unit of good 1. The slope of the line joining the two points show the cost of one unit of the good on the horizontal axis in terms of the other good (good 2). Algebraically: Slope = delta x_2/delta x_1.Since the PPF is assumed to be downward sloping, which means that the slope is expected to be negative, we don’t usually care much about the negative sign. To be strict, we should say that we only care about the absolute value of the slope: |delta x_2/delta x_1|. But to keep things simple, we may just drop the absolute value operator.Also note that, if you want to determine the cost of producing 1 extra unit of good 2 (in terms of good 1), you have to flip the axis or take the reciprocal of the slope. It’s as if you switched the axis, and the one that is horizontal became vertical and the vertical one became horizontal.Introductory textbooks usually say that the slope of the PPF, taken a two given points (or output bundles) is a measure of the *opportunity cost* of one unit of the good on the horizontal axis, cost in terms of the good on the vertical axis. Opportunity cost is fine, but technically the term economists use is “marginal rate of transformation” or MRT for short. It is the same thing: cost or MRT.The MRT is defined as the ratio of the marginal cost of good 1 to the marginal cost of good 2: MRT= MC_1/MC_2 = delta x_2/delta x_1. The marginal cost of any good is the cost of producing one additional unit of such good, expressed in monetary terms ($/unit of the good).To explain this in the simplest possible way, suppose good 2 is gold, and its units are ounces. Suppose also that it is determined (say, by the government) that 1 oz of gold = $1 and we use that as monetary units. The cost in $ of producing 1 oz of gold is obviously $1, by definition. In other words, the MC of 1 unit of good 2 is $1, by definition. Now imagine that the economy moves from bundle (4, 20.1) to bundle (14, 3), and let us express the changes (the gain of good 1 and the loss of good 2) in $. The gain of good 1 is 14 – 4 = 10 units of good 2. On the other hand, the loss in good 1 is 20.1 -3 units of good 2 or 17.1, which in $ is a loss of $17.1, because each oz. of gold is $1. We now ask what is the cost of 1 unit of good 1 if the economy is to make such a move. In economics, the cost of one extra unit of the good is called the *marginal cost*. The marginal cost of good 1 is then $17.1/10 units of good 1 = $1.71/unit of good 1. Now, since the marginal cost of good 2 is $1/oz of gold, then MRT=MC_1/MC_2 = ($1.71/unit of good 1)/($1/oz of gold) = 1.71 oz of gold/unit of good 1 = 1.71 units of good 2/unit of good 1. This is the same result we obtain by simply taking the slope of the PPF between those two output bundles.Now, let me call your attention to the curved or concave nature of this PPF. It is clear that the cost of producing one unit of good 1 varies depending on where you are at to begin with and where you’ll land at the end. For example, if you start at point (4, 20.1) and move to point (5, 19.6), the slope of the line joining those two points is: change in x_2/change in x_1 = (19.6 – 20.1)/(5-4) = -0.5/1. If the economy is making that move, the cost of producing one unit of good 1 will only cost it half a unit of good 2.However, if the economy is at point (13, 8.3) and it decides to move to point (14, 3), then the slope is: (3-8.3)/(14-13) = -5.3/1. If the economy makes this move, it will cost it 5.3 units of good 2 to produce one extra unit of good 1. What we observe then is that, as this economy produces more of good 1, it becomes increasingly expensive to produce the next unit of good 1. Similarly, the more of good 2 you produce, the more costly it is to produce an extra unit of such good, in terms of the sacrifice of good 1 that must be incurred.As I said, economists believe this to be the paradigmatic case. Why? Because, under the assumption that these goods remain goods (and don’t turn into bads at some point; in other words that the preference for these goods remains), if the production of one good didn’t entail (at some point) increasing sacrifice of other goods, then we would expect to see the economy producing only that good and none of the rest. Since we see that actual economies produce a large array of goods, then this suggests that increasing costs are common in the economy. This is not to say that, within limits, there are cases of goods whose production exhibits decreasing costs. That is the case of all networks – electric, gas, phone, cable, Internet distribution grids, e.g., once they are laid out – make it cheaper and cheaper to add subscribers. The limit to the supply of these goods is not the costs, which go down and down, but the size of the social appetite (backed by money) for them.Before moving to the next slide, let me say that -- ideally -- we would want to determine the MRT_1,2 at very small changes from one output bundle to another. In other words, we would want to determine the sacrifice of good 2 incurred when the output of good 1 increases just a tiny little bit. Graphically, we are talking about looking at the slope of the PPF at each given point. To do this rigorously, we need calculus. So I will not dwell on this. I just want the student to be mindful of the nature of the issues involved here.Let us now consider a simpler PPF…
