Download CE Examples

1
Calculating CE
The Edgeworth box figures indicate the general idea, below I present the explicit
calculations. I find market-clearing prices for x; recall the Walras’ law implies that
market clearing in all but one good is sufficient for an equilibrium. I also normalize
prices as discussed in class, typically by setting px or py or pk ... equal to 1.
1.1
Consumption / Exchange
Example 1 uA (x, y) = xy; uB (x, y) = x + y; x̄ = 4, ȳ = 3
x̄A = 3.5, ȳA = 3, x̄B = 0.5, ȳB = 0.
Solution 1: Find demand, set it equal to supply.
• First find the demand functions.
— For A we know that at any prices A will demand strictly positive amounts
of x and of y because if A consumers zero of ether good her utility is zero,
while by consuming a strictly positive amount of both, no matter how
small, then A’s utility is positive. Therefore we know that at any price
D
xD
Since both are strictly positive we know
A (px ) > 0 and yA (px ) > 0.
that the solution is always interior. Therefore the solution can be found
in one of two ways, as follows.
∗ max xy s.t. px x + py y = px x̄A + py ȳA . Substitute for y in max xy,
take the derivative and set equal to zero.
This will give xD
A =
(px x̄A + py ȳA ) /2px .
∗ At interior solutions to the consumer demand problem we know that
M RS A = px /py where in this case M RS A = yA /xA , and that the
budget constraint is satisfied. These two equations determine the two
unknowns. In this case we see from the first equation that px /py =
yA /xA , so that px xA = py yA , so that spending on the two goods is
equal, so that each gets half the income.
— B will buy the less expensive good, and will be indifferent as to how she
1
spends her income if prices are equal. That is,
⎧
0
if px > py
⎪
⎪
⎪
⎪
px x̄B + py ȳB /px
if px < py
⎨
anything
such
that
xD
(p
)
=
x
B
⎪
⎪
all income is spent if px = py
⎪
⎪
⎩
on the two goods
Normalize py = 1.
• Assume px > 1.
D
xD (px ) = xD
A (px ) + xB (px ) = IA / (2px ) + 0 = (px x̄A + ȳA )/ (2px ) = x̄A /2 +
ȳA / (2px ) = 7/4 + 3/ (2px ).
xS = 4
xD (px ) = xS =⇒ (3.5px + 3) / (2px ) = 4 =⇒ px = 23 . But as this is less then 1
there is no CE with px > 1.
• Assume px < 1.
D
xD (px ) = xD
A (px )+xB (px ) = IA / (2px )+IB = (px x̄A +ȳA )/ (2px )+(px x̄B + ȳB ) /px
= (3.5px + 3) / (2px ) + 0.5
xS = 4
xD (px ) = xS =⇒ (3.5px + 3) / (2px ) + 0.5 = 4 =⇒ px = 67 .
Conclusion 2 So there is a CE with px = 6/7.
What is the CE allocation? We calculated that x∗B = 0.5. If this is a CE then
the x market clears to x∗A = 4 − 0.5 = 3.5. (This can also be found by substituting
the CE price into xD
A (px ).) Since B spends all her income on x at this CE we know
∗
= 0. By Walras’ Law we know that the y market clears, so we have yA∗ = 3.
that yB
• The fact that we found a CE does not mean there are no others. To confirm
that there is no CE with px = 1 we need to solve for that case as well. If px = 1
S
then xD
A = 6.5/2 = 3.25. Since x = 4 this means that for the demand to equal
D
supply we need xB = 4 − 3.25 = 0.7. Is this a solution to B’s optimization?
Can xD
B = 0.7? No–B’s income is only 0.5 so B cannot afford this. Therefore
there is no CE with px = 1 .
2
Remark 3 Note that it is critical to avoid the following conceptual and technical
mistake. Do NOT say M RSB = 1 and since M RSB = px /py we have px = py as a
CE. Clearly as we can see above this is wrong and is not a CE. Moreover it indicates
severe misunderstanding of the concept of CE as it deduces from an individual what the
market prices will be, when those prices are determined by demand of all individuals.
Therefore, even if the answer happens to be 1, this method is incorrect and incomplete.
Remark 4 Note that in the CE we found above there is no trade. There still is
meaning to what is the CE, especially the CE prices. This is because at other prices
agents would want to trade (and markets would not clear). For example, in equilibrium I am not selling my car; the (non-stated) equilibrium price for the car is lower
than what I would sell it for and greater than what anyone would pay.
