Download An example of competitive equilibrium in production economy: U1

An example of competitive equilibrium in production economy:
√
√
U1 (·) = x1 + 4 y1 and U2 (·) = x2 + 2 y2 with endowment vector (4, 12), (8, 8)
1. Find a competitive equilibrium of the above pure exchange economy.
Lets extend it to a production economy. Suppose a firm exists which can produce (only)
good y using good x as input with production function y = αx. Each agent has equal share
in the profits.
2 Compute a competitive equilibrium (if any exist) which has no (extra) production of good
y i.e. availability of good y remains equal to the endowment.
3 Compute a competitive equilibrium (if any exist) which has positive production of good y
i.e. availability of y exceeds endowment and extra availability is produced in equilibrium.
Note: Throughout the answer I will assume:
Px
.
Py
2. CE as competitive equilibrium, CEA as competitive equilibrium allocation and CEP as
Competitive equilibrium price.
3. I have tried to make the answer very explanatory. With this I also wanted to give a
brief overview. So those students who are very confident about the concepts can leave the
explanatory note in the answer.
1. Px = p and Py = 1. The other interpretation of ‘p’ is that it is relative price, p =
Answer:
Case1: Production of y is not possible.
For any value of p both agents will equate their M RS equal to p (price ratio) to compute
their (optimal) demand bundles (why?).
So we have M RS1 = p =√ M RS2 in equilibrium, which leads to :
y1 √
M RS1 = M RS2 =⇒
= y2 =⇒ y1 = 4y2 .
2
A small digression: Some of you might want to conclude that equating MRSs i.e. M RS1 =
M RS2 gives the condition or characterize (all) the Pareto efficient points. It will be a good
exercise to check whether the equation y1 = 4y2 gives Pareto efficient points or not (why)?
Does it give ALL the
or some still left? Hint: What about allocations
Pareto efficient points
like (0, 0), (20, 12) & (20, 12), (0, 0)
An equilibrium must satisfy the equation “demand equals to supply” i.e. if y1 and y2 are
demanded quantities then it should add up to the availability i.e. y1 +y2 = 20. So along with
the equation y1 = 4y2 it gives y1 = 16 & y2 = 4. So what have we achieved so far? We have
computed the values of y1 & y2 which could be (part of) an equilibrium allocation. [Note:
Good x has not played any role yet. Why? Has it got to do with the type of preferences we are
dealing?] Now, for y1 = 16 & y2 = 4 to be a CEA we should set the CEP p in such a way that
Px
=⇒ M RS = 2 = p.
they become (optimal) demand for that p i.e. M RS1 (y1 = 16) =
Py
(Why only for agent 1?)
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Explanatory note: Are we done or something is still left?. If you think yes, because we
have cleared the market of good y and by Walras law it should also clear market for good
x. So it should complete the answer. This reasoning is not (completely) correct here. We
have one very plausible reason that why this reasoning may fail. If you have observed we
have not used individual endowments yet. So if our answer is complete then it implies that
fixing CEP p = 2 achieves CE irrespective of individual endowments. But if endowment
vector is like (0, 0), (20, 12) then income of agent 1 is 0 and he can’t afford to buy y1 = 16.
This seemingly paradoxical situation has arisen since, we have shown that the above level
of y1 = 16 and y2 = 4 are the actually demanded quantities. For this we have to find out
the actually demanded quantities subject to budget constraints,
i.e., whether the agents can
afford the above bundle given their (endowment) income.
8p + 4
Mi − Py .y
4p − 4
& x2 =
(why? xi =
where
The demand of good x will be x1 =
p
p
Px
Mi = p.ei ).
So for things to be in equilibrium, the (optimal)
demanded bundles xi (p) ≥ 0 ∀ i which
gives the condition that p ≥ 1(why?). Because if for any agent i we have xi < 0 then
it shows that
agent i doesn’t have sufficient income (based on the endowment) to buy the
required yi . By this we have checked that if p = 2 then given the endowments, agents have
sufficient income to buy the equilibrium distribution of good y. Alternatively one gets the
same condition by equating M1 ≥ 16 (why?).
So Competitive equilibrium : CEP : p = 2 and CEA : (2, 16), (10, 4)
Explanatory note: Is this way of computing CE is different from other previous questions,
eg. when both agents have Cobb-Douglas? Or why in this question we have to bother about
checking for demand side of good x when we have already cleared the market of good y? If
you have noticed, in some of the previous questions you have done, you calculate demand
of any one good (say x) of both agents and then add it to get aggregate demand and then
equate it to total supply(endowment) i.e. excess demand equals to zero, zx (p) = 0, and then
equilibrium price will emerge as the solution of that equation. So first we solve for CEP
and then we calculate CEA. But in this question first we (partially) calculated the CEA
y1 = 16 & y2 = 4. Then we compute the price which can sustain this allocation. Since
this price is not emerged as a solution of the equation which equates demand (based on
income from the endowment) equal to the endowment, therefore we can’t take this to be
an equilibrium
without checking for the condition on the other good or on the endowment
income.
Case2 : Production of y is possible. But no production at equilibrium.
