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%TCIDATA{Created=Thu Nov 06 21:35:40 2003}
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\newtheorem{theorem}{Theorem}
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\newtheorem{case}[theorem]{Case}
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\begin_body

\begin_layout Title
First Welfare Theorem in Production Economies
\end_layout

\begin_layout Author
Michael Peters
\end_layout

\begin_layout Date
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
today{}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Profit Maximization
\end_layout

\begin_layout Standard
Firms transform goods from one thing into another.
 If there are two goods, 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

, then a firm can transform 
\begin_inset Formula $x$
\end_inset

 into 
\begin_inset Formula $y$
\end_inset

 or 
\begin_inset Formula $y$
\end_inset

 into 
\begin_inset Formula $x$
\end_inset

 depending on what consumers want.
 The first figure below represents a simple production technology that the
 firms might use to do this.
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename firms_fig1.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Production Function
\begin_inset CommandInset label
LatexCommand label
name "fig1"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the figure, the function 
\begin_inset Formula $y=f\left(x\right)$
\end_inset

 represents the feasible production choices available to the firm.
 The horizontal axis measures an arbitrary good 
\begin_inset Formula $x$
\end_inset

 which could be either an input or an output.
 When 
\begin_inset Formula $x$
\end_inset

 is negative, interpret this to mean that it is being used as an input in
 the production of some good 
\begin_inset Formula $y$
\end_inset

 which is measured along the vertical axis.
 Any good could be an input.
 For example some firms use labor to produce parts for cars.
 The parts are an output for that firm.
 The same parts act as an input for the firm that makes cars.
 The distinction between inputs and outputs really isn't helpful here.
 A better idea is to think of a production technology that can transform
 one good into another.
 At a point on the production function like 
\begin_inset Formula $\left(x^{1},y^{1}\right)$
\end_inset

, the firm transforms 
\begin_inset Formula $x^{1}$
\end_inset

 units of good 
\begin_inset Formula $x$
\end_inset

 into 
\begin_inset Formula $y^{1}$
\end_inset

 units of good 
\begin_inset Formula $y$
\end_inset

.
 At this point, the input 
\begin_inset Formula $x$
\end_inset

 is negative and the output 
\begin_inset Formula $y$
\end_inset

 is positive.
 On the other hand, the firm could as easily use 
\begin_inset Formula $y$
\end_inset

 as an input (
\begin_inset Formula $y$
\end_inset

 is negative) and produce 
\begin_inset Formula $x$
\end_inset

 as an output as it does at the point 
\begin_inset Formula $\left(x^{0},y^{0}\right)$
\end_inset

.
 
\end_layout

\begin_layout Standard
The way that firms are incorporated into things is to assume that firms
 own all of the endowments of good 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

.
 They transform 
\begin_inset Formula $x$
\end_inset

 to 
\begin_inset Formula $y$
\end_inset

 or 
\begin_inset Formula $y$
\end_inset

 to 
\begin_inset Formula $x$
\end_inset

  in whatever way maximizes their profits.
 Consumers, in turn, own firms.
 To make things simple, assume there is only a single firm.
 There will also be two consumers, 
\begin_inset Formula $1$
\end_inset

 and 
\begin_inset Formula $2$
\end_inset

.
 Let 
\begin_inset Formula $\theta$
\end_inset

 be the proportion of the firm owned by consumer 
\begin_inset Formula $1$
\end_inset

 while 
\begin_inset Formula $\left(1-\theta\right)$
\end_inset

 is the proportion owned by consumer 
\begin_inset Formula $2.$
\end_inset

 Let 
\begin_inset Formula $\omega_{x}$
\end_inset

 and 
\begin_inset Formula $\omega_{y}$
\end_inset

 be the total amounts of good 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 that are available to the firm.
 