Exercise 5
• If we changed B’s endowment could there be a CE with px > 1?
No–B’s demand for x would not change from 0 in the first step above.
• If we decreased A’s endowment could there be a CE with px > 1? No–
decreasing A’s endowment could only lower A’s demand at any price for x hence
it will remain impossible to get an equilibrium.
Example 6 Same preferences as above x̄A = 3, ȳA = 0, x̄B = 1, ȳB = 3.
Solution 6:
• Assume px > 1.
D
xD (px ) = xD
A (px ) + xB (px ) = IA / (2px ) + 0 = (px x̄A + ȳA )/ (2px ) = (3px ) / (2px )
xS = 4
xD (px ) = xS =⇒ (3px ) / (2px ) = 4 =⇒ no solution.
So there is no CE with px > 1.
• Assume px < 1.
D
xD (px ) = xD
A (px )+xB (px ) = IA / (2px ) +IB = (px x̄A + ȳA )/ (2px )+px x̄B + ȳB =
(3px) / (2px ) + px + 3
xS = 4
xD (px ) = xS =⇒ (3px ) / (2px ) + px + 3 = 4.
As there is no non-negative solution, there is no CE with px < 1.
3
• Assume px = 1.
D
D
D
1. xD (px ) = xD
A (px ) + xB (px ) = IA / (2px ) + xB = 1.5 + xB
xS = 4
D
xD (px ) = xS =⇒ 1.5 + xD
B = 4 =⇒ xB = 2.5
Recall that B is indifferent and hence willing to consume any amounts of x and
y that are feasible and use all B’s income. B’s income is 4 so consuming 2.5
units of x at price 1 is feasible. So we have a CE.
Remark 7 Note that in this case saying MRSB = px /py would give the correct
answer, but the analysis is incomplete as it ignores the whole idea of CE ( market
demand equals supply) and hence is WRONG.
Example 8 uA = 2x + y, uB = x + 2y, the initial endowment is not given in numbers,just x̄A , x̄B , ȳA , ȳB , all of which are strictly positive. Find the CE (which depends
of course on the initial endowment).
Solution 8:
⎧
⎨
0
if px > 2py
px x̄A + py ȳA /px if px < 2py
(px ) =
⎩
“anything”
if px = 2py
⎧
0
if px > py /2
⎨
p
x̄
+
p
ȳ
/p
xD
(p
)
=
x B
y B
x if px < py /2
x
B
⎩
“anything”
if px = py /2
xD
A
where “anything” means anything such that the consumer spends all income in the
two goods. Normalize py = 1.
• Clearly there is no CE with px < 1/2 and none with px > 2. (In each case both
consumers buy only one good so there is excess supply in the other.)
• Can there be an equilibrium with 1/2 < px < 2? In this case A buys only x
and B buys only y. Therefore xD = px x̄Apx+ȳA , so if we solve xD = xS we have
x̄A + ȳpAx = x̄A + x̄B , or px = x̄B /ȳA . If x̄B /ȳA is between 1/2 and 2, that is, if
ȳA /2 < x̄B < 2ȳA then we have a CE price; otherwise there is no CE with price
in this range. What is the CE allocation? It is the lower-right corner of the
Edgeworth box, with A consuming only x and B only y.
4
• Can there be an equilibrium with px = 1/2? In this case A buys only x and B
is indifferent. So xD = x̄A + ȳA /px = x̄A + 2ȳA , and xS = x̄A + x̄B . Therefore
we have supply equal to demand if xB = xS − xD = x̄B − 2ȳA . If B can afford
this then we have an equilibrium. B can afford this if IB = px x̄B + py ȳB ≥
px (x̄B − 2ȳA ), that is if x̄B /2 + ȳB ≥ x̄B /2 − ȳA , that is if ȳB ≥ ȳA . What is
the CE allocation? A buys only x, and B buys all the y and the remaining x.
• Can there be an equilibrium with px = 2? I leave this as an exercise for you,
as well as drawing an Edgeworth box, and designating the regions in the box
where the CE is of each type.
Example 9 uA (x, y) = min {x, y}; uB (x, y) = x + y; x̄A = 2, ȳA = 1;x̄B = 2;ȳB = 2.
