Most
microeconomic analysis takes place within a partial equilibrium
framework. “Partial equilibrium” means that the
analysis is limited to examining the implications of the tendency of a subset of the economy to attain an equilibrium
status. This analysis rests on the assumption that the rest of the
economy is not affected by what goes on in the sector (often a
single market) being analyzed.
For many
purposes, it is reasonable to assume that what
goes inside a specific sector of the economy has no appreciable effect on the
rest of the economy. When this is true, no feedback effects need be
taken into account, since the implications of any change within the
sector remain confined to that sector. Effects
of changes within a particular sector are often small enough relative
to the rest of the economy to be ignored.
Such is not always the
case, however, so for some purposes partial equilibrium analysis must
be supplemented with a more general overview of the economy. The
present analysis relaxes the “partial” aspect of
partial equilibrium
analysis while maintaining the “equilibrium” part.
It examines the
nature of competitive equilibrium, and does so in a way that emphasizes
the interconnectedness of an economy's component parts. In particular,
it examines the
simultaneous nature of the determination of the prices of goods and of
resources. In doing so, it makes income endogenous, in contrast to the
partial equilibrium analysis of consumer behavior.
This introduction to general equilibrium analysis consists of three
parts. The file being viewed here develops the
logic of a simple general equilibrium model. It can be used alone or,
preferably, in conjunction with a similar development in a textbook or a
set of lectures. The second part of this set is an Excel workbook that
allows the user to change various values and trace the effects of such
changes through a general equilibrium system. Finally a workbook,
accompanied by a set of answers, provides guidance in working
through the model.
For information regarding downloading and implementing the workbook, click here or on the icon at the right.
The remainder of these notes are organized as follows. The next section
states the assumptions upon which a simple general equilibrium model
like the one used in the workbook are built. Then a summary of the
model shows more specifically the assumptions used for the
representation of the general equilibrium model that appears in the
workbook. The rest of the notes fills in the details. First, production
is considered and the nature of efficiency as it relates to production
is developed in detail―both
the nature of efficiency and the tendency of a competitive market
economy to achieve efficiency. Next, the distribution of output among
consumers is considered. Finally, these two are brought together in a
way that shows that considering either alone is not strictly correct.
Users preferring a hard copy may download this file in PDF format. To do so, click here
or on the Adobe icon below to print this document in a PDF format
(Acrobat Reader required). This provides a hard copy. The file does not
contain working links. The hard copy provides better control over printing the pages than the original HTML does.
The General Equilibrium Model's
Assumptions
The analysis herein addresses the functioning of a purely
competitive economy. This provides a point of departure for analyzing
models of imperfect competition.
The assumptions underlying the purely competitive model are these:
A
large number of buyers and sellers. No single decision-maker (buyer or
seller) can affect the market's demand or supply enough to
have a discernible effect on price.
All
potential buyers and sellers know the market
prices and product characteristics.
Consumers
choose to buy bundles of goods and services that
maximize their utility.
Producers choose to sell
quantities of their products that
maximize their profits.
Consumer
preferences and production technologies are given. That is, they do not
change as a result of any changes that the model is being used to
analyze. They may change for reasons that are outside the model ("exogenous" changes).
Both buyers and sellers may freely
enter or leave any
market.
In
addition to these assumptions, which underlie both partial-equilibrium
analysis and general-equilibrium analysis, our simple general-equilibrium
model requires one more assumption:
The
quantities of the
resources used in production are given. In terms of partial equilibrium
analysis, this means that the supply curves of resources are vertical
(have zero elasticity).
This assumption can be relaxed, but doing so dramatically increases the model's complexity.
Overview and Preview
This development of general equilibrium analysis is illustrated by a Microsoft Excel workbook. For instructions regarding the downloading of this workbook click here.
As you will see, this is a "2 x 2 x 2" model: two resources (labor and
capital) are used to produce two goods (Good X and Good Y), which are
consumed by two people (Anderson and Brooks). Anderson and Brooks also
provide the labor and the services of the capital.
The first sheet in the workbook provides a
nontechnical introduction to
the model. It contains a "Solver" routine that must be executed in
order to determine the equilibrium values. First consider the
components of the model: Two individuals
(Anderson and Brooks) own two sorts of resources (labor and capital)
that are used to produce two goods (X and Y). Exhibit 1
shows the values that must be entered in order to conduct the analysis.
The scroll bars at the right of the table entries allow user selection of values. The values
to be entered are these:
Coefficients of the two production functions:
XT = A(LX)α
(KX)(1-α) and YT
= B(LY)β(KY)(1-β).
These values determine the total output of X (XT) and of Y (YT).
The total amount of labor
(LT), which can be allocated to X production (LX units) or to Y
production (LY = LT - LX units). Likewise, for capital: KY = KT - KX.
The preceding conditions imply full
employment of both labor and capital.
Utility
functions for the two persons, named Anderson and Brooks:
UA = (XA)a(YA)(1-a) and UB =
(XB)b(YB)(1-b),
where XA is
the amount of X that Anderson consumes and likewise for YA, XB, and XB.
The share of
each resource
that each of the two producers/consumers owns. These ownership shares,
along with the solution of the general equilibrium system,
determine the income level of each of the two, which in turn determines
their demand curves.
Exhibit 2 shows results from a set of economic relationships
that
govern the system that is modeled here. XT and XT are generated by the
production functions defined above, the equations for which are entered into the cells that show the values 1942.1 and 291.34.
The "Price, Demand" entry is the demand price, the height of the demand
curve for X at the equilibrium values. The cell to the right of "Price,
Demand" contains a formula that yields the value $0.129 in this case.