- This PPF is linear – a straight line. Note that, if the production possibilities of an economy are as described in this PPF, then moving from output point (4, 17.6) to output point (14, 6.6) entails a cost of: (6.6 – 17.6)/(14-4) = -11/10 = -1.1, i.e. 1.1 units of good 2 per unit of good 1 must be sacrificed to expand production of good 1 from 4 to 14 units.Let us now consider that the economy is moving instead from output point (4, 17.6) to point (5, 16.5). In that case, the cost of one unit of good 1 is given by the slope of the line joining those two points: (16.5 – 17.6)/(5-4) = -1.1. And if the economy were to move from output point (13, 7.7) to point (14, 6.6), the cost of one unit of good 1 is given by the slope of the straight line joining those two points: (6.6 – 7.7)/(14-13) = -1.1. Also 1.1. Again, this is the MRT_1,2.When the PPF is a straight line, the cost of producing one unit of good 1 or the MRT_1,2 is constant. No matter where on the PPF you start and where you wind up, the MRT_1,2 or opportunity cost is always the same 1.1 units of good 2/unit of good 1.We are now ready to discuss the other element of the choice of output – the preferences or the welfare considerations of society. For simplicity, we will assume that the PPF is linear – i.e. constant costs.
- We now introduce an important concept in choice theory: the concept of indifference curves. An indifference curve is a set of combinations of good 1 and good 2 that yield the same level of welfare to society.We assume society figures out a way to aggregate its individual preferences into a set of collective preferences.We make the following assumptions regarding the collective preferences of society: (1) More is better than less: A is preferred to B if it has more of at least one of the goods. (2)Averages are better than extreme combinations: C is preferred to A and B, if C is a combination that averages A and B. And (3) preferences are consistent: if society prefers A to B and B to C, then society also prefers A to C.Note that every point in the 2-good space belongs to some indifference curve, even if we don’t draw that indifference curve. There is indeed – we assume -- an infinite number of indifference curves in the 2-good space: an indifference *map*. Obviously, we will only draw a few representative indifference curves in a given diagram.
- This diagram illustrates the assumption we make regarding preferences that “More is better than less.” The more of the goods you get, the better the combination, the more society prefers it to an alternative.Note that C has more of the two goods compared to B and B compared to C. Consequently, C is on a higher indifference curve than B and B on a higher indifference curve than A. One way to look at this is to imagine that at each bundle, the 2-good space gets split into 4 quadrants. The northeast quadrant will have preferences that are preferred to those of the given bundle. As long as a bundle has more of at least one of the goods, it will be preferred to the alternative. Bundles in the southwest quadrant are not preferred to the given bundle.Bundles that have more of one good, but less of the other are not necessarily better or worse – preferred or not. To determine that, we need to define the indifference curves. Bundles that are in higher indifference curves are always preferred to bundles in lower indifference curves. Note that U3 > U2 > U1.
- This diagram illustrates the 2nd assumption we make, namely that averages are better than extreme bundles. Extreme bundles are bundles that have a lot of one good but little of the other. Average bundles are those that have some of one good and some of the other. In the diagram, bundle C is an average of bundles A and B. Bundle A has 7 units of good 1 and B has 13 units. Bundle C has (7+13)/2 = 20/2 = 10 units of good 1. Also, bundle A has 15.7 units of good 2, bundle B has 8.5 units of good 2, and bundle C has the average amount of good 2: (15.7 + 8.5)/2 = 24.2/2 = 12.1. That average can be viewed as the midpoint of a straight line connecting bundles A and B, and it can be seen that C touches a higher indifference curve than the one to which A and B belong.
