Motivational Examples
() Exchange 1 / 35
Exchange Economy
Copyright By PowCoder代写 加微信 powcoder
Trade involves exchange.
Who buys what? Who sells what?
How are prices determined?
How is a consumerís income determined?
() Exchange 2 / 35
Budget Set
A consumer starts with endowment ω: this is a list of her resources. So it is a vector: for example
ω = (ω1,ω2) = (6,4)
What is the value of this endowment in some economy?
What consumption opportunities may it be exchanged for?
() Exchange 3 / 35
Budget Set
This depends on the prices or the prevaling rate of exchange for the two goods
then the endowment
p1 = 2; p2 = 3
ω = (ω1,ω2) = (6,4)
is worth 24, say $24.
So this now determines the consumerís budget.
() Exchange 4 / 35
Budget Set
More formally, given p1 and p2, the budget constraint for a consumer with endowment
ω = (ω1,ω2) = (6,4) p1x1 + p2x2 = p1ω1 + p2ω2
p(x ω) = 0invectornotation and the budget set is
B = f(x1,x2)jp1x1 +p2x2 p1ω1 +p2ω2,×1 0;x2 0g B = f(x1,x2)jp(x ω)0;x1 0;x2 0g
For every vector p, the endowment ω is always on B line. () Exchange
Budget line in exchange economy
H@ H H@
x1 Figure: Endowment ω is on the budget line with any vector of prices
() Exchange
Exchange Economy
Take again
p1 =2; p2 =3andω=(ω1,ω2)=(6,4) so the constraint is
and the price ratio is
2×1+3×2 =px=pω=24 p1 = 2
meaning that three units of good 1 trade for two units of good 2.
() Exchange
Exchange Economy
If quantities demanded by the some consumer (x1,x2) = (3,6) her net demands are
z1 =x1 ω1= 3and z2 =x2 ω2=2.
She sells 3 units of good 1 at $2 and buys 2 units of good 2 at $3. Rewrite the constraint
p (x ω) = 0
pz =p1z1 +p2z2 =2( 3)+32=0
The consumerís net demands weighted by the prices sum into zero.
() Exchange
Walras Law
This applies to any consumer involved in exchange and for any non-zero prices.
The result simply states that the consumer is choosing some point on the budget constraint.
Summing over all the consumers in the economy gives
Walras Law. The sum of all of the consumersínet demands weighted by the goods prices is zero.
() Exchange 9 / 35
Exchange Economy Equilibrium
This, however, does not pin down the ìrightîprices in any way. So what determines the prices?
The market: buyers and sellers make purchasing decisions given prices, and prices adjust.
The equilibrium prices clear the markets for goods.
At the equilibrium prices the total net demands are zeros.
() Exchange
Basic 2×2 model:
Two consumers, A and B. Two commodities, 1 and 2. Utilities:
Endowments:
uA xA1,xA2 = xA1α xA21 α
and uB xB1 ,xB2 = xB1 α xB2 1 α ω1A,ω2A and ω1B,ω2B.
() Exchange 11 / 35
Consumer Aís problem
max xA1 α xA2 1 α , s.to.
x A1 , x A2
p(xA ωA)=0
Consumer Bís problem
max xB1 α xB2 1 α , s.to.
x B1 , x B2
p(xB ωB)=0
Exchange 12 / 35
Given prices (p1,p2), FOCs of consumer A
α xA1α 1 xA21 α λp1 = 0
(1 α) xA1α xA2 α λp2 = 0 and budget constraint
p 1 x A1 ω 1A + p 2 x A2 ω 2A = 0
() Exchange 13 / 35
Divide one FOC on the other
1 αx1 =p =MRS12
This gives xA1 in terms of xA2 . Substitute into the constraint.
Finally obtain the demands
xA1 = pα p1ω1A+p2ω2A 1
xA2 = 1 α p1ω1A+p2ω2A p2
() Exchange 14 / 35
Similarly for consumer B:
xB1 =pα p1ω1B+p2ω2B 1
xB2 = 1 α p1ω1B+p2ω2B p2
Observe these are determined by preferences and prices.
The endowments e§ectively only scale them up: it matters for quantities but not for distribution.
The ratio xB1 /xB2 = const when the endowments change. Homothetic preferences. What do they guarantee?
() Exchange
Homothetic Preferences
Figure: Homothetic preferences
Exchange Economy Equilibrium
Now we want to compute the prices (p1,p2) that clear this economy. That is, given prices consumers A and B have expressed their
The question now is, what are these prices?
We know they must clear the market: (p1,p2) must be such that demand equals supply.
Another way to say this, a bit more precisely, is to claim that excess demand must be zero in equilibrium
() Exchange 17 / 35
Equilibrium
DeÖne the excess demand for commodity 1 as
z 1 ( p 1 , p 2 ) = x A1 + x B1 ω 1A + ω 1B
similarly for commodity 2
z 2 ( p 1 , p 2 ) = x A2 + x B2 ω 2A + ω 2B
A condition of equilibrium is
z1 (p1,p2) = 0 and z2 (p1,p2) = 0
() Exchange 18 / 35
Equilibrium
Recall that in the demand functions only the ratio of prices matter. We can set p2 = 1, this will be the numeraire good.
z1(p1,p2) = xA1+xB1 ω1A+ω1B=
= αω1A + p1 ω2A+αω1B + p1 ω2B ω1A +ω1B
11 Now solve for z1 (p1,p2) = 0.
() Exchange 19 / 35
Equilibrium
solve for z1 (p1,p2) = 0
p1= α ω2A+ω2B,andp2=1. 1 α ω 1A + ω 1B
Since (p1,p2) clear the market for good 1 by Walras law the market for good 2 also clears.