Likewise, "Price, Supply" is the
supply curve's height at this set of values. The price in question is a
relative price, PX/PY. The entry "|Demand-Supply|" is the absolute
difference of the vertical distance between the demand price and the
supply price. By minimizing this value (finding the approximate point
of intersection of the demand and supply curves), subject to a
technical
condition noted below, Excel's
Solver program generates a set of equilibrium values, as shown in Exhibit 3 below.
To find these values, the user should do two things.
First,
select a set of values for the parameters and variables listed in Exhibit 1, and
then execute the Solver program. Doing this requires selecting
"Solver" in the "Tools" menu and then selecting "Solve." (If Solver is
not installed, installation is straightforward. Click here
for more information.) Running Solver returns the values in
Exhibit 2 (the output levels and the relative product prices) and those
in Exhibit 3 (the employment levels of labor and capital in each of the
two industries). Actually, Solver returns the values of LX and LY; KX
and KY are derived from the conditions of full employment as noted
below.
For the set of values above, the demand and supply curve are
those in Exhibit 4. Again,
the price is PX/PY. A similar pair of curves could be derived for good
Y, but no additional information would result from doing so.
Actually, showing the demand and supply curves for which the
system is
to be "solved" is a bit misleading. The supply curve comes directly
from the technological information and can be drawn without reference
to the system's solution. Not so with the demand curve, for the
returns to owning resources are part of the solution. These rates of
return determine income levels, which in turn determine the demand
curve. The final sheet "AllTogether" shows more about how all of this
fits together.
[You
may skip this paragraph without loss of continuity.] Solver finds
the equilibrium values as indicated above, subject to a set of
constraints. Exhibit 5 shows how one of these constraints
appears in
Solver. That constraint refers to materials in cells AA17 and
AA18, which are the slopes of the isoquants in X production and Y
production. The slope of the X isoquant must equal that of the Y
isoquant. This implies that the ratio of the marginal product of labor
to the marginal product of capital must be the same in both industries.
Why this must be so is developed in later sheets. Two other constraints
are built into the spreadsheet. These are the full employment
constraints: LY = LT - LX and KY = KT - KX. These constraints appear
in cells F19 and F20 of the spreadsheet.]
The
Edgeworth
Production Box provides
the most direct way to
represent the
allocation of fixed quantities of two resources between two competing
uses. Generalization to more than two uses or more than two resources
would require algebraic representation and would not allow graphic
representation. Fortunately, most of the lessons learned from the
simplest cases can be generalized. Exhibit 6
shows an
Edgeworth
Box for two resources, labor and capital, and two goods, X and Y. For
now ignore the box itself and focus on the horizontal and vertical
axes. These two axes relate capital used in X production (KX) to labor used
in X production (LX). Thus, this part of the graph is the same as that used
earlier in your microeconomics course to analyze production in the long run. The Edgeworth Box
generalizes by allowing representation of a second good’s
production to
be superimposed on the same graph The upper right-hand corner of the
graph represents a
second origin—the
point at which the
amount of labor
and capital employed in producing good Y are both zero.
The
box, which will be used extensively throughout the remainder of this
analysis, provides a compact way of showing each of the following six
values in a glance:
LT, the
total amount of labor available (the distance
from the X origin to the right-hand edge of the box—700 in Exhibit 6),
KT, the
total amount of capital available (the distance
from the X origin to the upper edge of the box—50 in Exhibit 6),
LX, the
amount of labor allocated to X production (horizontal distance from the left vertical axis to the allocation point—350 in Exhibit 6),
LY, the
amount of labor allocated to Y production (horizontal distance from the allocation point to the right vertical axis—350 in Exhibit 6),
KX, the
amount of capital allocated to X production (vertical distance from the bottom horizontal axis to the allocation point—50 in Exhibit 6),
and
KY, the amount of capital allocated to Y production
(vertical distance from the allocation point to the top horizontal axis—10 in Exhibit 6).
The
box conforms to the assumption stated above, in that the quantities
of both resources are fixed
(independent of input prices).
To repeat and summarize: Exhibit 6 shows an Edgeworth box for
which the following
values pertain: LT (total labor) is 700 units, KT (total capital) is 50
units, LX (labor used in X) is 350, LY (labor used in Y) is also 350
units,
KX (capital used in X) is 40 units, and KY (capital used in Y) is 10
units.
Resource Allocation and
Output
Each
allocation of labor and capital implies a set of output rates for the
two goods, X and Y. The quantities produced, denoted XT and YT, depend on the production
functions for these goods. These functions have been defined earlier,
but are repeated here for convenience. This analysis employs
Cobb-Douglas production functions
for the two goods. These functions have the form:
XT
= A(LX)α
(KX)(1-α) (the production function for X)
and
YT = B(LY)β(KY)(1-β) (the production function for Y).
The
production of each good is, therefore, characterized by constant
returns to scale (the exponents sum to 1.0). The analysis treats
Good X the more labor-intensiveof
the two goods (α > β).
Exhibit 7 shows the implications
of the initial allocation of resources between the two goods.
Exhibit 7 incorporates the allocation information in Exhibit 6 with added production information.
Refer first to the XT1 isoquant. The point “Initial
Allocation
of L and
K” shows one of many ways to produce 2035.008 units of good
X.
The isoquant
shows many of the other ways that the same output level could be produced.
(The
isoquants can extend outside the Edgeworth Box, as indicated in the
figure. The isoquants show input combinations that obey the production function,
while the Edgeworth Box shows what resources are available.)
