www.inkpotonline.com
Facebook - @inkpotonline
Call Us - +91 9911680354
Unit-4 Exchange
Now we look at 2 consumer market, instead of one.
Assumptions:1. 2 agents that is they are consumers as well as producers & they want to
maximize profits & utility that is we have perfect competition.
2. Goods only.
3. Endowment is fixed & given. It is a case of pure exchange.
Edge worth Box
Graphical tool to analyze 2 consumer’s exchange problem simultaneously.
𝑐 𝑛 bundle (𝑥𝐴1 ,𝑥𝐴2 ) & 𝑋𝐵 =(𝑥𝐵1 ,𝑥𝐵2 )
A pair of bundles 𝑥𝐴 & 𝑥𝐵 is called allocation & it is a feasible allocation if total
amount of each good consumed is equal to total amount of each available.
𝑥𝐴1 +𝑥𝐵1 =𝑤𝐴1 +𝑤𝐵1
for good & similarly 2.
𝑤𝐴1 ,𝑤𝐴2 ,𝑤𝐵1 & 𝑤𝐵2 are initial endowments allocation which agent 1 & 2 trade to get
1
final allocation.
The x axis & line parallel to x represents good 1.
1
B
2
Initial allocation w.
At all points above that is for
m
A makes A better off & similarly
𝑤𝐴1
for B. Region M is where both
w
better of ∴ by trading they
2
will consume in that area.
A
1 1
𝑤𝐴
Both hare to be better off to trade with each other. Else there won’t be any
trade. To consume at any point in m. Each would have to sell one good & buy the
other.
B
2
m
w
A
M is pareto efficient. That is from
M, if one person has to be
Made better off the other has
To necessarily made worse off.
There is no exchange which is
Advantageous to both parties.
This is a pareto efficient allocation.
1
So, a pareto efficient allocation is one where:1. Where there is no way to make all people better off.
© 2016 Ink Pot Online Pvt. Ltd. Confidential, for limited circulation only.
www.inkpotonline.com
Facebook - @inkpotonline
Call Us - +91 9911680354
2. One indr. Cannot be made better off w/o making someone else worse off.
3. All gains from trade have been exhausted.
1 areas left.
4. There are no mutual advantageous trade
For this the ICS of 2 agents have to be tangent to each other. From the
tangency condition. We know that pareto efficient point is wee all gains have
been exhaust. The line containing all pareto efficient points is called contract
curve/pareto set it does not depend on initial allocation.
Market Trade
Till non, there were no prices but we need a set of prices 𝑃1 𝑃2 where demand of
a good=supplier of a good. Gross demand- a consumer wants of consume (𝑥1𝐴 𝑥1𝐵
𝑥2𝐴 𝑥2𝐴 ). Net demand- Gross demand- endowment (denoted by e)
∴ 𝑒𝐴1 =𝑥𝐴1 -𝑤𝐴1 (also called excess dd)
𝑥𝐴2
𝑥𝐴2
Excess
dd
𝑥𝐵2
At any price 𝑒𝐴1 ≠ 𝑒𝐵1 ,
like in this diagram,
there is excess demand of
good 2 & excess supplier of
good 1 & ∴ disequilibrium
now prices would change
& we will reach to equilibrium.
𝑥𝐴1
𝑥𝐵2
Excess SS
B
2
equilibrium
allocation
A
1
This is called market equalize
& competitor equalize &
walrasian equilibrium where the
net or excess demand of good
1 = 𝑛 supplier of good 1 &
similarly for good 2 at
a given set of prices.
If all consumes are
Choosing best bundle possible then,
MRSA=MRSB = -𝑃1⁄𝑃2
A+ prices (𝑃1 *,𝑃2 *)- competitive, Gross demand
for A&B=𝑥𝐴1 ,𝑥𝐴2 & 𝑥𝐵1 ,𝑥𝐵2 .
∴
𝑥𝐴1 + 𝑥𝐵1 = 𝑤𝐴1 + 𝑤𝐵1
𝑎𝑙𝑠𝑜 𝑥𝐴2 + 𝑥𝐵2 = 𝑤𝐴2 + 𝑤𝐵2
𝑇𝑜𝑡𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 = 𝑡𝑜𝑡𝑎𝑙 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟.
∴ [𝑥𝐴1 − 𝑤𝐴1 ] + [𝑥𝐵1 − 𝑤𝐵1 ] = 0
[𝑥𝐴2 − 𝑤𝐴2 ] + [𝑥𝐵2 − 𝑤𝐵2 ] = 0
&
(at prices 𝑃1 *&𝑃2 *)
(at prices 𝑃1 *&𝑃2 *)
© 2016 Ink Pot Online Pvt. Ltd. Confidential, for limited circulation only.
www.inkpotonline.com
Facebook - @inkpotonline
Call Us - +91 9911680354
Net demand of a good should be zero or the amount:A wants to but = B wants to sell.