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $z_{x}$
\end_inset

 and 
\begin_inset Formula $z_{y}$
\end_inset

 be the aggregate amounts of 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 that the firm chooses to make available in the economy.
 If the firm decides that it wants to produce good 
\begin_inset Formula $y$
\end_inset

 from good 
\begin_inset Formula $x$
\end_inset

, meaning that 
\begin_inset Formula $z_{y}>\omega_{y}$
\end_inset

 and 
\begin_inset Formula $z_{x}<\omega_{x}$
\end_inset

, then it needs to use up some of its endowment of good 
\begin_inset Formula $x$
\end_inset

 and use it in production of good 
\begin_inset Formula $y$
\end_inset

.
 So, 
\begin_inset Formula $z_{y}=\omega_{y}+f\left(z_{x}-\omega_{x}\right)$
\end_inset

.
 (Remember that to get 
\begin_inset Formula $y$
\end_inset

 out of the production process, we need a negative argument for good 
\begin_inset Formula $x$
\end_inset

 the way 
\begin_inset Formula $f$
\end_inset

 is drawn in Figure 
\begin_inset Formula $1$
\end_inset

.) On the other hand, if it wants to produce good 
\begin_inset Formula $x$
\end_inset

 (that is 
\begin_inset Formula $z_{x}>\omega_{x}$
\end_inset

), then it has to use up some of its endowment of good 
\begin_inset Formula $y$
\end_inset

 (so, 
\begin_inset Formula $z_{y}<\omega_{y}$
\end_inset

).
 
\end_layout

\begin_layout Standard
Now, imagine drawing a picture as in Figure 
\begin_inset Formula $2$
\end_inset

 of the function 
\begin_inset Formula 
\begin{equation}
z_{y}=\omega_{y}+f\left(z_{x}-\omega_{x}\right)\label{ppf}
\end{equation}

\end_inset

 This function is called the 
\shape italic
production possibilities frontier
\shape default
.
 It is given as the line segment 
\begin_inset Formula $CD$
\end_inset

 in Figure 2.
 The output of good 
\begin_inset Formula $y$
\end_inset

 varies between 
\begin_inset Formula $\omega_{y}+f(\omega_{x})$
\end_inset

 when the amount of good 
\begin_inset Formula $x$
\end_inset

 the firm chooses to produce is 0, to 
\begin_inset Formula $0$
\end_inset

 in the case where the firm chooses to produce an output 
\begin_inset Formula $z$
\end_inset

 such that 
\begin_inset Formula $f(z-\omega_{x})+\omega_{y}=0$
\end_inset

.
 
\end_layout

\begin_layout Standard
Now, suppose that the prices for 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 are given by 
\begin_inset Formula $p$
\end_inset

 and 
\begin_inset Formula $1$
\end_inset

, respectively (I keep using 
\begin_inset Formula $1$
\end_inset

 for the price of 
\begin_inset Formula $y$
\end_inset

 because it is only the relative price of good 
\begin_inset Formula $x$
\end_inset

 that makes a difference to the firm or the consumers).
 Whatever the firm chooses to produce it can sell to the consumers at prices
 
\begin_inset Formula $p$
\end_inset

 and 
\begin_inset Formula $1$
\end_inset

.
 So, the profit, or revenue, of the firm is just 
\begin_inset Formula $pz_{x}+z_{y}$
\end_inset

, if it produces 
\begin_inset Formula $\left(z_{x},z_{y}\right)$
\end_inset

.
 An iso-profit locus is a collection of productions that give the same profit.
 The line segment 
\begin_inset Formula $AB$
\end_inset

 in Figure 
\begin_inset Formula $2$
\end_inset

 gives part of one such production locus.
 There are a family of such loci--all of the lines that are parallel to
 
\begin_inset Formula $AB$
\end_inset

.
 If the firm maximizes profits, it picks the highest iso-profit curve that
 touches its production possibilities frontier.
 This gives the production choice 
\begin_inset Formula $\left(z_{1},z_{2}\right)$
\end_inset

 in Figure 
\begin_inset Formula $2$
\end_inset

.
 