The
YT1
isoquant shows ways of combining labor and
capital to produce 121.615 units of good Y. It also passes through the
initial allocation point. The levels of resource use should be
read read off
the top of the Edgeworth Box (labor) and the right-hand side axis
(capital).
The XT1
and YT1 isoquants intersect twice, approximately at LX = 350
and at LX = 686. Between the two intersection points,
the
isoquants define a lens-shaped area, one that is quite important. Any
reallocation represented by movement to a point within that area
represents
a
situation in which either XT or YT increases (or both do). Consider a
move
to the southeast along the XT1 isoquant. By definition of an isoquant,
XT is being held constant, but the level of YT increases. Likewise a
movement to the
southeast
along the YT1 isoquant represents a situation in which YT is held
constant while XT grows. Finally, any movement into the interior of the
lens
represents a situation in which both XT and YT increase without either
improved technology or increased resource supplies.
The next
section discusses the condition that must apply for all potential gains from resource
reallocation to be realized. Once this condition is satisfied, no
further resource reallocation can add to the production of either of the
two goods without simultaneously decreasing the production of the
other good. Satisfaction of such a condition is one criterion for economic
efficiency.
Efficient Resource
Allocation
As
noted
above, it is unlikely that an arbitrary allocation of resources among
the production of a set of goods will be efficient. Specifically in the
case above, we saw that movement from the initial allocation to one
within a specific lens-shaped area could result in increased production of both
goods (or more of either one of the two goods without reducing
production of the other good). We now examine the nature of efficiency
more closely.
Notice that, at the original allocation point, the YT1 isoquant is
flatter than the XT1 isoquant. Suppose that, at the initial endowment, the slope of the XT1
isoquant
is -5 and that of the YT1 isoquant is -2. This implies that increasing
LX by 1 unit can be accompanied by a 5 unit decrease in KX. At the same
time, decreasing LY by 1 unit requires only a 2 unit increase in KY.
Thus, the same output could be produced, and 3 units of K could be
retired. Of course, we're not going to retire the input; rather, we reallocate it.
Consider
another possibility: Increase LX by 1 unit and decrease KX by only 4
units. Then XT must increase since it would have stayed constant if KX
had fallen by 5 units. Now decrease LY by one unit while increasing KY
by 4 units. Since a 2 unit increase in KY would have maintained YT at
its initial level, YT must increase. (The number 4 is not
special; any number between 2 and 5 would have yielded the same
qualitative result: More XT and more YT at the same time.) Exhibit 8 summarizes this. The
notes in the last column explain why the effects are positive.
Effect of a Hypothetical Endowment Change Given Isoquant Slopes
Good
Isoquant Slope
Change in L
Change in K
Effect on Output (sign)
Rationale for Sign
X
-5
+1
-4
+
(∆KX = -5 would leave XT unchanged)
Y
-2
-1
+4
+
(∆KY = +2 would leave YT unchanged)
Exhibit 8
It is a simple matter to generalize the result above.
Whenever an
allocation of two inputs between two goods is such that the isoquants
of the two goods have different slopes, then a reallocation like the
one sketched above can increase the output of both goods (or of one
good without reducing the output of the other one). This implies that
efficiency occurs only when the isoquants are tangent to each other.
(Actually, tangency requires the satisfaction of two conditions, the
equality of the slopes of the two isoquants and the full employment of
labor and capital.)
Exhibit 9 allows another look at the nature of
efficiency. The graph shows the two initial isoquant, passing through
the point at which LX = 350 and KX = 40. It also
shows two other isoquants, for XT = 2416.259 and YT = 225.4133. These
two isoquants are tangent to each other when LX = 500 units and KX =
17.470 units. Once that allocation is achieved, no further
reallocations can simultaneously increase both XT and YT. This is one
point on the contract
curve.
The contract curve consists of all allocations that result in
economic
efficiency (characterized by the tangency of isoquants). Given the
assumption that the input levels are infinitesimally divisible, an
arbitrarily large number of such tangency points are possible, so the
contract curve is a continuous line. (This is an assumption of
convenience. Relaxing it would make the graphic representation much
harder, but would not appreciably affect the logical content of the
analysis.)
Briefly consider the entry Min_LX = 517.708. The value Min_LX is roughly
the value
of LX that corresponds to the intersection of the contract curve and
the XT1 isoquant. Setting LX at that level yields a point on the
contract curve at which XT does not change (actually, it does change
very slightly, because Excel
cannot deal with infinitesimal change). All of the gain in
efficiency accrues to the production of good Y. LX can be set at
smaller numbers (or at numbers above Max_LX). For any value of LX, the
program returns a value of KX that is on the contract curve. Given this
LX, KX pair, the output levels for XT and YT are dictated by the facts
that LT = LX + LY and KT = KX + KY (i.
e., fixed input supplies and full employment.
Values of LX below Min_LX result in points on the
contract curve
for which XT decreases. A movement from the initial allocation to one
of
these cannot be readily judged "good" or "bad" in a way that we might
judge a move to points within the lens-shaped area, including its
boundaries. Making a move to a point within the lens-shaped area increases both XT and YT or increases one of the
two without decreasing the others. Such moves enhance economic
efficiency, providing the potential for
the improved welfare of those who are part of the economy. Moves to
points on the contract curve but outside the lens-shaped area increase
the output of one good while decreasing the output of the other.
Whether such a move is a good one depends on the relative values of X
and Y.