Solution to Example 9. Set py = 1. A’s demand cannot be calculated using
derivatives, however it should be clear that A will spend her income in a way that leads
D
D
D
to equal consumption of x and y. If xD
A (px ) = yA (px ) and px xA + yA = IA ≡ px 2 + 1
(where I do not explicitly write (px ) to denote the dependence of the demand on the
D
price of x as it clutters up the notation and should be clear) then px xD
A +xA = px 2+1,
2px +1
D
D
so xA = px +1 = yA . B’s demand as before is to spend all income on the less expensive
good, and if px = 1 then B is indifferent.
D
S
• Can px > 1? If so then xD = xD
A (since xB = 0) and as x = 4 this implies
2px +1
3
= 4, which implies px = − 2 , which cannot be an equilibrium.
px +1
2px +1
2px +1
2px +2
IB
D
• Can px < 1? If so then xD = xD
and as
A + xB = px +1 + px = px +1 + px
2p
+1
2p
+2
xS = 4 this implies pxx+1 + pxx = 4, and then px = −2, which cannot be an
equilibrium.
D
D
• Can px = 1? xD
A + xB = 4 implies xB = 4 −
afford this? Yes, as B’s income is 4.
2px +1
px +1
= 4 − 3/2 = 2.5. Can B
1. This implies that we have a CE with px = 1 and with allocation xA = 3/2,
xB = 2.5, yA = 3/2, yB = IB − px xB = 1.5.
We can see that the y market clears as well and maximizes players utility at
this allocation which confirms our calculations. (We don’t really need to check
the y market because of Walras’ law.)
Example 10 uA (x, y) = y +2x1/2 , uB (x, y) = y +ln (x) ,let x̄A , ȳA ,x̄, ȳB be variables.
5
Solution to Example 10. Set py = 1.
A’s demand can be calculated by solving maxx,y uA (x, y) s.t. px x + y = px x̄A + ȳA ,
and at an interior solution we know that this solves M RS A = px /py , so 1/x1/2 = px
2
or xD
A = 1/px . (We could also solve the maximization problem–I do so in another
example with quasilinear preferences below.) The remainder of A’s income will be
2
spent on y. If A’s income is not enough to purchase xD
A = 1/px then A will consume
no y and spend all of her income on x, so that xD
A = (px x̄A + ȳA ) /px = x̄A + ȳA /px .
Similarly at interior solution B’s demand can be seen to solve 1/x = px or xD
B =
1/px. At a corner solution in which B does not have enough income to purchase
1/px units of x then B will consume no y and xD
B = x̄B + ȳB /px .
We do not know that the solution is interior; but we will work under that hypothesis for now and if there is a contradiction we will turn to corner solutions.
Thus, if there is a CE that is interior it will satisfy 1/px +1/p2x = xS = x̄ = x̄A +x̄B .
2
Solving ³we get 1/px + 1/p
´ x = x̄, which has only one non-negative solution which is
p
1
px = 2x̄
1 + (1 + 4x̄) . If at this price both agents have an interior solution, which
means that their income px x̄A + ȳA and px x̄B + ȳB is sufficient for their demand for x,
then we have found a CE. If ȳA and ȳB are both large enough then this will be the case
since increasing their endowment of y will increase their income without
µ changing their
¶
2
2
demand for x. What is the allocation in this case? xD
A = 1/px =
which can be seen to equal.
³
√2x̄
1+
´
(1+4x̄)
If the above is not feasible because the agents cannot afford this interior allocation
we know we are at a corner solution. In any case we can check for corner solutions
as follows.
First assume that only A is at a corner solution. Then xD = x̄A + ȳA /px + 1/px
(where the first two terms are A’s demand for x and the last term is B’s demand).
We find an equilibrium price by setting demand equal to supply, x̄A + ȳA /px + 1/px =
x̄A + x̄B , and the solution is px = (ȳA + 1) /x̄B . We can then substitute this back
into B’s demand, 1/px , and if B can afford this we have found a CE.
Next one can assume that only B is at a corner solution, and proceed as above.
If at the resulting price A can afford his demand 1/p2x , we will have a CE.
Lastly, one can consider the case that both are at a corner solution.
Example 11 uA (x, y) = x + ln y, uB (x, y) = xy, x̄A = x̄B = ȳA = ȳB = 2.
6
Solution:
• We first find A’s demand, which is the solution to max uA (x, y) s.t. px x + py y =
IA . That is we solve max x + ln y s.t. px x + py y = 2px + 2py .
Normalize px = 1.