\end_layout

\begin_layout Standard
Once the firm has produced this output, it distributes its profits back
 to its shareholders: the fraction 
\begin_inset Formula $\theta$
\end_inset

 goes to consumer 
\begin_inset Formula $1$
\end_inset

, and 
\begin_inset Formula $\left(1-\theta\right)$
\end_inset

 goes to consumer 
\begin_inset Formula $2$
\end_inset

.
 Since the consumers use these profits to finance their purchases, both
 of them would prefer that the firm choose the aggregate production that
 maximizes its profits since that will always provide them with their highest
 income for consumption.
 The income of consumer 
\begin_inset Formula $1$
\end_inset

 is 
\begin_inset Formula $\theta\left(pz_{1}+z_{2}\right)$
\end_inset

.
 This means that the budget line that consumer 
\begin_inset Formula $1$
\end_inset

 faces is the one that intersects the 
\begin_inset Formula $x$
\end_inset

-axis at the point 
\begin_inset Formula $\theta\left(pz_{1}+z_{2}\right)/p$
\end_inset

 (since that is the maximum quantity of good 
\begin_inset Formula $x$
\end_inset

 he would be able to purchase with that income).
 
\end_layout

\begin_layout Standard
The outcome at an 
\emph on
arbitrary
\emph default
 (i.e.
 not equilibrium) price pair 
\begin_inset Formula $\left(p,1\right)$
\end_inset

 is given in Figure 
\begin_inset Formula $2.$
\end_inset


\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename firms_fig2.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Out of Equilibrium
\begin_inset CommandInset label
LatexCommand label
name "fig2"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Each of the two consumers chooses the best point in his or her budget set
 (which occurs where their indifference curves are tangent to their correspondin
g budget lines).
 The choice by consumer 
\begin_inset Formula $1$
\end_inset

 is the consumption bundle 
\begin_inset Formula $\left(x_{1},y_{1}\right)$
\end_inset

 in the figure.
 Consumer 
\begin_inset Formula $2$
\end_inset

's choice should be read with respect to the coordinate system that starts
 at the point 
\begin_inset Formula $\left(z_{1},z_{2}\right)$
\end_inset

.
 So, the quantity of good 
\begin_inset Formula $x$
\end_inset

 that consumer 
\begin_inset Formula $2$
\end_inset

 wants is given by the horizontal distance between the point 
\begin_inset Formula $x^{\prime}$
\end_inset

 and the point 
\begin_inset Formula $z_{1}$
\end_inset

.
 From this, you can see that the total demand for good 
\begin_inset Formula $x$
\end_inset

 (which is given by 
\begin_inset Formula $x_{1}+\left(z_{1}-x^{\prime}\right)$
\end_inset

) exceeds the total amount of good 
\begin_inset Formula $x$
\end_inset

 that is produced by the firm.
 So, the relative price of 
\begin_inset Formula $x$
\end_inset

 should rise.
 
\end_layout

\begin_layout Standard
As the relative price of 
\begin_inset Formula $x$
\end_inset

 rises, the family of iso-profit curves faced by the firm will all get steeper.
 This will cause the firm to choose a profit-maximizing level of output
 on its production possibilities frontier that involves more 
\begin_inset Formula $x$
\end_inset

 and less 
\begin_inset Formula $y$
\end_inset

.
 As one might expect, the rising price of good 
\begin_inset Formula $x$
\end_inset

 will cause both consumers to demand a little less 
\begin_inset Formula $x$
\end_inset

 and a little more 
\begin_inset Formula $y$
\end_inset

.
 Eventually, the increase in supply of 
\begin_inset Formula $x$
\end_inset

 and the reduction in demand will bring the market to a state of equilibrium.
 
\end_layout

\begin_layout Section
Competitive (Walrasian) Equilibrium
\end_layout

\begin_layout Standard
A competitive (Walrasian) equilibrium is a pair of consumption choices 
\begin_inset Formula $\left(x_{1}^{\ast},y_{1}^{\ast}\right)$
\end_inset

 for consumer 
\begin_inset Formula $1$
\end_inset

 and 
\begin_inset Formula $\left(x_{2}^{\ast},y_{2}^{\ast}\right)$
\end_inset

 for consumer 
\begin_inset Formula $2$
\end_inset

, and a production plan 
\begin_inset Formula $\left(z_{1}^{\ast},z_{2}^{\ast}\right)$
\end_inset

 for the firm such that there is a price 
\begin_inset Formula $p^{\prime}$
\end_inset

 for good 
\begin_inset Formula $x$
\end_inset

 for which the following things are true: 
\end_layout

\begin_layout Enumerate
\begin_inset Formula $x_{1}^{\ast}+x_{2}^{\ast}=z_{1}^{\ast}$
\end_inset