This same story can be related in terms of output,
as Exhibit 10 demonstrates. The
initial resource allocation (LX = 350, KX = 40, LY = 350, and LY = 10)
results in a point inside the production possibilities curve, implying
that more of one good can be produced without reducing production of
the other good. A movement to the production possibilities curve results a simultaneous increase in both XT
and YT.
If LX were set at Min_LX, the move would have been a vertical one. All
of the efficiency gain would have been allocated to the production of
more of Good Y. Likewise, if LX were set at Max_LX, the move would have
been a horizontal one, with all of the efficiency gain going to the
production of more of Good X.
Markets and
Equilibrium: The Role of Input Prices
Continue
to take the allocation of labor and capital as they were in
the preceding sections: LX = 580, LY = 120, KX = 17.470, and KY =
52.530. So far, we have just asserted these values and have
said nothing about any tendency for them to be realized via the
operations of a system. We persist in this vein for a while, but now we
insert an important question: What must be the relative prices of the
inputs if this set of values (this point on the contract curve) is to
be consistent with market equilibrium?
From
the analysis of firms' behavior (Chapter 8 in Nicholson), we know
that the following must be true for each firm: Its isoquant must be
tangent to its isocost line. In the current setting, this implies that
the line in Exhibit 11 with the slope equal to w/r (w is the per-unit cost of labor and r is
the per-unit cost of capital, so w/r is the slope of each firm's
isocost line) must be tangent to both the XT2 isoquant and the YT2
isoquant. (The second "w/r = 0.090" entry in the figure is intentional. The value of w/r must
be the same whether expressed as a function of MPLX/MPKX or MPLY/MPKY.
Both are reported.)
Furthermore, the common slope must occur on the contract
curve,
where all labor and capital are demanded. This constitutes an equilibrium condition toward which resources markets tend to move—that the demand for labor equals the supply of labor and likewise for capital. Suppose
that w/r were higher
than 0.090. Both
producers of X and producers of Y would attempt to substitute capital
for labor. This means an excess demand for capital and an excess supply
of labor would push w/r back toward the equilibrium position.
Thus, the specified input mix (and the implied output mix) is
consistent with equilibrium in the input markets—firms are at
equilibrium (w/r = slope of isoquants) and markets are in equilibrium
(quantity demanded equals quantity supplied for both labor and capital).
Determining
Relative Product Prices
According to the preceding section, the input markets determine relative
input prices (w/r). This section shows that product markets determine
relative product prices. The
actual determination of the price of X relative to that of Y appears
graphically as the slope of a line, as was the case of the isocost line
in the Exhibit 11. Exhibit 12 shows a production
possibilities curve. A point on that curve has been selected as before:
A level of LX is specified; this, along with the contract curve,
determines KX, LY, KY, XT, and YT.
To see that the slope of the
production possibilities curve does
reflect prices requires some algebra and some recollection. First the
algebra:
∆XT = MPLX*∆LX (change in X).
This
follows
from the
definition of marginal product. The reasoning is
straightforward: The change in XT is the product of the change in XT
per unit change in LX times the change in the number of units of LX.
Likewise,
∆YT = MPLY*∆LY (change in Y).
To
convert this to a statement about the slope of the production
possibilities curve requires simple division:
∆YT/∆XT = (MPLY*∆LY)/(MPLX*∆LX) (change in Y per one-unit change in X).
Now rearrangement of the terms coupled with the recognition that ∆LX = -∆LY (full employment) yields
this result:
∆YT/∆XT = -MPLY/MPLX, or
-∆YT/∆XT = MPLY/MPLX (the slope of the production possibilities curve, stated as a positive number).
Now, move
from a statement about relative marginal products to relative prices.
Recall that equilibrium in a competitive industry, like X, requires the
following condition:
PX
= MCX,
where
PX is the
equilibrium product price and MCX is the marginal cost of the good.
Marginal cost, however, is intimately related to
MPLX. The relationship
is this:
MCX =
w/MPLX (marginal cost of Good X).
Thus
the equilibrium condition that PX = MCX implies that PX =
w/MPLX. Likewise
equilibrium in the Y market requires that PY =
w/MPLY.
Therefore,
we can deduce that PX/PY =
MCX/MCY = MPLY/MPLX,
which
is the same as the slope of the production possibilities curve.
One more aspect of this graph warrants attention. The line
tangent to
the production possibilities curve, whose slope is -PX/PY, is referred
to as an "IsoValue" line. Since we are examining the workings of
markets, we look to markets for measures of value. The height of a
demand curve provides a measure of value, the marginal willingness to
pay,
for the good. Using this measure of value, the total value at the point
of tangency of the production possibilities curve and the IsoValue
curve is the following:
Value0 =
PX*XT0 + PY*YT0 (value of a specified bundle of Good X and Good Y).
Value0 is this simple economy's "Gross Domestic Product" (GDP). The
"zero" subscripts are added as a reminder that the quantities to
which we refer are the quantities dictated by the arbitrary selection of LX, coupled
with the efficiency criterion developed previously. By construction,
this value is the same at all points on the IsoValue line. Thus, the
Y-intercept is Value/PY, so
the Y-intercept is (PX/PY)*XT0
+ YT0,
and the
X-intercept is XT0
+ (PY/PX)*YT0.
More
generally, any point on the line can be defined as
YT =
[(PX/PY)*XT0 + YT0]
- (PX/PY)XT.
The
"value" defined above corresponds to the GDP—the total value of
final goods and services (X & Y here) produced by the economy.
The term inside the square bracket is a constant. The slope of the IsoValue line is –PX/PY, or -∆YT/∆XT = PX/PY.