The budget constraint is pxx + py y = 2px + 2py ⇐⇒ x + py y = 2 + 2py ⇐⇒
py y = 2 + 2py − x ⇐⇒ y = (2 + 2py − x) /py .
Substituting into the maximization problem we get max x+ln y s.t.y = (2 + 2py − x) /py ,
or max x + ln ((2 + 2py − x) /py ).
1
Dx (x + ln ((2 + 2py − x) /py )) = 1 − 2py −x+2
= 0; the solution is: xD
A = 2py + 1.
¢
¡
D
D
So yA = IA − px xA /py = (2 + 2py − (2py + 1)) /py = 1/py .
D
Thus at an interior solution xD
A = 2py + 1, yA = 1/py .
• B’s demand: max uB (x, y) s.t. px x+py y = IB . That is, max xy s.t. px x+py y =
2px + 2py .
Remember we set px = 1.
max xy s.t. x + py y = 2 + 2py , or max xy s.t. y = (2 + 2py − x) /py , or
max x ((2 + 2py − x) /py ).
Dx x ((2 + 2py − x) /py ) = p2y − 2 pxy + 2 = 0, the solution is: xD
B = 1 + py .
¡
¢
So yBD = IB − xD
B /py = (2 + 2py − (py + 1)) /py = 1 + 1/py . We know that
B will consume at an interior point (see argument in example 1 above).
D
So B’s demand is xD
B = py + 1, yB = 1 + 1/py .
[Shortcut: Cobb-Douglass utility ⇒ spend shares of income according to exponents’ ratio. In this case: equal exponents ⇒ equal shares ⇒ xD
B = IB / (2px ) =
D
= IB / (2py ) = (2 + 2py ) / (2py ).]
(2 + 2py ) /2 and yB
• Markets clear: xD = xS .
D
xD = xD
A + xB = 2py + 1 + 1 + py = 2 + 3py
xS = 4
2 + 3py = 4, Solution is: py =
2
3
2
D
Solution: py = 2/3, so¡ x¢D
A = 2 × 3 + 1 = 7/3, xB =
2
D
1.5, and yB = 1 + 1/ 3 = 2.5.
2
3
+ 1 = 5/3, yAD = 1/
¡2¢
3
=
• By Walras’ law we know this is an equilibrium so long as demands are nonnegative as they are. We can check by considering the market for y: y D = y S .
D
y D = yA
+ yBD = 1/py + 1 + 1/py = 1 + 2/py
7
yS = 4
1 + 2/py = 4, the solution is: py =
2
3
as above.
Example 12 uA (xA , yA ) = xA yA and uB (xB , yB ) = xB yB .
x̄B = 3.
x̄A = ȳB = 1, ȳA =
Solution: We want to solve max xA yA s.t. px xA + py yA = I = pxx̄A + py ȳA .
We can normalize and choose py = 1, and then
px
py
= px .
We then know how to show that A’s demand functions are xD
A (px ) = I/ (2px ) =
D
(px x̄A + ȳA ) / (2px) = (px + 3) / (2px ) and yA (px ) = I/2 = (px + 3) /2. Similarly we
know how to show that B’s demand functions are xD
B (px ) = (3px + 1) / (2px ) and
D
yB
(px ) = (3px + 1) /2.
The supply of x is simply what the two consumers are endowed with: x̄A + x̄B = 4,
D
and similarly for y. Solving then the market-clearing equation for y we get yAD +yB
=
(px + 3) /2 + (3px + 1) /2 = 4, or px = 1.
The equilibrium is then:
px
py
= 1; xA = yA = xB = yB = 2.
We can check that this is correct by using the market-clearing equation for x and
D
substituting in the price we found. At px = 1 we have xD
A + xB = (px + 3) / (2px )
+ (3px + 1) / (2px) = 4 which is the supply.X
2
2.1
Production
Review of producer theory
Producers maximize profits. The solution of the maximization problem (typically)
determines the optimal amount of inputs and output given the price ratios. That is,
it determines the demand functions for inputs and the supply function of output as a
function of the price ratio. This can also be used to calculate the firms profits. The
case of constant returns to scale is a little different, and is discussed separately at the
end below.
An interior solution is characterized by setting the value of the marginal product,
which is the price of the output, say px , times the marginal productivity of the input,
8
say M Pk , equal to the cost of the input. (The intuition is that M Pk px is the increased
revenue from using an additional unit of the input, and pk is the additional cost of
this input. If the revenue is greater the firm should increase production, while if the
cost is greater the firm should decrease production.) This can be rewritten as
Marginal productivity of input = price of input/price of output
where we see that only the price ratio determine the optimal decision.