; 
\begin_inset Formula $y_{1}^{\ast}+y_{2}^{\ast}=z_{2}^{\ast}$
\end_inset

 (the markets clear); 
\end_layout

\begin_layout Enumerate
\begin_inset Formula $p^{\prime}z_{1}^{\ast}+z_{2}^{\ast}\geq p^{\prime}z_{1}+z_{2}$
\end_inset

 for any pair 
\begin_inset Formula $\left(z_{1},z_{2}\right)$
\end_inset

 on the firm's production possibilities frontier; and 
\end_layout

\begin_layout Enumerate
\begin_inset Formula $u_{1}\left(x_{1}^{\ast},y_{1}^{\ast}\right)\geq u_{1}\left(x_{1},y_{1}\right)$
\end_inset

 for all 
\begin_inset Formula $\left(x_{1},y_{1}\right):p^{\prime}x_{1}+y_{1}\leq\theta\left(p^{\prime}z_{1}^{\ast}+z_{2}^{\ast}\right)$
\end_inset

 and 
\begin_inset Formula $u_{2}\left(x_{2}^{\ast},y_{2}^{\ast}\right)\geq u_{2}\left(x_{2},y_{2}\right)$
\end_inset

 for all 
\begin_inset Formula $\left(x_{2},y_{2}\right):p^{\prime}x_{2}+y_{2}\leq\left(1-\theta\right)\left(p^{\prime}z_{1}^{\ast}+z_{2}^{\ast}\right)$
\end_inset

.
 
\end_layout

\begin_layout Standard
You can see what happens after the price of good 
\begin_inset Formula $x$
\end_inset

 rises (to 
\begin_inset Formula $p^{\prime}$
\end_inset

) in Figure 
\begin_inset Formula $3.$
\end_inset


\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename firms_fig3.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Equilibrium
\begin_inset CommandInset label
LatexCommand label
name "fig3"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the picture, consumer 
\begin_inset Formula $1$
\end_inset

 now has income 
\begin_inset Formula $\theta\left(p^{\prime}z_{1}^{\ast}+z_{2}^{\ast}\right)$
\end_inset

 which he uses to buy 
\begin_inset Formula $x_{1}^{\ast}$
\end_inset

 units of good 
\begin_inset Formula $x$
\end_inset

.
 Now, consumer 
\begin_inset Formula $2$
\end_inset

 chooses to buy 
\begin_inset Formula $z_{1}^{\ast}-x_{1}^{\ast}$
\end_inset

 units of good 
\begin_inset Formula $x$
\end_inset

, and the markets clear.
 
\end_layout

\begin_layout Section
First Welfare Theorem
\end_layout

\begin_layout Standard
The 
\emph on
first welfare theorem
\emph default
 is one of the most important contributions of classical microeconomic theory.
 It says that no 
\emph on
feasible
\emph default
 allocation exists in which all consumers are better off than they are in
 the competitive equilibrium.
 This is similar to the argument that we made for an exchange economy: consumer
 indifference curves must be tangent at any equilibrium.
 However, production adds another wrinkle.
 It might be true that both consumers could be made better off if the firm
 would just behave in a different fashion and pick some production plan
 that doesn't necessarily maximize profits.
 