Later, we use the fact that only relative prices matter and define
units of Y such that PY identically equals 1.000, so the IsoValue
line's equation becomes the simpler:
YT = PX*(XT0
- XT) + YT0,
and
the "GDP" is stated in Y-unit equivalents. To see that the slop of the IsoValue line, -∆YT/∆XT, is PX/PY (actually PX, since PY is set to 1.0), rearrange the terms in the equation above:
The term in the parentheses is a constant, so the change in YT per one-unit change in XT is -PX.
The
Supply of Good X
The
preceding analysis shows that, given the production possibilities
curve, any specific combination of goods X and Y is uniquely related to
a unique relative price.
That section considered a single price ratio, but this insight
is easily generalized. If we list all possible PX/PY values and trace
the associated quantities of X, then we have derived a supply curve for
good X like the one in Exhibit 13. (Of course, a similar supply curve could be derived for good
Y.) Mechanically, the inversesupply
curve is developed. That is, each value of XT is associated with
the relative price necessary to bring forth that quantity, that
relative price being the height of the supply curve at the specified
quantity. This approach, rather that stating prices and determining the
resulting quantities, is dictated by the use of the spreadsheet. The
resulting supply curve is the same either way.
This supply curve differs from the one developed using
partial
equilibrium analysis in one important respect. In that analysis, "other things equal" were
technology and the supply curves of resources to the industry
under consideration. (Recall that a horizontal supply of inputs to the
industry is a necessary but not sufficient condition for the product supply curve to be
infinitely elastic.) In the present model, as in its partial
equilibrium counterpart, technology is among the "other things" that
are equal. In this case, however, the aggregate supply of labor and
capital are taken as fixed (vertical supply curves). Both resources are
assumed to flow among firms in the two industries. Thus, the supply
curve facing any single firm in either industry is a horizontal line,
as it is in the partial-equilibrium model, but no input supply curves can be
defined for either of the two industries. Each firm, in each industry, buys resource services in economy-wide resource markets.
Distributing the
Economy's Output: The Edgeworth Consumption Box
Exhibit 14 is essentially the same as the one above that
shows a point on the production possibilities curve. The only
difference involves emphasis: Now we focus on the box that is defined by the
selection of a point on the production possibilities curve (which is
effected by selecting a value of LX and then imposing
the conditions necessary to have L and K allocated between X
and Y production in an efficient fashion). Once XT = 2416.259 and YT =
225.413 (or whatever values) are selected, the next question to arise
is, By whom shall these products be consumed? (A related question
regards the institutional framework within which the answer to this
question is determined. We postpone that
question for now.)
For purposes of
exposition,
we illustrate the answer the question of who consumes the products by
allowing only two consumers. These are formally introduced momentarily.
A point at the origin (the southwest corner of the box) corresponds no
consumption for one consumer with the other getting all of the output
of both goods. Likewise, the northwest corner represents a situation in which the first consumer
gets everything and the second consumer gets nothing. The
similarity of this box to the Edgeworth Production Box with which the
workbook began should be apparent. We refer to this box as the
Edgeworth Consumption Box.
Every point in the Edgeworth Consumption Box represents
levels of
consumption for the two consumers. Exhibit 15 is
the same as Exhibit 14, except
that the production possibilities
curve is removed and a distribution of the output between the two
consumers,
now named Anderson and Brooks, is added. The distribution is
arbitrary. We have selected to give 1537.328 units of good X and 75.806
units of good Y to Anderson, with Brooks getting the remainder. As noted above, the origin (Good X = 0 and Good Y = 0)
would correspond to Anderson getting nothing and Brooks getting
2223.359 units of X and 253.483 units of Y. The box's northeast corner
is Brook's origin. If the distribution point were at the northeast
corner, Anderson would get all of both goods and Brooks, none.
Assigning an arbitrary distribution is temporary. After examining the
nature of
efficiency given a fixed quantity, we "close the system" by allowing
Anderson's and Brooks's income levels to be determined by their
ownership of labor and capital and by the relative prices of these inputs. Given their income levels, their demand
curves for good X (and implicitly for good Y) are determined, as is the
market demand curve. Setting demand equal to supply (defined above)
results in equilibrium values for PX, PY, w, and r as well as the income levels
of the two consumers/producers.
Output
Distribution
and Utility Levels
Any
distribution of goods X and Y between the two individuals generates an associated set of
utility levels for these two. Just
as isoquants previously represented output levels, indifference
curves now represent utility levels. (Unlike output levels, the
numbers associated with the utility levels have no meaning. The
rankings of these numbers are meaningful. UA = 200 implies that
Anderson is better off than if UA = 100, but we cannot say that
Anderson is twice as well off.) Given Anderson's and Brooks's
utility functions, the initial distribution of goods X and Y yields the
utility levels shown in Exhibit 16. The graph in Exhibit 16
shows that the indifference curves associated with this distribution of
the goods intersect each other at the designated distribution of Good X
and Good Y. That is, Anderson's and Brooks's indifference curves are
not tangent.
An indifference curve's slope shows the rate at
which the
individual is willing to exchange good X for good Y. In the current
example, Anderson is willing to give up less Y per additional unit of X
than is Brooks. At the point of intersection (the southeast corner of
the lens-shaped area in Exhibit 16), the relevant slopes are about
0.021 for Anderson and 0.397 for Brooks. Thus,
Anderson would
gain by giving up 50 units of X for, say, 2 units of Y (0.04 units of Y
per unit of X). Brooks would be
willing to exchange the 2 units of Y for the much fewer than 50
units of X (0.04 is well below 0.397, the maximum number of units of Y
that Brooks would willingly exchange for an additional unit of X).