We now solve this explicitly two cases: where there is only one input and where
there are two inputs.
2.1.1
Profit maximization with one input
max px x − pl l s.t. x = f (l)
Substituting for x we get max pxf (l) − pl l. Now taking the derivative with respect
to l and setting equal to zero we get
f 0 (l) = pl /px .
³ ´
pl
. Substipx
³ ´
S
tuting this amount of l in the production function gives the supply of x, x ppxl =
³ ³ ´´
f lD ppxl . Nominal profits are then π = px xS − pl lD , and real profits, measured
Solving this equation for l would give the demand function for l, lD
in units of output, are
π
px
= xS − lD ppxl . Thus the preceding equation gives us
µ ¶
µ ¶
pl
pl
π
D
S
l
,x
, and
px
px
px
We can draw these in a figure. The horizontal axis is l, the vertical is x. The
firm maximizes profits by setting f 0 = ppxl . The curve in the figure is f , the straight
lines have slope pl /px , and the point on f with this slope is the small square in the
figure. The height there is the supply xS , the horizontal distance is the demand for
labor, lD . The line through the origin with slope ppxl is used to draw the real cost of
labor: lD ppxl . The difference between the total produced (which is the real revenue
[measured in units of x]) and the real labor cost, is the real profits. Thus, we can see
how the demand for input, supply of output, and real profits, all change as the price
ratio changes.
All the straight lines are lines of constant real profits: π = px x − pl l, so x =
π/px + lpx/pl . For different values of π these are different lines of constant profit.
9
The highest feasible line is the one that maximizes profits; this is the line that is tangent to f at the small square. The intercept of each line is the real profit generated
along that line. One can see that the difference between the line through the origin
and the line tangent to f is the real profits and is the intercept of the tangent line.
6
5
4
3
2
1
0
5
10
x
15
20
Example 13 (One Input)
f (k) = ln (k + 1) .
We want to solve maxk ln (k + 1) px − kpk . The optimal solution satisfies M Pk =
1
= ppxk , so
k+1
px
kD =
− 1.
pk
Note: This is an example of a point at which you need to be careful. Inputs will
not be negative, so what does the producer do if ppxk ≤ 1? The producer will set in this
case x = k = 0. Otherwise,
µ ¶
px
S
x = ln
.
pk
³ ´
³
´
Real profits in units of x are 0 if ppxk ≤ 1, and they are ln ppxk − ppkx ppxk − 1 otherwise.
Example 14 (One Input)
f (k) =
√
k.
We want to solve maxk px k 0.5 − pk k. The optimal solution satisfies M Pk = 2√1 k = ppxk .
In this case then
s µ ¶
µ ¶2
µ ¶2
2
1 px
1 px
1 px
π
1 px 1 px
pk
1 px
D
S
k =
, x =
=
, and
=
−
=
.
4 pk
4 pk
2 pk
px
2 pk 4 pk
px
4 pk
10
2.1.2
Profit maximization with two inputs
max pxx − pk k − pl l s.t. x = f (k, l)
Substituting for x in the objective function yields max px f (k, l) − pk k − pl l. Taking
derivatives with respect to the inputs k and l gives MPk = ppkx and M Pl = ppxl . Solving
these two ³
equations
with
´
³ k´and l as unknowns we get the demand functions for the
D pk
D pl
inputs k
and l
. These can be substituted into the production function
px
px
to get the supply function xS , and, as before, we can calculate profits (nominal and
real).
Example 15
f (k, l) = k 1/3 l1/3 .
M Pk = 13 l1/3 /k 2/3 = pk /px and M Pl = 13 k 1/3 /l2/3 = pl /px . Dividing one equation by
the other we get M RT S = M Pk /M Pl = l/k = pk /pl . So l = kpk /pl . Substituting
1/3
k /pl )
this into the equations gives (kp3k
= ppxk , and the solution is
2/3
kD =
S
This gives x =
³
1 p3x
27p2k pl
2
1 p3x
1 p3x
D
.
Similarly
we
find
l
=
.
