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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 proﬁtmaximizing supply from the ﬁrm. Increasing returns to scale is an example of a nonconvexity. 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.
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 Fall '09
 JeffreyA.Miron
 Robinson Crusoe

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