- This diagram illustrates the 3rd assumption we impose on preferences – that they have to be consistent.What consistency means here is something very specific: transitivity. If you say that you prefer 1 slice of pizza to a scoop of ice cream and that you prefer one scoop of ice cream to a hamburger, then you must necessarily prefer 1 slice of pizza to 1 hamburger. Otherwise, you’d be contradicting yourself. Now, this is not to say that people are always or necessarily consistent in their choices. We are just pretending here that people tend to be consistent, and drawing the conclusions that flow from such assumption. People are complicated.Now, if people are consistent, then it is not possible for indifference curves to cross. This assumption rules out the possibility of indifference curves crossing.
- Before we put the PPF and society’s preferences over the goods together, and see how society chooses its optimal level of output of the 2 goods, let us introduce the important concept of the marginal rate of transformation of good 1 for good 2, and what the convexity of the indifference curves we are assuming implies on that regard.The marginal rate of substitution of good 1 for good 2 is the amount of good 2 that society is willing to sacrifice for the sake of an additional unit of good 1. It is calculated with the formula:MRS_1,2 = \\Delta x_2/\\Delta x_1Note that this represents the willingness of society with regards to its desires or wants or needs or preferences. This should be separate from the issue, which we covered when we discussed the PPF, of what society must sacrifice of good 2 for the sake of one extra unit of good 1, i.e. the MRT_1,2. The MRT is determined by the production possibilities, available productive resources and technology. The MRS, on the other hand, has to do with what society likes and values.That said, it should be clear that, when we draw indifference curves as convex curves, we are assuming that the MRS_1,2 is decreasing in x_1 – i.e. the MRS goes down as society has more of good 1 and less of good 2. As society gets more of good 1, then it appreciates or values or likes good 1 less and less, while good 2 gains a higher appreciation from society. In brief, just as we specified production possibilities in such a way that the MRT (or cost) was increasing, we now say that the MRS is decreasing.With all that in place, we are ready to study how an economy (under our assumptions) chooses its output of the two goods.
- The output bundle (the amounts of good 1 and good 2 produced) will be determined by the interaction between the production possibilities of the economy captured by its PPF and the collective preferences over the two goods captured by its indifference curves.Our focus is on the mechanics of choice. Once we define an equilibrium (or “optimal output choice”), we’ll want to shock the model to see what happens to this optimal bundle if the economic environment changes.Again, if this economy does not trade, the output bundle chosen is also necessarily the consumption bundle.
- This diagram shows society’s PPF, which is linear, i.e. it exhibits constant costs or constant MRT_1,2, and 3 representative indifference curves to illustrate how society chooses its optimal output bundle.Note that bundle C is not attainable. It falls outside of the PPF. So C is out of the question. A, on the other hand, is attainable, and it is on the PPF. So, from the viewpoint of production possibilities, this is an efficient choice of output. However, it is not optimal, once we take into consideration the preferences of society. At A, the MRS_1,2 > MRT_1,2. This means that, at A, society is required to sacrifice less of good 2 per unit of good 1 than it is willing to sacrifice. Society could move up to a higher level of welfare, a higher indifference curve, by choosing to produce another point on the PPF to the right of A, i.e. to produce more of good 1.Now, if instead of being at A, society were on the PPF but down further along the PPF, beyond B, then the opposite problem would exist. At that point, the MRS would be less than the MRT – i.e. society would be sacrificing too many units of good 2 for the sake of one unit of good 1. Society would have gone too far, and it would be better off but cutting some of its good 1 output and thus releasing resources to produce more of good 2 along the PPF.This reasoning leads us to conclude that the optimal choice of output can only be B, where the MRS = MRT. At B, both what society must sacrifice and what society is willing to sacrifice of good 2 for the sake of one extra unit of good 1 coincide. This pins down then the optimal level of output of both goods: good 1 and good 2.Let us now see what happens to this optimal choice if the PPF were to change.