𝑒𝐴1 + 𝑒𝐵1 = 0 & 𝑒𝐴2 + 𝑒𝐵2 = 0
Aggregate excess demand for good 1 = z1
𝑧1 = 𝑒𝐴1 + 𝑒𝐵1
&
𝑥2 = 𝑒𝐴2 + 𝑒𝐵2
Also at equilibruim prices.
𝑧1 (𝑃1∗ 𝑃2∗ ) = 0
&
𝑧2 (𝑃1∗ 𝑃2∗ ) = 0
We prove this through walrus law.
Walrus law
𝑃1 𝑧1 + 𝑃2 𝑧2 = 0
Walrus law status that the value of aggregate. Demand is identically here, that
is zero for all possible prices & not just equalize price.
Proof: demand because it satisfies the budget constraint.
𝑃1 𝑥𝐴2 + 𝑃2 𝑥𝐴2 = 𝑃1 𝑤𝐴1 + 𝑃2 𝑤𝐴2
Or
𝑃1 [𝑥𝐴1 − 𝑤𝐴1 ] + 𝑃2 [𝑥𝐴2 − 𝑤𝐴2 ] = 0
𝑃1 𝑒𝐴1 + 𝑃2 𝑒𝐴2 = 0 -1) {at any price𝑃1 𝑃2 }
Value of agent’s net demand must be equal to zero at any price (𝑃1 𝑃2 )
Similarly for B
𝑃1 𝑒𝐵1 + 𝑃2 𝑒𝐵2 = 0 -2)
Adding 1+2
𝑃1 (𝑒𝐴1 + 𝑒𝐵1 ) + 𝑃2 (𝑒𝐴2 + 𝑒𝐵2 ) = 0
Or
𝑃1 𝑧1 + 𝑃2 𝑧2 = 0
{for any price}
And, if demand=supplier in one market that is if 𝑧1 = 0
Then 𝑧2 is necessarily equal to zero. {only for eqm prices}
So, if there are K markets them we need to find prices till K-1 market because
if -1 markets all in equilibrium them with K market would also be in equilibrium.
Relative prices
Walrus law implies that there are only K-1 independent markets & therefore
only K-1 equilibrium & therefore if prices (𝑃1∗ 𝑃2∗ ) are multiplied by plud yhrm ig
t>0 then all prices change in same manner.
so if 𝑡 = 1⁄𝑃1 therefore we only need to find K-1 prices.
Existence of equilibrium (aggregate excess demand & function is a continuous
function.
Every equilibrium may not have the existence of a competitive equilibrium. But
the additional equilibrium of a continuous function can be used to establish the
© 2016 Ink Pot Online Pvt. Ltd. Confidential, for limited circulation only.
www.inkpotonline.com
Facebook - @inkpotonline
Call Us - +91 9911680354
existence of equilibrium that is small changes in price should had to small
changes in demand.
Equilibrium & efficiency (A competitive equilibrium is always pareto &
efficient)
Any allocate on is pareto efficient if A bundles don’t induct with B’s bundles.
But at market, A’s pref. should be tangent to his price ratio & similarly with B &
therefore the 2 won’t induces therefore,
there is no allocation either agent would prefer more than equilibrium bundle,
therefore it is pareto efficient.
𝑥 − 𝑒𝑞𝑚
𝑦 − 𝑝. 𝑒.
Proof: (𝑦𝐴1 𝑦𝐴2 ) (𝑦𝐵1 𝑦𝐵2 ) allocation which is a competitive equilibrium but not
pareto efficient
We know 𝑦𝐴1 + 𝑦𝐵1 = 𝑤𝐴1 + 𝑤𝐵1
-1)
2
2
2
2
Also
𝑦𝐴 + 𝑦𝐵 = 𝑤𝐴 + 𝑤𝐵
-2)
1
2
1
2
(𝑦𝐴 𝑦𝐴 ) > (𝑥𝐴 𝑥𝐴 )
&
(𝑦𝐵1 𝑦𝐵2 ) > (𝑥𝐵1 𝑥𝐵2 )
]because y > x
We know market equilibrium means an affordable allocation but y is better than
x, therefore it should cost more than x, therefore,
𝑃1 𝑦𝐴1 + 𝑃2 𝑦𝐴2 > 𝑃1 𝑤𝐴1 + 𝑃2 𝑤𝐴2 -3)
&
𝑃1 𝑦𝐵1 + 𝑃2 𝑦𝐵2 > 𝑃1 𝑤𝐵1 + 𝑃2 𝑤𝐵2 -4)
Adding 3 & 4
𝑃1 (𝑦𝐴1 + 𝑦𝐵1 ) + 𝑃2 (𝑦𝐴2 + 𝑦𝐵2 ) > 𝑃1 (𝑤𝐴1 + 𝑤𝐵1 ) + (𝑤𝐴2 + 𝑤𝐵2 )
Comparing with 1 & 2
𝑃2 (𝑤𝐴1 + 𝑤𝐵1 ) + 𝑃2 (𝑤𝐴2 + 𝑤𝐵2 ) > 𝑃1 (𝑤𝐴1 + 𝑤𝐵1 ) + 𝑃2 (𝑤𝐴2 + 𝑤𝐵2 )
∴ another condition is that agents should hare convex preferences then there
will always be a set of prices that each pareto efficient is a market equilibrium
This is not possible & therefore this assumption must be wrong. This is the
first theorem of welfare economics. A competitive equilibrium will necessarily
be pareto efficient. It does tell us who gets the gain, just that all gains are
exhausted.