\end_layout

\begin_layout Standard
Actually we can argue that if we take any pair of consumption bundles where
 
\emph on
both
\emph default
 consumers are better off, then there can be no production plan that makes
 this feasible.
 To see this, suppose that 
\begin_inset Formula $\left(x_{1}^{\prime},y_{1}^{\prime}\right)$
\end_inset

 and 
\begin_inset Formula $\left(x_{2}^{\prime},y_{2}^{\prime}\right)$
\end_inset

 are consumption bundles such that
\begin_inset Formula 
\[
u_{1}\left(x_{1}^{\prime},y_{1}^{\prime}\right)>u_{1}\left(x_{1}^{\ast},y_{1}^{\ast}\right)
\]

\end_inset

 and
\begin_inset Formula 
\[
u_{2}\left(x_{2}^{\prime},y_{2}^{\prime}\right)>u_{2}\left(x_{1}^{\ast},y_{2}^{\ast}\right)
\]

\end_inset

 One observation is immediate.
 Whenever this is true, it must be that
\begin_inset Formula 
\[
p^{\prime}x_{1}^{\prime}+y_{1}^{\prime}>\theta\left(p^{\prime}z_{1}^{\ast}+z_{2}^{\ast}\right)
\]

\end_inset

 and
\begin_inset Formula 
\[
p^{\prime}x_{2}^{\prime}+y_{2}^{\prime}>\left(1-\theta\right)\left(p^{\prime}z_{1}^{\ast}+z_{2}^{\ast}\right)
\]

\end_inset

 The reason for this is that consumers choose the very best consumption
 bundles that they can afford with their income.
 If they could have afforded 
\begin_inset Formula $\left(x_{1}^{\prime}.y_{1}^{\prime}\right)$
\end_inset

 or 
\begin_inset Formula $\left(x_{2}^{\prime},y_{2}^{\prime}\right)$
\end_inset

, then they certainly would have chosen them.
 
\end_layout

\begin_layout Standard
Now, if the firm is maximizing its profits
\begin_inset Formula 
\[
p^{\prime}z_{1}^{\ast}+z_{2}^{\ast}\geq p^{\prime}z_{1}^{\prime}+z_{2}^{\prime}
\]

\end_inset

 for any 
\begin_inset Formula $\left(z_{1}^{\prime},z_{2}^{\prime}\right)$
\end_inset

 along the firm's production possibilities frontier.
 So, it must be that
\begin_inset Formula 
\[
p^{\prime}x_{1}^{\prime}+y_{1}^{\prime}>\theta\left(p^{\prime}z_{1}^{\prime}+z_{2}^{\prime}\right)
\]

\end_inset

 and
\begin_inset Formula 
\[
p^{\prime}x_{2}^{\prime}+y_{2}^{\prime}>\left(1-\theta\right)\left(p^{\prime}z_{1}^{\prime}+z_{2}^{\prime}\right)
\]

\end_inset

 Then, if we add these two inequalities together, we get
\begin_inset Formula 
\[
p^{\prime}\left(x_{1}^{\prime}+x_{2}^{\prime}-z_{1}^{\prime}\right)+\left(y_{1}^{\prime}+y_{2}^{\prime}-z_{2}^{\prime}\right)>0
\]

\end_inset

 If prices are positive, then at least one of the two expressions 
\begin_inset Formula $\left(x_{1}^{\prime}+x_{2}^{\prime}-z_{1}^{\prime}\right)$
\end_inset

 and 
\begin_inset Formula $\left(y_{1}^{\prime}+y_{2}^{\prime}-z_{2}^{\prime}\right)$
\end_inset

 are strictly positive, which means the firm simply can't produce enough
 to supply what consumers want.
 
\end_layout

\begin_layout Standard
So, it is good for firms to maximize profits in two senses.
 First, if the firm were to propose some alternate production plan which
 didn't involve profit maximization, both consumers would expect their income
 to fall (notice that this is partly because they don't expect the change
 in production plan to have any effect on prices).
 So, the shareholders of the firm would unanimously vote against such a
 change.
 Second, even if the firm could change its production plan, and even if
 prices do change, the alternate plan can't possibly make both consumers
 better off.
 Notice that when we make either of these arguments, we don't value profits
 of firms for their own sake.
 We are only concerned with the utility of consumers.
 