Therefore, trade can make both Anderson and
Brooks better off.
As with
production, a consumption contract curve defines all combinations of XA and YA
(Anderson's X and Y) and XB and YB (Brooks's X and Y) for
which no further
redistribution can make
one person better
off without simultaneously reducing the other's utility level. Exhibit
17 shows a move to a point on the contract curve from
the initial distribution. Both Anderson and Brooks gain utility as a
result of the exchange. In this case, Anderson trades about 845
units of X in exchange for about 79 units of Y, an
exchange
rate of around 10.9 to 1, or the number of units of Y traded per unit
of X is about 0.09. This value, 0.09, lies between the slopes of the two
indifference curves at the initial distribution point, so both Anderson
and Brooks gain from the redistribution of the two goods.
To summarize, Anderson starts with 1537.328 units of Good
X and trades away 845.263 units, leaving 878.931 units. In return
Brooks gives 78.841 units of Good Y to Anderson, reducing Brooks's
consumption of Y from 149.607 units to 70.766. The rate of exchange for the two products is 0.093 (= 78.841/845.263) units of Y per unit of X.
As the table above the graph in Exhibit 17 shows, utility levels rise for both
Anderson and Brooks. The graph reflects this, showing that both Anderson and Brooks
are on higher indifference curves than before trade occurred.
Product
Prices Revisited
So
far, the distribution of X and Y between A and B has been
treated without reference to any institutional arrangement. Most such
exchanges occur via markets, so we next consider how such exchanges
take
place in a market setting. As in the case of the markets for
resources, prices (relative prices) direct the distribution of the
goods among individuals. In this particular case, the outcome depicted
in the previous paragraph could be effected in a market setting with a
relative price of about 0.096 (i.e., about 10.4 units of X per unit of Y).
Exhibit 18 shows a price line with this slope passing through a
point on the contract
curve.
Consider the price line in more detail. The price line goes
through the
point (692, 155)—approximately; we round to simplify the math . With a relative price of
0.096 (which differs from 0.093 shown earlier due to rounding),
we can directly construct Anderson's budget line. Giving Anderson 692
units of X and 155 units of Y is the same as giving Anderson about 221
units of Y (155 + 0.096*692 = 221.432) and allowing trade along the
price line. (Or, for that matter, giving Anderson about 2307 units of X
and allowing such trade—692 + 155/0.096 = 2306.583. Or, giving Anderson about $221.43.) The same line,
using Brooks's origin as the point of reference, defines Brooks's
budget line.
Completing
the Model: Determining Income, Demand, and Prices
One item remains before the system is complete: income determination.
Before turning to this feature of the model, however, we
review some
salient points.
The
"givens" so far have been these:
quantities of resources (L
&
K), technology (production functions for X and Y), the allocation of L
and K between the production of X and of Y,
the distribution
of
the resulting X and Y between two consumers, A(nderson) and B(rooks),
and
the utility functions of these two consumers.
Given
the values above, two sets of prices emerge: PX/PY and w/r. These
are the relative prices that must occur if the relative prices are
determined in competitive product and resource markets—that is, in
markets where each individual is a price taker.
The
next step is to remove some of these "givens." In particular, we wish
to see how markets simultaneously determine PX/PY and w/r without an
arbitrary specification of how many resources are allocated to each
good
and how much of each good is distributed to each individual. Doing this
turns out to be quite direct. These values emerge from the model once we incorporate
the specification that each
consumer is also a producer whose income depends on the amounts of L
and K that she/he sells in the resource markets.
We approach this analysis in two steps, each related to a
figure taken
from the final worksheet of the workbook. The first excerpt shows the
equilibrium values; we discuss how these are determined. The second
excerpt, shown below, considers the nature of supply and demand in a
general equilibrium system.
Exhibit 19 shows the results of the particular set of
parameter values used in preceding sheets. The ownership of labor and
capital are as indicated in the first sheet. (We can drop the
assumption that Anderson and Brooks are
individuals. The model really requires that many Andersons and many
Brookses offer labor and capital services so that each is a price
taker. One could think of 1,000,000 Andersons, each with one
one-millionth of the labor (or any other number) and likewise for
Brooks. For the sake of relating the story, however, we refer to these
two as individuals.)
Once the ownership of resources is determined, so is the
relative
intensity of demand for X and Y and, thereby, the
relative price. Given the relative price, however, only one point on the
production possibilities curve (or, equivalently, on the supply curve
for X) is consistent with equilibrium. Determining a point on the
production possibilities curves, coupled with the demand for X and for
Y, determines how much L and K are allocated to each good's production.
This implies w/r, which completes the model. (We set PY = 1,
identically, so that PX/PY is just PX.) Input prices are easily
determined as follows:
PY*MPLY = w, so (equilibrium in the market for Resource X)
w = MPLY (or, w = PX*MPLX) and
PY*MPKY = r, so (equilibrium in the market for Resource Y)
r = MPKY (or r = PX*MPKX).
