Lecture_34_Production

Lecture_34_Production - Lecture 34: The Robinson Crusoe...

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Unformatted text preview: Lecture 34: The Robinson Crusoe Economy © 2008 Jeffrey A. Miron Outline 1. Introduction 2. The Robinson Crusoe Economy 3. Crusoe, Inc. 4. The Firm 5. Robinson’s Problem 6. Putting Them Together 7. Different Technologies 8. Production and the First Welfare Theorem 9. Production and the Second Welfare Theorem 10. Production Possibilities 11. Comparative Advantage 12. Pareto Efficiency 13. Castaways, Inc. 14. Robinson and Friday as Consumers 15. Decentralized Resource Allocation 1 Introduction The analysis we examined earlier on general equilibrium assumed a pure exchange economy. This was convenient and fruitful. It kept the set of economic interactions to a bare minimum, and it yielded key insights about the efficiency properties of general equilibrium. In particular, we examined both the First and Second Welfare Theorems of economics. It is also interesting, however, to consider general equilibrium issues in any econ— omy with production. This has two benefits. First, it allows us to see under what conditions the two welfare theorems still apply once we introduce production. Second, the introduction of production produces some new insights that we cannot obtain in the pure exchange model because these insights are fundamentally about production itself. Because we will be making the model more complicated in one direction (adding production), we will make it simpler in another direction: We will initially assume the economy has exactly one consumer and one firm. As before, the economy has two goods. We then adopt the “trick” of assuming that the one consumer and the one firm behave competitively, i.e., as price takers, even though that would be unlikely if an economy really had just one consumer and one firm. This approach can be thought of as assuming that the consumer and firm are representative of a large number of identical consumers and firms. This model is known as the Robinson Crusoe economy, after the famous book by Daniel Defoe. 2 The Robinson Crusoe Economy In this economy, Robinson Crusoe (RC) plays a dual role: he is both a consumer and a producer. As a consumer, he chooses between two goods: leisure and coconuts. If he sits on the beach watching the ocean, he is consuming leisure. If he spends his time gathering coconuts, he has less time for leisure but gets to eat the coconuts. We can depict RCs production opportunities and preferences over the two goods as in Figure 32.1. The technological constraints that RC faces are summarized by the production function relating coconuts consumed to hours of time spent gather coconuts. Since the technology uses only one good, labor, we can depict the production function in the two—dimensional graph. Note that as drawn it displays diminishing returns to scale and a diminishing marginal product of labor. The graph contains the only two consumption goods, coconuts and leisure (leisure is RCs total endowment of hours minus the hours spent gathering coconuts, labor). We can therefore depict RC’s preferences. Since we have labor on the horizontal axis, the indifference curves are positively sloped, as shown. The utility maximizing choice for RC must be the point at which the highest indifference curve just touches the production function.1 Why? At any point inside the production function, RC could choose a different point that involved less labor and/or more coconuts. Given that he prefers to be on higher indifference curves, he should choose the highest one that is possible. This means an indifference curve that just touches the production function. Assuming the production function and the indifference curves are both differ- entiable at this point, we can conclude that at the optimum choices for labor and coconuts, the marginal product of labor equals the marginal rate of substitution between leisure and coconuts. This makes sense. The marginal product of labor is the extra amount of coconuts RC would get from giving up one unit of leisure. The M R8 is the marginal utility gets from coconuts per unit of marginal utility from leisure. So, imagine that initially the MRS waw greater than the MP. Then think about what happens if RC spends one less hour gathering coconuts. His consumption of coconuts goes down by the MP of labor. His utility from leisure goes up by the marginal utility of leisure. His utility from coconuts goes down by the marginal utility of coconuts.divided by the marginal product of leisure. Under the assumption that the MRS initially exceeds the MP, this causes a net increase in utility, so the initial choice of labor and coconuts could not have been optimal. 11f the production function and indifference curves are smooth, then the point at which they just touch will also be a point at which the slope of the indifference curve equals the slope of the production function. If one or both of these curves is not smooth, however, then there is still a point at which they just touch, but the slope of one or both might not be defined at that point. 