27p2k pl
27p2l pk
´1/3 ³
1 p3x
27p2l pk
3
´1/3
3
1 px
1 px
pk k D − pl lD = px 9ppkxpl − pk 27p
2 p − pl 27p2 p
l
k
k
p2x
. Finally,
9pk pl
1 p3x
= 27
.
pk pl
=
l
profits are π = px xS −
Remark 16 Note that everything we just calculated, except profits,³can´be
in
³ written
´2
px
px
1
D
D
terms of price ratios rather than absolute prices. That is k = 27 pl
,l =
pk
³ ´ ³ ´2
³ ´³ ´
px
px
px
1
. Thus multiplying all prices by a constant does
, xS = 19 ppxk
27 pk
pl
pl
1 px px
not effect the firms decision. Of course it does effect nominal profits π = 27
p .
pk pl x
The last element in this last equation is not a price ratio but an absolute price level.
1 px px
However, real profits depend only on price ratios: pπx = 27
is the profits measured
pk pl
in terms of how many units of x it can buy, and depends only on price levels.
2.1.3
Constant Returns to Scale
If the production function has constant returns to scale then profits are zero and
either the level of production is zero or the level of production is indeterminate.
That is, either the firm does not want to produce or the firm is indifferent between all
11
levels of production. (Of course it is possible that the firm will want to produce an
infinite amount, but we do not discuss such a case as obviously it cannot happen in
equilibrium.) The price ratios will determine the optimal relative use of the inputs,
but not the total. This is because if MPk px = pk and MPl px = pl at some levels of
inputs k, l, then the equality holds when the inputs are both multiplied by the same
constant.
Example 17
f (k) = 3k.
Then if ppkx > 3 the firm will not want to produce at all. If ppxk < 3 then the firm will
want to produce an infinite amount, so this can never occur in equilibrium. If ppxk = 3
then the firm is willing to produce any amount.
Example 18
f (k, l) = k 1/2 l1/2 .
Then M Pk = 12 l1/2 /k 1/2 = pk /px and M Pl = 12 k 1/2 /l1/2 = pl /px . Dividing one
equation by the other we get M RT S = MPk /M Pl = l/k = pk /pl . So l = kpk /pl .
Thus the relative use of the inputs is determined. But the total demand and supply
is not: substituting this into the MP equations does not give a solution to the total
quantities. Substituting l = k ppkl into profits we get
0.5 0.5
π = pxk l
− kpk − lpl = px k
r
pk
− 2kpk .
pl
q
√
Therefore, π > 0 ⇐⇒ px ppkl − 2pk > 0 ⇐⇒ px > 2 pk pl . In this case the
firm will want to produce an infinite amount, which cannot happen in equilibrium. If
√
√
px < 2 pk pl then the firm will not produce at all. If px = 2 pk pl then the firm will
be willing to produce any amount.
2.2
2.2.1
Calculating CE
One producer, one consumer, and one input which is also consumed.
Labor is denoted by l, x = f (l), ¯l is the amount of l the agent has, u (x, ) is the
agent’s utility function, where l + = ¯l. We denote leisure by , so the demand for
leisure is D , and the net supply of labor is lS = ¯l − D .
12
0
The firm ³solves
´ max px f (l) − pl l, so we get f = pl /px . Denote the firm’s real
profits by pπx ppxl .
The consumer solves max u (x, ) s.t. px x + pl = pl ¯l + π, or x +
pl
px
=
pl ¯
l + pπx .
px
Note that the consumer’s income includes the profits.
Therefore, in equilibrium M RS = ppxl , and MRT = f 0 = ppxl , so M RS = M RT ,
that is, the equilibrium is Pareto efficient–another instance of the first welfare theorem.
We can draw this as follows. As usual the budget constraint of the consumer is
given by a downward sloping line with (absolute value of) the slope equal to pl /px ,
and (at an interior solution) the consumer chooses a point on that budget constraint
which is tangent to the indifference curve. The height of the budget constraint is
given by the profits received by the consumer: π/px .
Now copy the figure for firm drawn above but draw it in "reverse", with leisure
increasing to the right and hence labor, the input, increasing to the left. That is
given in black in the figure. Next add the budget constraint: a line with slope −pl /px
shifted up by the real profits. It is indicated by the dotted red line (which coincides
with the constant-profit line of the firm). The consumer chooses the best point on
this line. If that point is the square, then we have a CE; if not then we are not at a
CE and either there is excess supply of labor and excess demand of x (if the consumer
chooses to the left of the square) or the opposite.