Monopoly
If one consumer tries to act as a monopolist that is A. A knows B’s demand
curve & therefore will choose a set of prices that makes A as well off as B given
B’s demand at each price. This is given by B’s price offer curve. If we draw a
budget line for B, then the line where it intersects B’s offer curve represents
B’s optimal 𝐶 𝑛 .
© 2016 Ink Pot Online Pvt. Ltd. Confidential, for limited circulation only.
www.inkpotonline.com
Facebook - @inkpotonline
Call Us - +91 9911680354
So, A has chosen that point on B’s
price offer curve which offers him
highest utility possible. So A’s that is
tangent to B’s offer curve & ∴ x
+
A’s
ic
B’s
Price offer curve
Through x a budget line is drawn & we see
that it passes through A’s that is also B’s
that is which is tangent to price line shows
B’s
price line shows an area of mutual trade.
offer
W
B’s ie
curve
∴ x is not pareto efficient. A would like to sell
more at equilibrium price, ∴ price of good 1 has to be lowered.
A’s
ic
But a perfectly discriminating monopolist sit is pareto efficient because in that
case, each unit is sold at a different price.
1
B
2
A’s ic
W
B’s ic
His that is remains the one with
endowment as he is offered his
max willing as price ∴ his welfare
is constant charged a price for
each unit at which he is indifferent
between buying & not buying & not
buying.
1
A
Efficiency & equilibrium
-2nd welfare theorem
Given pareto efficiency, we can find a set of prices which makes it a competitive
equilibrium.
B
A pareto efficient allocation is where
A’s that is tangent to B’s that is K if we
draw a straight line through the
Tangency, we get equilibrium prices which
gives us efficient market.
W
A’s ic
W
B’s ic
A
B
But this is not always possible.
X is pareto optimal but a budget
A’s ic
x
© 2016 Ink Pot Online Pvt. Ltd. Confidential, for limited circulation only.
B’s ic
y
A
www.inkpotonline.com
Facebook - @inkpotonline
Call Us - +91 9911680354
line can’t give us equilibrium & ∴ there
are no prices at which A & B
will want to consume at x.
Implications of first welfare theorem
Assumptions of this:1. No 𝐶 𝑛 externality because when they are present then a competitive
equilibrium is not pareto efficient. Because even after reaching they
may trade because of externality.
2. Agents actually behave competitively – This may not happen & if not then
agents would not take the prices as given.
3. Perfect competition- without this equilibrium won’t exist.
Importance of first welfare theorem is that it gives a general mechanismcompetitive market-that we can use to ensure pareto efficient out comes. This
is true for multiple people & goods. When prices are given, each consumer can
determine his demand & the market will function well enough to determine
competitive prices & therefore an efficient out come.
Implications of second welfare theorem
The problems of distribution & efficiency can be separated. The market
mechanism distributes goods & prices allocate them by indicating their relative
scarcities.
Policy makes sometimes & intervene in price decisions to get distributional
equity.
But this leads to distortion as one achieves an efficient allocation by
making prices as reflecting their relative scarcity ∴ in perfect completion, mv=p
decides the amount of good to be consumed.
The amount a person consumes depends upon his wealth & ∴ his endowment &
giving it to other, the state can achieve distributing role.
So, tax based on endowment is non distortionary but tax based on choices is
distortionary.
© 2016 Ink Pot Online Pvt. Ltd. Confidential, for limited circulation only.
www.inkpotonline.com
Facebook - @inkpotonline
Call Us - +91 9911680354
But the problem is measurement of people’s endowment as most people’s
endowment is labor that is the labor they can sell & not the labor they used up
selling. Taxing labor that they sell is like tax based on choices & is ∴
disortionary.
Suppose a state says that each consumes was required his labor time as tax.
Non this is not based on endowment as we don’t care how many hours a person
actually works. This lumpsum tax is non distortionary.
∴ second welfare theorem says prices should show searcity & all taxes should be
lumpsum.
© 2016 Ink Pot Online Pvt. Ltd. Confidential, for limited circulation only.