\end_layout

\begin_layout Section
Distribution
\end_layout

\begin_layout Standard
One thing you should notice about this entire construction is that firms
 don't, in any sense, create wealth or goods.
 The ability to create is embedded in the production possibilities frontier
 which is taken as a given.
\begin_inset Foot
status open

\begin_layout Plain Layout
The traditional theory of the firm has nothing to say about where this productio
n possibilities frontier comes from.
 This makes it pretty useless in thinking about things like economic development
, or economic growth.
\end_layout

\end_inset

 All firms do is decide how to allocate this wealth.
 Many potential ways to choose among alternate production plans exist.
 For example, the government could choose the entire production plan.
 This was the model used in centrally-planned economies like the old Soviet
 Union.
 Alternatively, one could imagine a mixture of private, profit-maximizing
 firms and publicly-regulated companies, similar to what happens in most
 Western economies.
 
\end_layout

\begin_layout Standard
Profit maximization isn't necessarily good for everyone.
 To see this, consider the following example in which one of the consumers
 (say consumer 
\begin_inset Formula $1$
\end_inset

) simply has an endowment of the goods 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 but does not own shares.
 The other, consumer 2, owns a firm that can transform 
\begin_inset Formula $x$
\end_inset

 into 
\begin_inset Formula $y$
\end_inset

 (and conversely)-- possibly by buying some of the endowment of consumer
 
\begin_inset Formula $1$
\end_inset

.
 The firm that is owned by consumer 
\begin_inset Formula $2$
\end_inset

 starts with some endowment of goods.
 
\end_layout

\begin_layout Standard
To begin, suppose that the government declares that the firm is not allowed
 to produce anything and that it simply has to give its endowment to consumer
 
\begin_inset Formula $2$
\end_inset

 who can use the endowment to finance the best consumption plan possible.
 Restrictions like this are pretty common.
 An example might be a zoning restriction that prevents a homeowner from
 turning her house into an apartment building, or a farm owner who is not
 allowed to build housing on his farm land.
 A possible outcome is shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig4"

\end_inset


\begin_inset Formula $.$
\end_inset


\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename firms_fig4.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Distribution
\begin_inset CommandInset label
LatexCommand label
name "fig4"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In Figure 
\begin_inset Formula $4$
\end_inset

, consumer 
\begin_inset Formula $1$
\end_inset

 starts with an endowment equal to 
\begin_inset Formula $\left(\omega_{x}^{1},\omega_{y}^{1}\right)$
\end_inset

.
 Consumer 
\begin_inset Formula $1$
\end_inset

 owns no shares of the firm.
 The firm owns an endowment 
\begin_inset Formula $\left(\omega_{x}^{2},\omega_{y}^{2}\right)$
\end_inset

 and this firm is in turn owned by consumer 
\begin_inset Formula $2.$
\end_inset

 If the firm simply offers its endowment for sale on the market then the
 feasible set of trades is given by the wide flat box whose corners are
 at the origin, and at the point where the line segment 
\begin_inset Formula $A^{\prime}B^{\prime}$
\end_inset

 intersects the production possibilities frontier.
 Prices adjust until the relative price of 
\begin_inset Formula $x$
\end_inset

 is 
\begin_inset Formula $p^{\prime}$
\end_inset

.
 In the associated exchange equilibrium, consumer 
\begin_inset Formula $1$
\end_inset

 receives the allocation 
\begin_inset Formula $\left(x_{1}^{\ast},y_{1}^{\ast}\right)$
\end_inset

.
 Consumer 
\begin_inset Formula $2$
\end_inset

's indifference curve is tangent to this point, so conditional on the production
 decision of the firm, no allocation can make both consumer 
\begin_inset Formula $1$
\end_inset

 and consumer 
\begin_inset Formula $2$
\end_inset

 better off.
 Notice that in this equilibrium, consumer 
\begin_inset Formula $1$
\end_inset

 is selling some of his endowment of good 
\begin_inset Formula $y$
\end_inset

 in order to acquire good 
\begin_inset Formula $x$
\end_inset

.
 Of course, consumer 
\begin_inset Formula $2$
\end_inset

 is doing the opposite: selling off the good 
\begin_inset Formula $x$
\end_inset

 that the firm provides in order to acquire good 
\begin_inset Formula $y$
\end_inset

.
 