3 3 Crusoe Inc. So far we have looked at RC’s problem in a way that takes into account both his producer role and his consumer role. Now let’s think about what happens if RC decides to alternate between his two roles. One day he behaves as a producer, while the next day he behaves as a consumer . Imagine that RC sets up a labor market and a coconut market. RC also creates a firm, which he owns. The firms uses labor to gather coconuts, which it sells in the coconut market. The firm will consider the prices for labor and coconuts and then decide how much labor to hire and how many coconuts to produce. The firm’s decisions will be determined by profit maximization. RC, as a worker, earns wages from the firm. RC, as the owner of the firm, gets profits. RC, as a consumer, decides how much of the firm’s output to purchase. To keep track of these transactions, RC invents a currency, called dollars. Assume that we set the price of coconuts at $1 a piece; that is, we make coconuts the numeraire. This means we only need to determine the wage rate. We will think about this from the perspective of the firm (Crusoe, Inc.), and then from the perspective the consumer (RC). Specifically, we want to derive the equilibria in the markets for labor and coconuts. 4 The Firm Each evening, Crusoe, Inc. (CI) decides how much labor to hire for the next day, or, equivalently, how many coconuts produce (since the production technology uses only one input). Given the price of 1 for coconuts and a wage of w, the solution to the firm’s problem can be illustrated as in Figure 32.2. To see why, consider all combinations 4 of coconuts and labor that yield a given level of profits, 7r. Any such combination must satisfy the equation 7r = C’ -— wL Solving for C, we have C = 7T + wL This equation shows the relation between coconuts and labor for this particular level of profits. There is an infinity of such lines corresponding to all possible levels of profits. These equations are known as isoprofit lines. The vertical intercept measures the level of profits. If CI generates 7r*, this money can buy 7r* coconuts, since the price of coconuts is 1. CI will choose a combination of coconuts and labor that is feasible~which means on or inside the production function~and that maximizes profits. This means choos— ing the isoprofit line that is just tangent to the production function. The production function and the isoprofit line therefore have the same slope at the profit maximizing choice of labor and coconuts. The slope of the isoprofit line is w, the wage rate, and the slope of the production function is the marginal product of labor, MP1,. So, we get the usual condition for profit maximization: the marginal benefit of using a bit more labor must equal its marginal cost. So, CI has done its job. Given the wage w, it has determined how much labor it wants to hire, how many coconuts to produce, and what profits it will generate. So, CI declares a profit of 7r* and sends this to the owner, RC. 5 Robinson’s Problem The next day RC wakes up and gets his profits check of 7r" dollars. While eating his breakfast of coconuts, he contemplates how much he wants to work and consume. He might consider just consuming his endowment: spend his profits on coconuts and consume his endowment of leisure. Alternatively, he might work for some number of hours. In that case, RC trudges down to CI and starts to gather coconuts. We can describe this using standard indifference curve analysis, where we put labor on one axis and coconuts on the other. Since labor is a bad, the indifference curves are positively sloped. Higher indifference curves are more preferred to lower ones. RC’s budget line has a slope of w. and passes through the point (0, 7r*). Given this budget line, RC choose how much to work and how many coconuts to consume. At the optimal bundle, the MRS between cocunuts and labor must equal the wage. See Figure 32.3. 6 Putting Them Together Now we can superimpose the two figures to get Figure 32.4. It turns out that the equilibrium of this decentralized economy is exactly the same as that of the centralized economy in which RC makes all the decisions at once. In this equilibrium, the MRS equals the wage, and the M PL equals the wage, so MRS = M PL. That is, the slope of the indifference curve equals the slope of the production function. The decentralized interpretation of equilibrium might not seem that persuasive or interesting if there is only one person. Imagine, however, that an economy has many identical consumers and many identical firms. Then seperating production and consumption decisions is useful because it saves information gathering and processing costs. Each consumer only has to know his own income, the price of coconuts, and and the market wage. Likewise, each producer only needs to know the price of output and the price of inputs. Each decision—making unit only has to know the approprate prices to make the right (efficient) choices. This is because prices measure marginal resource costs. 