We can now determine the value of goods and services
produced: "GDP" =
YT + PX*XT. Likewise, we can determine the cost of "GDP" which is the
income received by resource owners: "GDP" = w*L + r*K. These two values
appear in Exhibit 19. (A slight discrepancy can occur from the fact that we
solved for these values using numerical approximations rather than by
direct analytical computation.) The income distribution between
Anderson and Brooks is also indicated. The final
"GDP" entry is redundant: it is just the sum of Anderson's and Brooks's
incomes. These income levels are determined as follows:
MA = w*LA + r*KA (Anderson's money income)
where LA is the flow of labor
services that Anderson provides and KA is the flow of capital services
that Anderson provides. Likewise,
MB = w*LB + r*KB. (Brooks's money income)
Since Anderson and Brooks own all resources, the following must be true:
LT = LA + LB (labor ownership)
and
KT = KA + KB. (capital ownership)
Finally, consider demand. The simple utility functions
that we
employ yield correspondingly simple individual demand curves for good X:
XA =
a*MA/PX (Anderson's demand for X)
and
XB =
b*MB/PX. (Brooks's demand for X)
Such
individual demand curves exhibit unitary price elasticity and income elasticity.
MA and MB are the money income levels of Anderson and Brooks. The
values of a and b are those in the utility functions. (The general
equilibrium model does not depend on this specific sort of utility
function and resulting demand curve, but building a spreadsheet does
require a specific, necessarily arbitrary specification.) The market
demand curve is the sum of the individual demand curves:
XT =
a*MA/PX + b*MB/PX, or
XT = (a*MA
+ b*MB)/PX. (demand for X)
It
is a straightforward matter to confirm that, given MA, Anderson's
demand curve slopes downward. Likewise with Brooks. Notice, however,
that the market demand curve shown in Exhibit 20 has a U-shape. (The U is more
pronounced if Anderson supplies more of the capital and Brooks more of
the labor.) That is, it
slopes upward over a range of values for good X.
Is this really possible? It
is, because the income levels MA and MB are not given at the general
equilibrium level,
even though each individual Anderson and each individual Brooks takes
his/her income level as
given (likewise with prices). Rather, they are determined as part of
the overall set of equilibrium values. As it happens, we selected
parameter
values such that both Anderson's and Brook's incomes (measured in terms
of units of the numeraire Good Y) increase as X production increases
and
Y production decreases. Given the current parameter values, over some
range of XT this income effect more than offsets the substitution
effect.
Do not take
from this analysis the conclusion that demand curves are probably not
downward sloping. We selected parameter values that would yield this
result. This lesson is that what is held constant in partial
equilibrium analysis
may not be properly held constant in a general-equilibrium framework
and that different "other things equal" specification can yield quite
different results.
As an aside, consider one more cautionary note. As XT increases, so
does total income. Thus, the economy would maximize income ("GDP") when only
Good X is produced. This would not, however, generate maximum value.
Beyond XT = 1942, the amount that an Anderson or a Brooks is willing to
pay for an additional unit of X is less than the cost of producing the
additional X. Thus, value is maximized at XT = 1942, while "GDP" is
maximized when only X is produced (XT = 3620, approximately). Treat
this as one more statement that GDP and related measures of value
should be interpreted with care.
Conclusion
This
overview and the workbook on which it is based show how a simple
economic system can establish equilibrium via a set of relative prices,
prices of two goods and two resources. The system is complete, in that
all market values are determined within the system. This includes
income levels for resource owners.
One important question that is beyond the scope of this relatively simple
illustration of
general equilibrium analysis is the generality of what has been shown.
In particular, when the number of goods and resources is large, and
when the production and utility functions are not of the functional
forms used here for simplicity, can we have confidence that the system
still has an equilibrium and that it will tend toward any such
equilibrium?
The answer to this question was long in coming. Leon Walras
posed the
question in the 1870s and offered a tentative answer in the
affirmative. In the 1950s, Gerard
Debreu established that Walras was correct.
Notes
Second-Best
Considerations
The
classic reference is R. G. Lipsey and Kelvin Lancaster,
“The General
Theory of Second Best,” The
Review of Economic Studies, Vol.
24, No. 1
(1956 - 1957), 11-32. Subsequent articles have applied the
Lipsey/Lancaster analysis to cases involving monopoly, external
effects, and other institutional constraints.
Text Reference
This
treatment follows that of Walter Nicholson, Microeconomic
Theory: Basic Principles and Extensions, 9th
Edition, Thomson-SouthWestern, Chapter 12. It begins with
production and
treats consumption later. Many textbooks begin with a fixed set of
quantities and extend the analysis to incorporate production. The use
of spreadsheets dictates beginning with production.
Implications of equal
exponents
If the two
exponents are equal, then the two goods use labor and
capital in the same ratio, and both the contract curve (developed
below) and the production possibilities curve
(also developed below) are linear.
Caveat
Note
the word "potential." A move that increases both XT and YT does
increase total physical output. Such a move might, however, affect
income distributions, so that some people involved in the economy are
made worse off. If this happens, one cannot unambiguously say that such
a move is a good one, even though it does make the economy more
efficient.
Marginal Product and
Marginal Cost
To see this
relationship, consider the following:
∆XT = MPLX*∆LX (Change in the amount of X produced),
and
∆CX = w*∆LX. (Change in the amount of Y produced)
The first
statement appears above, and just says that the change in
output from adding labor to X production is the product of the amount
added and labor's marginal product. The second statement says that the
change in the total cost of producing the X when LX changes equals the
wage per unit of LX times
the change in the amount employed.
Marginal cost of X is defined as ∆CX/∆XT, which
equals w/MPLX.
The Excel Workbook
We recommend saving the workbook file to a disk and then opening it. Important consideration: The workbook contains macros. Activating these macros requires that Excel's
security level be set a medium or lower before the workbook is opened.
The default is high; this setting will not allow the opening of macros.
Under "Tools/ Macro .../
Security" set the security level to medium.