12
10
8
6
4
2
0
2
4
6
8
10
12
x
14
16
18
20
22
24
Remark 19 The firm and consumer solve their choices separately, even though the
firm is owned by the consumer. The firm maximizes profits given prices; the consumer
maximizes utility given prices and income (taking profits of the firm as given).
13
Example 20
u (x, ) = x , ¯l = 24, f (l) = l1/3 .
Normalize prices by setting px = 1.
Profit maximizing implies f = pl , so l
¡ ¢1/3
xS = lD
³
³
´3/2
/3 = pl , so l =
= 3√1 3 √1 3 , and
pl
³ ´1/2
³ ´3/2 ³
´
1 √1
1
1
1
1
S
D
√
√
√
= 3 pl , and π = x − pl l = 3pl
− pl 3pl
=
− 27 √1pl .
3
0
Utility maximization implies xD =
´
√1 − √1
× √1pl .
3
27
D
−2/3
I
,
2
D
=
1
3pl
where I = ¯lpl + π = 24pl +
I
2pl
³
³
Competitive equilibrium is therefore the solution to x = x , or 24pl + √13 −
¢2 √
¡√
3
1
4 3 3.
× 12 = √13 √1pl , and the solution is pl = 36
D
We can check this using the labor market: lD +
³
1
√ √
pl 3
−
1
√ √
pl 27
³
√1
3
√1
27
+
−
´´
´
q
÷ (2pl ) = 24. Substituting we get
1
√
2√
3
1
4) 3 3
(
36
¶
D
S
= ¯l, or
µ
³
1
3pl
´3/2
1
³
√ 2√ ´
1
3 36
( 3 4) 3 3
³ ³ ¡ √ ¢2 √ ´´
3
1
÷ 2 36
4 33
= 24.0. X
√1
27
¶3/2
and ¯l is the amount of l available to the consumer. Normalize px = 1. The firm’s
behavior we found above. The consumer’s behavior is as follows. If pl < 1 then the
consumer will consume only leisure, D = ¯l, xD = 0. If pl > 1 then the consumer
will consume only x, xD = ¯lpl , D = 0. If pl = 1 then the consumer is indifferent
among all (non-negative) allocations of and x that satisfy his budget constraint.
Clearly if px > 0 and pl < ∞ then the firm will demand a positive amount of l.
On the other hand, if pl < 1 the consumer will not supply any labor. Therefore pl < 1
cannot be pat of any equilibrium.
14
´
³ ³ ¡ √ ¢2 √ ´
3
1
+ 24 36
4 33
u (x, ) = x + , f (l) = l1/3 ,
• First assume that ¯l = 0.1.
√1
pl
+ (24pl +
Example 21
Now we will consider two possibilities.
´
We just saw that pl < 1 cannot be the equilibrium price.
What about pl = 1? Then lD = 3√1 3 . However, since ¯l = 0.1 < lD , even if the
consumer consumes no leisure, he will not supply enough labor to satisfy the
firm’s demand for labor.
So what about pl > 1?
D
=
1
√
3 3
√1 3 = ¯l, so
pl
Then lD =
1 √1
√
3 3
p3
1
√
3 3
√1 3 , and
pl
D
= 0, so we have lD +
= 0.1, and pl = 1. 547 2.
Markets clear with
l
lD = ¯l = 0.1, xD = xS = 0.11/3 . The consumer will consume as much x as he
can with the income that is generated by his labor and the profits.
(The x market can be used as a check.)
• Second, assume that ¯l = 24.
Now let’s try again pl = 1. Then lD = 3√1 3 < 24. So we can solve lD + D = 24
by setting pl = 1, lD = 3√1 3 , D = 24 − 3√1 3 . Then the x market must clear with
xD = xS = √13 .
³
´
1
1
1
D
D
S
√
√
√
− 27 +
We can check by verifying that x = 3 . In fact, x = π+l pl =
3
´
³
√
24 − 24 − 3√1 3 = 13 3. X
Example 22 u (x, y) = xy, y = f (x) = ln (1 + x), the consumer owns 10 units of x
• Consumer demand:
max u (x, y) s.t. px x + py y = I; max xy s.t. px x + py y = π + 10px
Normalize px = 1.