\end_layout

\begin_layout Standard
The iso-profit curves faced by the firm are all parallel straight lines
 whose slopes are equal to 
\begin_inset Formula $-p^{\prime}.$
\end_inset

 Plotting the aggregate endowment point on the production possibilities
 frontier shows that simply selling off the endowment does not maximize
 the firm's profits.
 The iso-profit line (with slope 
\begin_inset Formula $-p$
\end_inset

) is flatter than the production possibilities frontier.
 Consumer 
\begin_inset Formula $2$
\end_inset

 will do better if the firm alters its production plan to produce some additiona
l 
\begin_inset Formula $y$
\end_inset

 from the endowment of good 
\begin_inset Formula $x$
\end_inset

 because this will increase the income that consumer 
\begin_inset Formula $2$
\end_inset

 takes to the market.
 To put is in a slightly different way, observe that the steep production
 possibilities curve means that consumer 
\begin_inset Formula $2$
\end_inset

 can acquire good 
\begin_inset Formula $y$
\end_inset

 much more cheaply by having the firm produce it than she can by buying
 it from consumer 
\begin_inset Formula $1$
\end_inset

.
 
\end_layout

\begin_layout Standard
If the firm is now free to raise its output of 
\begin_inset Formula $y$
\end_inset

, it will create an excess supply of good 
\begin_inset Formula $y$
\end_inset

 that will make 
\begin_inset Formula $y$
\end_inset

's relative price fall, or make 
\begin_inset Formula $x$
\end_inset

's relative price rise.
 Both these things are bad for consumer 
\begin_inset Formula $1$
\end_inset

 who is buying 
\begin_inset Formula $x$
\end_inset

 and selling 
\begin_inset Formula $y$
\end_inset

.
 When markets finally clear, the relative price of good 
\begin_inset Formula $x$
\end_inset

 will level off at 
\begin_inset Formula $p$
\end_inset

, and consumer 
\begin_inset Formula $1$
\end_inset

 will end up at point 
\begin_inset Formula $E$
\end_inset

 where he is much worse off than he was in the original equilibrium.
 
\end_layout

\begin_layout Standard
Oddly enough, this new equilibrium is 
\emph on
Pareto optimal
\emph default
.
 There is no way to make 
\emph on
both of the consumers
\emph default
 better off by changing the firm's production plan.
 How does one reconcile this with the fact that consumer 
\begin_inset Formula $1$
\end_inset

 was so much better off in the original equilibrium? 
\end_layout

\begin_layout Standard
One possible answer that may have occurred to you is that the original equilibri
um is also Pareto optimal.
 This is partly right and partly wrong.
 The absolute value of the slope of the production possibilities curve,
 at the point where the iso-profit curve 
\begin_inset Formula $A^{\prime}B^{\prime}$
\end_inset

 crosses it, is called the 
\emph on
marginal rate of transformation
\emph default
 between 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

.
 If you think of using one small unit 
\begin_inset Formula $dx$
\end_inset

 of 
\begin_inset Formula $x$
\end_inset

 as the input, then the slope gives the amount of 
\begin_inset Formula $y$
\end_inset

 that you get back out for each such unit.
 At the point 
\begin_inset Formula $\left(x_{1}^{\ast},y_{1}^{\ast}\right)$
\end_inset

 where the indifference curves for the consumers are tangent, the indifference
 curves both have slope 
\begin_inset Formula $-p^{\prime}$
\end_inset

.
 The absolute value of this is less than the marginal rate of transformation
 along the production possibilities frontier.
 