We repeat: Failing to set the security level below its default level
will cause Excel to load the workbook but to strip it of all macros.
To open the workbook, click here or click on
the icon at the right.
Solver
To add Solver to Excel,
select "Add-Ins" in the "Tools" menu. Then
click on the "Solver Add-In" and "OK."
A
technical point regarding the solution in this workbook is in order.
Solver has been instructed to minimize the absolute difference between
the demand price and the supply price. It could have been instructed to
set that value equal to zero. Our experience has been that Solver does
not converge to a solution as reliably when the second option is
chosen. As a practical matter, the solutions are the same.
One other point: Minimizing the vertical distance (the difference
between supply prices) rather than the horizontal distance (the
difference between quantity demanded and quantity supplied) is dictated
by the mechanics of the worksheet.
Debreu Gerard
Debreu, Theory of Value. New York: John Wiley & Sons, 1959.
This
demonstration was the basis for Debreu's receipt of the Nobel Prize. A
good introductory treatment of the approaches used by Walras and Debreu
is found in Nicholson. Also, Nicholson provides a good annotated
bibliography.
Exercises
A set of
questions accompanies this worksheet. Click here
or on the icon immediately below to open the workbook. A set of answers is also
provided.
Print the document
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uals
(Anderson and Brooks) own two sorts of resources (labor and capital)
that are used to produce two goods (X and Y). Exhibit 1
shows the values that must be entered in order to conduct the analysis.
The scroll bars at the right of the table entries allow user selection of values. The values
to be entered are these:
above, the equations for which are entered into the cells that show the values 1942.1 and 291.34.
parameters and variables listed in Exhibit 1, and
then execute the Solver program. Doing this requires selecting
"Solver" in the "Tools" menu and then selecting "Solve." (If Solver is
not installed, installation is straightforward. Click here
for more information.) Running Solver returns the values in
Exhibit 2 (the output levels and the relative product prices) and those
in Exhibit 3 (the employment levels of labor and capital in each of the
two industries). Actually, Solver returns the values of LX and LY; KX
and KY are derived from the conditions of full employment as noted
below.
but no additional information would result from doing so.
aterials in cells AA17 and
AA18, which are the slopes of the isoquants in X production and Y
production. The slope of the X isoquant must equal that of the Y
isoquant. This implies that the ratio of the marginal product of labor
to the marginal product of capital must be the same in both industries.
Why this must be so is developed in later sheets. Two other constraints
are built into the spreadsheet. These are the full employment
constraints: LY = LT - LX and KY = KT - KX. These constraints appear
in cells F19 and F20 of the spreadsheet.]
resents a
second origin—the
point at which the
amount of labor
and capital employed in producing good Y are both zero. 
of
the two goods (α > β).
Exhibit 7 shows the implications
of the initial allocation of resources between the two goods.
It also
shows two other isoquants, for XT = 2416.259 and YT = 225.4133. These
two isoquants are tangent to each other when LX = 500 units and KX =
17.470 units. Once that allocation is achieved, no further
reallocations can simultaneously increase both XT and YT. This is one
point on the contract
curve.
part of the economy. Moves to
points on the contract curve but outside the lens-shaped area increase
the output of one good while decreasing the output of the other.
Whether such a move is a good one depends on the relative values of X
and Y.
To see that the slope of the
production possibilities curve does
reflect prices requires some algebra and some recollection. First the
algebra:
That section considered a single price ratio, but this insight
is easily generalized. If we list all possible PX/PY values and trace
the associated quantities of X, then we have derived a supply curve for
good X like the one in Exhibit 13. (Of course, a similar supply curve could be derived for good
Y.) Mechanically, the inversesupply
curve is developed. That is, each value of XT is associated with
the relative price necessary to bring forth that quantity, that
relative price being the height of the supply curve at the specified
quantity. This approach, rather that stating prices and determining the
resulting quantities, is dictated by the use of the spreadsheet. The
resulting supply curve is the same either way.
imposing
the conditions necessary to have L and K allocated between X
and Y production in an efficient fashion). Once XT = 2416.259 and YT =
225.413 (or whatever values) are selected, the next question to arise
is, By whom shall these products be consumed? (A related question
regards the institutional framework within which the answer to this
question is determined. We postpone that
question for now.) 
. The graph in Exhibit 16
shows that the indifference curves associated with this distribution of
the goods intersect each other at the designated distribution of Good X
and Good Y. That is, Anderson's and Brooks's indifference curves are
not tangent. 
The
next step is to remove some of these "givens." In particular, we wish
to see how markets simultaneously determine PX/PY and w/r without an
arbitrary specification of how many resources are allocated to each
good
and how much of each good is distributed to each individual. Doing this
turns out to be quite direct. These values emerge from the model once we incorporate
the specification that each
consumer is also a producer whose income depends on the amounts of L
and K that she/he sells in the resource markets. 
Such
individual demand curves exhibit unitary price elasticity and income elasticity.
MA and MB are the money income levels of Anderson and Brooks. The
values of a and b are those in the utility functions. (The general
equilibrium model does not depend on this specific sort of utility
function and resulting demand curve, but building a spreadsheet does
require a specific, necessarily arbitrary specification.) The market
demand curve is the sum of the individual demand curves:
A
technical point regarding the solution in this workbook is in order.
Solver has been instructed to minimize the absolute difference between
the demand price and the supply price. It could have been instructed to
set that value equal to zero. Our experience has been that Solver does
not converge to a solution as reliably when the second option is
chosen. As a practical matter, the solutions are the same. 