7 Different Technologies In the discussion so far, we have assumed that the technology had a diminishing marginal product of labor. Since labor was the only input, this was equivalent to 6 assuming dimirninishing returns to scale. This is a reasonable assumption for many activities (as one spends more hours in a given day gathering coconuts, one gets tired and cannot gather as many coconuts per hour), but it is nevertheless interesting to consider other possibilities. Suppoe first that the technology has constant returns to scale. For a one input production funciton, this means the graph of the production function is a straight line through the origin, as shown in Figure 32.5. The assumption of constant returns to scale also means that CI must earn exactly zero profits. Otherwise, it would either shut down (if it were earning negative profits) of expand indefinitley (if it were earning positive profits). Thus RC’s endowment consists of zero profits and T, his initial endowment of time. RC’s budget set coincides with the production set, and the equilibrium is otherwise as before. The situation is different if the technology available to CI has increasing returns to scale, as illustrated in Figure 32.6. When production exhibits increasing returns, we can still determine an optimal production and consumption point, as shown, and this must still be a tangency between the indifference curve and the production function. The problem is trying to support this outcome as the choice of a profit—maximizing firm. If a firm were faced with the price ratio given by RC’S MRS, it would want to produce more than RC would demand. This is because if the firm exhibits increasing returns to scale, then average costs exceed the marginal costs, which means negative profits. The goal of profit maximization would then lead the firm to want to expand output; but this would be inconsistent with the demand for output and the supply of inputs from consumers. In the case depicted, there is in fact no price at which the utilitywmaximizing demand by the consumer will equal the profit-maximizing supply from the firm. Increasing returns to scale is an example of a non-convexity. In this case it is the production set that is not convex. This means there cannot be a tangency that separates the preferred points from the possible points. 7 Non—convexities pose difficulties for competitive markets. In these case, prices do not convey the information necessary to choose an efficient allocation. This is not necessarily an issue if the increasing returns are local and small. For example, almost any productive activity has some start—up cost, which means the we expect increasing returns starting from zero production. In many cases, however, this level of production is small relative to the market, so firms will operate in regions of their production functions where returns are non—increasing. If increasing returns are large or global, however, competitive equilibria may not exist. 8 Production and the First Welfare Theorem We learned earlier that in a pure exchange economy, a competitive equilibrium is Pareto efficient. A natural question is whether the same conclusion holds in a production economy? The answer turns out to be yes (see the Appendix to Chapter 32 in Varian). This conclusion assumes there are not substantial non—convexities.and no production or consumption externalities It also says nothing about distribution. 9 Production and the Second Welfare Theorem We showed earlier that every Pareto efficient allocation can be supported as a com— petitive equilibrium, assuming convex preferences. The same holds in a produciton economy, assuming production is also convex 10 Production Possibilities We now generalize this discussion to an economy with several inputs and outputs. We assume two goods for convenience, but the analysis applies even with many goods. Assume that RC can produce a second good, like fish, so he must choose how to divided his time between fishing and gathering coconuts. The graph Figure 32.7 shows the difierent possibile combinations of fish and coconuts that RC might produce. This graph is known as the production possibilities set, and the border is the production possibilities frontier. The shape of the production possibilities frontier depends on the underlying tech nologies. Under constant returns to scale, the production functions for both good are linear, as is the production possibilities frontier. The slope of the production possibilities frontier is the marginal rate of transformation. Assume, for example, that RC can produce 10 pounds of fish per hour or 20 pounds of coconuts per hour. This means the production technology is as follows: F: 10Lf 0220116 LC+Lf = 10 where F is RC’s production of fish, 0 is RC’s production of coconuts, LC is the time RC spends gathering coconuts, L f is the time RC spends fishing, and 10 is RC’s total endowment of time. These equations jointly imply that the relation between L and F is 2F+C=200 The PPF is easy to derive in this case because we have exactly on way of producing each good. 11 Comparative Advantage Now suppose there is a second worker on the island who can also produce fish and coconuts but who has different skills at the two activitives than RC. Call this other worker Friday. Assume that Friday can produce 20 pounds of fish per hour, or 10 pounds of coconuts. Thus, if Friday works for 10 hours, his PPF will be 9 F+ZC=200 This is depicted in Figure 32.8. The marginal rate of transformation for Friday is ~1/2; for RC ~2. In this situation, we say that RC has a comparative advantage in coconuts and Friday has a comparative advantage in fish. If RC shifts his labor from fish to coconut by enough to reduce