\end_layout

\begin_layout Standard
So, what is the slope of consumers 
\begin_inset Formula $1$
\end_inset

's indifference curve? His marginal rate of substitution is the amount of
 good 
\begin_inset Formula $y$
\end_inset

 you would need to give him to compensate him when you take away a little
 (
\begin_inset Formula $dx$
\end_inset

) of his good 
\begin_inset Formula $x$
\end_inset

.
 Consumer 
\begin_inset Formula $2$
\end_inset

's indifference curve is tangent at this point.
 That means if you do a tiny transfer of good 
\begin_inset Formula $x$
\end_inset

, say 
\begin_inset Formula $dx$
\end_inset

, from consumer 
\begin_inset Formula $1$
\end_inset

 to consumer 
\begin_inset Formula $2$
\end_inset

, and consumer 
\begin_inset Formula $2$
\end_inset

 compensates 
\begin_inset Formula $1$
\end_inset

 by giving him 
\begin_inset Formula $dy$
\end_inset

 in exchange, where 
\begin_inset Formula $dy$
\end_inset

 is 
\begin_inset Formula $1$
\end_inset

's (and 
\begin_inset Formula $2$
\end_inset

's) marginal rate of substitution, neither 
\begin_inset Formula $1$
\end_inset

 or 
\begin_inset Formula $2$
\end_inset

 are any better off.
 
\end_layout

\begin_layout Standard
Instead of transferring good 
\begin_inset Formula $x$
\end_inset

 from consumer 
\begin_inset Formula $2$
\end_inset

 to consumer 
\begin_inset Formula $1$
\end_inset

, suppose that 
\begin_inset Formula $1$
\end_inset

 transfers a tiny bit of good 
\begin_inset Formula $x$
\end_inset

 to 
\begin_inset Formula $2$
\end_inset

 who uses it to produce additional 
\begin_inset Formula $y$
\end_inset

 using the production function.
 The production possibilities frontier is steeper than 
\begin_inset Formula $1$
\end_inset

's indifference curve, so this will give 
\begin_inset Formula $2$
\end_inset

 more than enough output to pay 
\begin_inset Formula $1$
\end_inset

 his marginal rate of substitution and maintain his utility.
 But then, all the residual output will be left over for consumer 
\begin_inset Formula $2$
\end_inset

 to enjoy.
 In other words, when the production possibilities frontier is steeper than
 both consumer's indifference curves, 2 can take a little of 
\begin_inset Formula $1$
\end_inset

's good 
\begin_inset Formula $x$
\end_inset

 and use it to produce 
\begin_inset Formula $y$
\end_inset

, which he uses to pay 
\begin_inset Formula $1$
\end_inset

 back for the good 
\begin_inset Formula $x$
\end_inset

.
 Then, 
\begin_inset Formula $2$
\end_inset

 will have some output left over for himself.
 The endowment point can't be Pareto optimal.
 
\end_layout

\begin_layout Standard
On the other hand, if Pareto optimality is the only objective, it can be
 achieved at the initial endowment point simply by moving along the contract
 curve until the consumers' marginal rates of substitution are equal to
 the marginal rate of transformation in production.
 Figure 5 shows how this might be accomplished.
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename firms_fig5.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
An Alternative 
\begin_inset CommandInset label
LatexCommand label
name "fig5"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Notice that at the point 
\begin_inset Formula $E^{\prime}$
\end_inset

 in Figure 
\begin_inset Formula $5$
\end_inset

, the indifference curves are both tangent and have the same slope as the
 production possibilities curve at the endowment point.
 So, everything is Pareto optimal.
 This might be achieved by having the government regulate the firm's output
 choice, tax away some of the firm's profit (a tax on dividends or capital
 gains) and then redistribute the proceeds to consumer 
\begin_inset Formula $1$
\end_inset

.
 Consumer 
\begin_inset Formula $1$
\end_inset

 will love this alternative plan, and consumer 
\begin_inset Formula $2$
\end_inset

 will hate it, but it will produce a Pareto optimal outcome.
 
\end_layout

\begin_layout Standard
Pareto optimality is a perfectly sensible objective for economic policy
 to try to accomplish.
 Profit maximization by firms is one way to achieve this, but you need to
 remember that it is a means to a goal, not a goal in itself.
 You should also try to remember, as Figure 
\begin_inset Formula $5$
\end_inset

 illustrates, that there are alternative ways of achieving Pareto optimality.
 Different methods lead to different distributional consequences - so, even
 though all consumers will agree that they want the outcome to be Pareto
 optimal, they may sensibly disagree about how this is accomplished.
 
\end_layout

\end_body
\end_document