- Consider now the effect of a change in society’s production possibilities. Say that the weather conditions turn adverse and disrupt the production of good 1. This represent a negative technological shock or change, since society – fully employing its resources – is now capable of turning out less goods of type 1 than before the weather conditions worsened.Initially, the PPF was given by the equation: x_2 = 22 – 1.1 x_1, but as a result of the adverse technological change, the PPF is now x_2 = 22 – 1.51 x_1, which indicates that good 1 has become significantly more costly in terms of the amount of good 2 given up. As a result of this change in production possibilities, bundle A – which used to be the optimal choice of output – is now beyond the economy’s productive possibilities: it is unfeasible or unattainable. Society must tighten its belt accordingly. The new PPF touches a lower indifference curve at point B=(7.3, 11). That is the bundle that equalizes the PPF’s MRT = 1.5125 and the slope of this indifference curve at this point, the MRS which must also be 1.5125.You may be asking yourself how I know these values, and the reason is that I set up the graphs so that I get these numbers. At this point, you should be concerned with understanding the logic here. Not with how I came up with these particular numbers. Just follow the reasoning.To summarize: the optimal output choice (the optimal bundle or combination of good 1 and good 2) this society will produce is determined at the point where the MRT and the MRS coincide.The MRT is the slope of the PPF. Since we are assuming a linear PPF, the slope of the PPF is constant. The MRS is the slope of the indifference curve at each particular point. At the optimal bundle point, the MRT and the MRS are equal. If we now that the choice of output is optimal and we know the MRT, then we know the MRS, because it must be equal to the MRT.The MRS is not constant, because we have drawn the indifference curves as convex curves. They are not straight lines. So, the slope at each point is going to be different. When the production of good 1 is low, the MRS is very high, but as society produces more of good 1, the MRS goes down and down. The tangent line to the indifference curve at each point becomes flatter and flatter as more of good 1 is produced. If the PPF and the indifference curve cross, instead of touching each other at a single point, then the MRT is not equal to the MRS. For the MRT and MRS to be equal, the PPF must be a tangent line touching the indifference curve at a single point. That point, a combination of good 1 and good 2, is the optimal output choice or bundle.If the PPF swivels inwards as a result of a decrease in society’s resources or a negative technological change, then society must adjust and reduce its output of at least one good, if not both of them. That will depend on the location of the indifference curves. In our example, by chance, the optimal output level of one of the goods (good 2) remains constant, but the optimal output level went down.If, contrariwise, the PPF swivels outwards, because society’s resources expand or an improvement in technology, then society adjusts to a higher level of welfare (a higher indifference curve). The level of output of at least one good will increase. Try and play with these graphical model by yourself to make sure you understand the implications of different changes in the PPF.Finally, let us a consider a resource or technological shock in the economy that makes it possible to produce more of both goods in the economy…
- Say that the amount of resources now available are greater or that technology has improved, and – as a result – the PPF shifts outwards to a new location given by: x_2 = 24 – 1.1 x_1. Note that the intercept has increased, although the slope of the PPF remains the same. This, again, is what we call a shift of the PPF.The old optimal bundle is now inside of the feasible set. It is not productively efficient any longer, because it is not on the PPF. Society can do better than that with its new production possibilities. In fact, society can now reach a higher level of welfare or indifference curve. It can reach and touch U_3 at point C. That is a tangency point. The PPF becomes tangent to the indifference curve U_3 at that point alone. The MRT of the PPF (1.1) is here equal to the MRS of the indifference. This is true only at this point on the indifference curve, point C=(10.9, 12), that is, 10.9 units of good 1 and 12 units of good 2.This is a case that we could characterize as economic growth. The productive potential of the economy expands and as a result, society produces more of both goods and enjoys a higher level of welfare.With this analytical tools, we are now ready to tackle trade in a general equilibrium framework. It’s a modest general equilibrium framework, but it is general equilibrium, in that we can keep track of the effects of trade in a whole economy, even though our economy here only produces two goods. Again, there’s no reason to be sad about this, since we can always construe the two goods as one good of interest (good 1) and all other goods combined (good 2).And this is the end of this presentation. I hope it was helpful to you.