() Exchange 20 / 35
Equilibrium Summary
xA1 = αω1A+p1ω2A
1 xA2 =(1 α)p1ω1A+ω2A
xB1 =αω1B+p1ω2B 1
xB2 =(1 α)p1ω1A+ω2A
They are expressed in terms of endowments and prices, and reáect
preferences.
() Exchange
Equilibrium Summary
p1= α ω2A+ω2B,andp2=1. 1 α ω 1A + ω 1B
Are expressed solely in terms of endowments and preferences.
Prices are used to re-allocate the aggregate endowment according to
preferences.
() Exchange 22 / 35
Edgeworth Box representation
() Exchange 23 / 35
General equilibrium: nice properties.
First Theorem of Welfare Economics: A competitive equilibrium is Pareto e¢ cient.
() Exchange 24 / 35
Insuring, lending and borrowing
Why do we need those?
In static optimization problem
maxfu(x) jsubjecttoBC:p(x ω)=0g x
There is no tomorrow, and There is no uncertainty.
() Exchange
Intertemporal smoothing and insurance
Without a tomorrow there is
no need to save, and no capacity to borrow
Without uncertainty there is no need for insurance.
So there is no need for Önancial markets.
We need a model that accommodates these aspects of real life. Dynamic horizon: today and tomorrow at least
Uncertainty: at least two states tomorrow
() Exchange
Intertemporal smoothing
Suppose a consumer lives for two periods
In each period can consume the same good x
And has preferences represented by and increasing and concave utility function
U = u(x1,x2), where (x1,x2) are consumptions at 1 and 2.
The consumer receives her endowment ω = (ω1,ω2) at dates 1 and 2.
If there is no way to save or borrow the consumer will simply eat x1 =ω1 andx2 =ω2.
() Exchange 27 / 35
Budget Constraint
There is a saving technology (banks or bonds) that return 1 tomorrow for each 1 saved today and allows the reverse as well.
Since ω may be very uneven this generates savings/borrowings motive. What are optimal (x1,x2) now?
Given some consumption x1 at price p1, savings (or borrowings) are s = p1 (ω1 x1),
Then in period 2, one may consume
x2 :p2x2 =s+p2ω2.
Putting this together as intertemporal budget constraint p2x2 p2ω2 +p1 (ω1 x1)
Note that this is one constraint and hence only one Lagrange multiplier
() Exchange
Intertemporal smoothing
Hence we solve
subject to
Lagrangean for this problem
maxu(x1,x2) x1, x2
p1x1 + p2x2 p1ω1 + p2ω2
L(x1,x2,λ) = u(x1,x2) λ[p1x1 +p2x2 p1ω1 p2ω2]
() Exchange
Intertemporal smoothing
Optimality conditions
∂L (x1, x2, λ) ∂x1
∂L (x1, x2, λ) ∂x2
p1ω1 +p2ω2 p1x1 p2x2 = 0 p1x1 +p2x2 = p1ω1 +p2ω2 and
ru(x) = λp.
= ux1 λp1=0 = ux2 λp2=0
Exchange 30 / 35
Euler Equation
Eliminating λ from the Örst two equations ux1(x) = p1.
This is Euler equation Alternatively
ux2 (x) p2 ux1 (x) = ux2 (x)
Exchange 31 / 35
Insurance smoothing
Suppose a consumer lives for one period but her endowment is uncertain
It is either
ωb at boom or ωr < ωb at recession
The preferences are represented by an increasing and concave utility function
U =u(xb,xr)
where xb is consumption at boom, and xr at recession.
() Exchange 32 / 35
Budget Constraint
There is a technology (insurance company) that allows to transfer part of the endowment at boom into a recession state.
This is a contingent transfer, in case of recession there is a payout, in case of a boom there is a payment.
Since (ω1,ω2) may be very uneven this generates insurance motive. What are the optimal (xb , xr )?
Suppose s (of money) is transferred from the boom into the recession Then, in the recession one consumes
xr :prxr =s+prωr. In the boom one consumes
Eliminating s
xb :pbxb =pbωb s. prxr +pbxb =prωr +pbωb
This is again one constraint and hence one Lagrange multiplier
() Exchange
Insurance smoothing
Lagrangean for this problem
L(xb,xr,λ) = u(xb,xr) λ[prxr +pbxb prωr pbωb] Optimality conditions
∂L(xb,xr,λ) ∂xb ∂L(xb,xr,λ) ∂xr
prxr+pbxb =prωr+pbωband ru(x) = λp.
= uxb λpb=0 = uxr λpr =0
Euler Equation
Eliminating λ from the Örst two equations uxr (x) = uxb (x)
The optimality is achieved by a contingent transfer, in case of recession there is a payout, in case of a boom there is a payment.
() Exchange
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com