max xy s.t. x + py y = I = π + 10, or max xy s.t. (π + 10 − x) /py = y, or
max x (π + 10 − x) /py
1
Dx (x (π + 10 − x) /py ) = p1y (π − x + 10) − pxy = 0, Solution is: xD
c = 2π + 5
¡
¡
¡1
¢¢
¡1
¢
¢
=
π
+
10
−
π
+
5
/p
=
π
+
5
/py
ycD = I − xD
/p
y
y
c
2
2
¡
¢
1
1
D
xD
c = 2 π + 5, yc = 2 π + 5 /py
• Firm demand
max π s.t. y = f (x)
max py y − px x s.t. y = ln (1 + x)
max py ln (1 + x) − x
Dx (py ln (1 + x) − x) =
py
x+1
− 1 = 0, Solution is: xD
f = py − 1
15
¡
¢
yfS = ln 1 + xD
= ln (py )
f
π = py y − x = py y − (py − 1) = py ln py − py + 1
S
xD
f = py − 1, yf = ln (py ) , π = py ln py − py + 1
• Markets clear: xD = xS
D
xD = xD
c + xf = π/2 + 5 + py − 1 = (py ln py − py + 1) /2 + 5 + py − 1
xS = 10
(py ln py − py + 1) /2+5+py −1 = 10, or (py ln py − py + 1) /2+5+py −1−10 =
1
p + 12 py ln py − 11
=0
2 y
2
1
• Solution: py = 4.4, so π =
4.4 ln (4.4)
− 4.4 + 1 = 3.2, xD
c = 2 3.2 + 5 = 6.6,
¡
¢
1
D
S
xD
f = 4.4 − 1 = 3.4, yc = 2 3.2 + 5 /4.4 = 1.5, yf = ln (4.4) = 1.5.
By Walras’ law we know this is an equilibrium so long as demands are nonnegative. We can check by considering the market for y.
2.2.2
Two outputs with one firm producing each; and one input owned
by two consumers.
x = f (kx ) , y = g (ky ) .
Profit maximization will lead the firm producing x to satisfy M Pkx = pk /px . The
firm producing y will satisfy M Pky = pk /py . Therefore, profit maximization will lead
MP y
to MPkx = ppxy . We know that consumer will satisfy M RS = ppxy , and we remember that
MPky
MPkx
k
= M RT , so again we see that MRS = MRT holds at a competitive equilibrium.
Example 23
x=
p
kx , y = ky .
Note: y has constant returns to scale. Normalize pk = 1.
We saw above that profit maximization by the firm producing x implies kxD = 14 p2x ,
x = p2x , and πpxx = p4x .
S
Since y has CRTS with y = ky we know that if py > 1 then the firm demands
an infinite amount of k so there cannot be an equilibrium. If py < 1 then the firm
will produce 0 units of y. If py = 1 then the firm is indifferent among all levels of
production. Either way its profits are zero.
The above implies that if in equilibrium there is a positive amount of y then py = 1
and kyD = k̄ − kxD .
16
Assume that there are two consumers each with half a unit of k and each of whom
2
owns half of each of the firms and with utility functions uA (xA , yA ) = xA yA
, and
2
uB (xB , yB ) = xB yB .
We know how to solve for consumer demand: xD
A =
D
yB
=
IA
,
3px
A
B
I
I
yAD = 2 3p
, xD
B = 2 3px ,
y
IB
.
3py
Incomes are calculated as follows. I A = k̄A pk + π x /2 + π y /2 = 1/2 + 18 p2x . I B is
the same.
Equilibrium then requires that we solve for px , py , kx , ky , xA , xB , yA , yB using the
D
equations we have derived plus two of the three market clearing equations: xD
A + xB =
D
D
xS , yA
+ yB
= y S , and kxD + kyD = k̄.
The market-clearing equation for x is standard:
√
solution of which gives the price for px = 2/ 3.
1/2+p2x /8
3px
2
x /8
+ 2 1/2+p
=
3px
px
,
2
the
The other two market-clearing equations are a little different since y is CRTS. We
know that if y is positive in equilibrium then py = 1, so we know that the demand for
2 /8
2 /8
A
IB
x
x
y is yAD +yBD = 2I
+ 3p
= 2 1/2+p
+ 1/2+p
. Thus the supply of y is determined by
3py
3
3
y
the demand. Substituting the price we found for x, we get yAD = 4/9 and yBD = 2/9,
so y S = 6/9 = 2/3, and therefore kyD = 2/3 (the firm will demand as much k as it
needs to produce the y = 2/3).
The third market-clearing equation can be used to check that we have not made a
2
mistake. kxD = 14 p2x , k̄ = 1, so kyD = 1 − p4x = 2/3. X
17