Comparative Statics

Econ 452
Comparative Statics
Comparative Static Analysis
1. What?
- compare equilibria
- model gives equilibrium conditions
- first order conditions
- frequently in implicit form
- if exog. variables change, how does
new eq'm compare to original?
- check changes in endog.
variables wrt exog. variables
through use of eq'm conditions .
2. Why?
- comparative dynamics are hard
- belief that economic system is in
equilibrium
Example: basic consumer theory:
- 2 goods, (x,y);
- budget set given by { p x , p y , M }
- choose x,y to max u(x,y) s.t.
px x + p y y ≤ M
- properties of demand functions?
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Econ 452
Comparative Statics
1. set up Lagrangean, obtain FOC's
ux ( x, y) − λ px = 0
u y ( x, y) − λ py = 0
px x + p y y − M = 0
These 3 equations implicitly define
Marshallian/Walrasian demand
functions x ( p x , p y , M ), y ( p x , p y , M ), and
the marginal utility of income,
λ ( p x , p y , M ).
2. Totally differentiate equations wrt
exog variables ( px , p y , M ) and endog
variables ( x, y,λ ) ; write in matrix form.
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Econ 452
Comparative Statics
 u xx

 u yx
 − px

u xy
u yy
− py

− px   dx  
λ dpx
  

− p y   dy  = 
λ dp y

0   d λ   xdpxx + ydp y − dM 
Determinant of LHS matrix can be signed:
SOC for constrained maximization > 0
3. Can find slopes of demand curves and
engel curves using Cramer's rule.
Alternative (if only two choice variables):
- use constraint-as-equality (from
theory) to eliminate one variable
here: y = (Μ − px x)/ py
so
u( x, y) = u( x, y( x)) = u( x, M −p px x )
y
- treat as unconstrained max'm
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Econ 452
Comparative Statics
problem: FOC is
du
dy
= u x ( x, y ) + u y ( x, y )
dx
dx
px
= u x ( x, y ) + u y ( x, y )(− ) = 0
py
ux
px
=
This yields FOC for max: u
py
y
- once again, can derive properties of
demand functions, recognizing that
FOC defines x ( p x , p y , M ) , and
x ( p x , p y , M ) = ( M − px x ( px , p y , M )) / p y
Sometimes, need to do more…:
Demand for insurance - with no
asymmetric information:
Contingent commodities / state space
Consider the standard insurance model
- accident occurs with prob. 0 < p < 1
- loss of L
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Econ 452
Comparative Statics
- Individual's endowment point, E, (if
buys no insurance):
- if accident (state 1), ind'l has
y = y − L;
1
0
- if no accident(state 2), ind'l has
y2 = y0
- If individual buys insurance paying
out I in state 1, at cost δ per dollar:
- state 1: has
y1 = y0 − L −δ I + I = y0 − L + (1−δ )I
- state 2: has y = y −δ I
2
0
EU without insurance?
EU = pU ( y0 − L) + (1− p)U ( y0)
EU with insurance?
EU = pU ( y0 − L + (1−δ )I ) + (1− p)U ( y0 −δ I )
Optimal insurance? Choose I to max
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Econ 452
Comparative Statics
∂EU = pU '( y )(1−δ ) −δ (1− p)U '( y ) = 0
1
2
∂I
I* satisfies
pU '( y1)
= δ
(1− p)U '( y2) 1−δ
What is value of delta?
Cost of insurance:
Assume: 1. identical individuals
2. risk neutral firms
3. competitive insurance markets
4. no administrative costs
(1) + (2) implies firm max's expected
profit on representative contract;
p(δ I − I ) + (1− p)δ I
= I (− p(1−δ ) + (1− p)δ )
Eprofit =
(3) implies Eprofit = 0
Therefore: in competitive insurance
market,
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Econ 452
Comparative Statics
p = δ
(1− p) (1−δ )
Sub'g this back into FOC for consumer,
tells us
I* satisfies
U '( y1)
=1 : RA individual
U '( y2)
will buy full insurance if "fair insurance" is
available.
Suppose insurance is unfair, so δ > p .
Then Eprofit > 0 on the average contract
(assuming no fixed costs).
Will consumer still want full insurance?
Comparative statics of insurance:
From above, FOC for optimal insurance is
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Econ 452
Comparative Statics
pU '( y0 − L + (1−δ )I *)(1−δ ) −δ (1− p)U '( y −δ I *) = 0
0
Rearranging:
(1 − δ )U '( y0 − L + (1 − δ ) I *) (1 − p )
=
p
δ U '( y0 − δ I *)
This defines demand for insurance as the
implicit function I * = f ( y , L,δ , p) .
0
Properties of this function?
1. How much insurance?
i) if δ = p , ind'l buys full insurance at
all income levels.
ii) if δ > p , so insurance is unfair (and
ins. firm makes strictly positive
profit), then ind'l buys less than
full insurance, so y < y .
1
2
2. How does insurance vary with y?
- totally differentiate FOC wrt I* and
endogenous & exogenous variables
- look at partial wrt initial income:
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Econ 452
Comparative Statics
dI * = δ (1− p)U "( y2) − p(1−δ )U "( y1)
dy0 (1−δ )2 pU "( y1) + δ 2(1− p)U "( y2)
Sign? denominator <0, numerator….?
- sign depends on relative sizes of U",
evaluated at different income levels.
- coeff. of ARA useful
- to use, need to bring in U'( ) - how?
i)
rearrange FOC to solve for
p(1−δ )U '( y1)
δ (1− p) =
U '( y2)
(dI */ dy0) to obtain
U "( y2) U "( y1)
p(1−δ )U '( y1)[
]
−
U '( y2) U '( y1)
ii) sub. into
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Econ 452
Comparative Statics
=
p(1−δ )U '( y1)[ A( y1) − A( y2)]
iii) sign depends on term in [ ],
which is coeff. of ARA at two
values;
- have y > y , so…
2
1
- CARA - no change in
insurance
- DARA - I* down as y up
- IARA - I* up as y up
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