his production of fish by one unit, he gets two coconuts. If Friday shifts his labor from fish to coconuts enough to reduce his production of fish by one unit, he gets one half extra coconut. The situation is reversed for shifting from coconuts to fish. We can now consider a joint PFF that combines the best of both workers. If both allocate all their time to one activity, they can jointly produce either 300 coconuts or 300 fish. If they want to produce a positive amount of fish and accept fewer coconuts, it makes sense to first shift Friday into fish since an hour of his time will produce 20 fish and only cost 10 coconuts. There is therefore a section of the PPF starting from (0,300) that has a slope of negative -—1 / 2, up to the point (200, 200). Once the economy gets to that point, it can only produce more than 200 fish by shifting RC into fish, which means giving up two coconuts for every fish produced. Thus, the PPF has a slope of —-2 from that point up to complete specialization in fish, at (300,0). Thus the PPF for the whole economy is not one straight line but a combination of two. If we had many workers, each with a different set productivities, the PPF would have more segments and would gradually resemble the smooth PPF we drew earlier. 12 Pareto Efficiency Now that we have described the PPF, the question is how can we choose among the feasible consumption bundles in a Pareto efficient way? Let the aggregate consumption bundles be indicated by (X1, X2). That is, X1 is the total amount of good 1 produced and available for consumption, and X 2 is the total amount of good 2 produced and available for consumption. 10 We can think of this bundle as a point on the PPF, as shown in Figure 32.9. Once we have this point, can draw an Edgeworth box as earlier (also shown in the figure). We can do this for each possible point on the PPF. This allows us to analyze efficient consumption allocations. Every efficient consumption allocation, given the Edgeworth box, must equate the MRS’s of the two consumers. The question is then, which choices from the contract curve will be Pareto effi- cient? The answer is, those for which the MRS = M RT. An example is shown in the figure. At this point, the common M RS of the two consumers equals the M RT because the slope of the indifference curve tangency is equal to the slope of the PPF. A Pareto efficient equilibrium therefore has two properties. For a given amount produced, the consumption decisions must be eificient, meaning the the two con— sumers must have the same MRS. Otherwise, as discussed earlier, there would be a way to reallocate consumption goods that made at least one person better off without making the other worse off. With production, another condition must also hold: this MRS must equal the M RT. Why? Assume not. If the two ratios are not equal, then there is way to make at least one consumer better off by altering the production pattern. Trade between consumers is one way to get fish in exchange for coconuts. But choosing a different production pattern is also a way to transform fish into coconuts, and vice versa. So, if one consumer’s MRS does not equal the MRT, this consumer could be better off by reallocating production in the direction of the good whose marginal value to the consumer exceeds the marginal production costs measured in terms of reduced production of the less valued good. 13 Castaways, Inc. Now we want to think about how a decentralized resource allocation system works. Our economy has two consumers and two goods. The two consumers are also both shareholds of the firm, now known as Castaways, Inc. These two people are both the consumers and the producers, but we will again think about their roles separately. The profit maximization problem for the firm is 11 max 1000 +ppF —— wCLC —- prF subject to the technological constraints on production of coconuts and fish described earlier. This profit maximization problem leads to some optimal choices for the amounts of labor for both Robinson and Friday. These optimal choices imply that the production of coconuts and fish will lie on the PPF, at a point where its slope equals the price ratio. This is shown in Figure: 32.10. 14 JF, RC as Consumers As consumers, Robinson and Friday earn income equal to their profits from ownership of the firm. They then make individual consumption choices based on this income, taking the market prices for coconuts and fish as given. 15 Decentrliazed Resource Allocation Finally, under certain conditions, we can show that when the individual firms pursue profit maximization, given prices, and the individual consumers maximize utility given prices, an equilibrium set of prices exist at which all supplies equal demands. Further, this equilibrium is a Pareto efficient allocation. Plus, any Pareto efficient allocation can be supported as the outcome of a competitive market, if the initial endowments can be suitably redistributed. 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This note was uploaded on 09/16/2009 for the course ECONOMICS 1010A taught by Professor Jeffreya.miron during the Fall '09 term at Harvard.

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Lecture_34_Production - Lecture 34: The Robinson Crusoe...

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