Some of you might be tempted to answer this part by saying “same as part 1 ”. This answer
is only partially correct (that too from calculation perspective). Part 1 & 2 are fundamentally different questions. The statement “I cannot produce” is different from the statement
“I can produce but not producing is optimal ”. In part 1 there is no possibility of production
at all but in part 2 production is possible but we want that in equilibrium there is zero
production. If there is no production that will mean that supply will be identical to the
endowment. So the consumption side remain as part 1 because the availability(supply) of
2
goods remain as the endowment. What about the production side?
Production decision is made by the firm. So the equilibrium conditions should be such that
firm doesn’t find it optimal to produce any quantity of y. When production function is CRS
type then the condition for zero production will be p ≤ M C (why ‘=’ ?, if you can’t tell this,
2
=⇒ α ≤ 2.
then it is good time to revise the production theory first). So we have 1 ≤
α
So Competitive equilibrium : CEP : p = 2 and CEA : (2, 16), (10, 4) along with the condition α ≤ 2
Case3 : Positive production in equilibrium.
With the production technology of CRS type a positive (and finite) production is possible
if p = M C and firm earns a zero profit. So p = M C =⇒ p = α.
Let µ be the (extra) units of good y produced in the equilibrium (over and above 20), which
makes availability of good y = 20 + µ.
.
Using the condition y1 = 4y2 (derived in part 1) we get y2 = 20+µ
5
√
µ
To compute the CEP, we use the condition M RS = p =⇒ p = 4 + ε where ε = .
5
Good x used in production of y : µ = αx =⇒ x = αµ
12p − 5ε
µ
=
.
Good x left for (direct) consumption: 12 −
α
p
4p − 4 − 4ε
8p + 4 − ε
Demand side of x: x1 =
& x2 =
p
p
Again for p to be an equilibrium we must have xi ≥ 0 ∀i. From this we get x1 ≥ 0 =⇒ p ≥
ε−4
. So we need the first inequality to hold in equilibrium (why
1 + ε and x2 ≥ 0 =⇒ p ≥
8 √
not second?), which gives p = 4 + ε ≥ 1 + ε. If you solve this inequality and you will
get ε ∈ [−2.3, 1.3] (these numbers are approximated). Since production can’t be negative.
So as long as extra production is maximum upto 5ε with ε ∈ [0, 1.3] this characterize the
equilibrium.
This completes the answer or do we have to check one more constraint(?). Do we need to
check that input demand of x (used in producing y) should remain below 12 units?.
Not really because of Walras law (why?). If some (cautious) students want to check for it
5ε
5ε
≤ 12 =⇒ √
≤ 12 =⇒ ε ∈ [−2.7, 8.5] which is already satisfied by
they we will get
p
4+ ε
the ε in the previous the interval.
√
So Competitive equilibrium
√ : CEP : p = 4 + ε. Production of y = 5ε where ε ∈ (0, 1.3]
with the condition α = 4 + ε and CEA mentioned above.
Extension of Ques4: Problem set-2.
4 (d) : Does the consequent part of second welfare theorem hold for this economy i.e. every
Pareto efficient allocation can be achieved as competitive equilibrium with suitable transfers.
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Answer :
Yes, we can achieve any point on the line segment OA by fixing CEP p = 1 because agent 1
calculates his optimal demand by equating M RS to the price ratio and every point on OA
has x1 = y1 which makes M RS = 1. What about Pareto efficient points on the line segment
AB. One can achieve even these points as CEA by suitable price ratio. So for any allocation
z ∈ AB one should calculate M RS1 and fix p = M RS1 . Eg. take z = (5.5, 5), (0.5, 0) at
10
= p. One can check that because p < 1 agent 2 will consume only
this point M RS1 = 11
good x. Are we done?
No, we have just provided the condition (price ratio) which is necessary for a particular
Pareto efficient allocation to be a CEA. Now we need to ensure that agents will indeed consume that particular allocation by providing appropriate transfers
(sufficiency of income).
So take any Pareto efficient allocation z = (x̂1 , ŷ1 ), (x̂2 , ŷ2 ) ∈ OAB. One can achieve it as
CEA by fixing CEP, p = xy11 with appropriate transfers which are Ti = (p.x̂i + ŷi ) − (p.eix + eiy )
for i = 1, 2. One can check that with this price ratio and the transfers each agent i will
demand (x̂i , ŷi ).
The first expression in Ti is the exact income required to afford (x̂i , ŷi ) and second expression is the actual income of agent i given his endowment. So the difference is the transfer
required by agent i (if positive
then it is subsidy, tax otherwise). So if we take the example
48
& T2 = −48
[To start with income of agent 1 is less
of z = (5.5, 5), (0.5, 0) then T1 = 11
11
than what is required to buy the bundle (5.5, 5), so he gets the difference as subsidy and the
opposite holds for agent 2 so he will be taxed].
One can even make further argument and can characterize the structure of transfers. We
know that from the initial endowment without
equilibrium
any transfers, the competitive
exists with CEP = 1 & CEA = (3,3),(3,2) point R in the diagram above (you are asked
to compute this in first part of the question). So to achieve any allocation on OAB the
structure of transfers given the endowment in the question, will be as follows :

at any point below R
 <0
=0
at point R
T1 =

>0
at any point above R
And T2 = −T1 (why?). The quantum of Ti ’s as mentioned above.
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