ECON 505 - Introduction to Game Theory
Professor: Jorge Lemus
Fall 2016
Quiz # 1
Wednesday, September 7th.
Duration: 35 minutes
P1. Ann and Bob have the goal of maximizing their expected payoff.
Anns payoff is uA (x) = x2 and Bobs payoff is uB (x) = x. Co
Interest rate and budget line
Higher interest rate implies that cash flow in the future is
discounted at a higher rate, thus less valuable today.
The interest rate is the relative price between consumption now and
the future
One unit of consumption now wo
A Simple Model: Summary
1) Economy grows indefinitely because of human capital
accumulation
2) Rate of growth determined by intensity and efficiency of human
capital accumulation
A Simple Model: Policy
What are good policies?
1) Increase schooling
2) Incr
Two Period Model: Consumer
Consumers live two periods
New trade-off: consumption today vs consumption tomorrow
(add this to the consumption leisure trade-off studied in part I)
New instrument: bonds s
1) All bonds are identical
2) All bonds are safe
3) No
Growth Lessons
Malthus:
No growth in per-capita income (even with productivity growth)
Growth in population and aggregate income driven by sustained
increases in productivity
Solow:
With only population growth, no growth in per-capita income, but
growth i
Using the Endogenous Growth Model
Some numbers:
Growth rate of about 3%
Employment to Population ratio: .634 in Dec. 2006, .597 in May
2009
Growth rate of model economy is b(1 u) 1 = .03
Let u = .634 then b = 2.814
Using the Endogenous Growth Model
1 time
Schooling: Spending
Country
Mongolia
Switzerland (U.S)
Zambia
Ecuador
Source: United Nations
How about private spending?
How about spending per student?
Effect of an Increase in Education
when u is smaller, more people work on producing H and less on
prod
2008-2009
Human Capital and the Recession of
One Consequence of the Recession of 2008-2009
Large decline in employment and the labor force
What did these newly unemployed and out of the labor force
people do?
Some went back to school increase their human
How to find.
Steady state level of capital stock for given saving rate s when there
is population growth n and technology grows at rate g?
sf (k ) = (d + n + g)k
Equilibrium law of motion for k
Growth in capital per effective worker:
= (K ) (N ) (H )
K
NH
Determination of the Equilibrium Real Wage
A Simple Model: The Equilibrium
Market clearing imposes: u H s = u H d = u H
So:
C = zu H
(note w = z) and
H 0 = b(1 u)H
0
H = (H ) = b(1 u) 1
1
H
Computing growth rates of consumption:
(C ) = (zu H ) = (H ) = |c
A Simple Model: The Consumer
Suppose :
1) There is no leisure:
Workers split their time between work
and human capital accumulation.
Let u denote time at work. (1 u) time at school
In principle, u is endogenous, but we will take it as fixed
2) Human capit
Assumptions on utility functions
As before we assume that u(c) is increasing
Household, in any period, prefers more consumption to less
Another way of saying this is that marginal utility, u0 (c),
is positive
We assume diminishing marginal utility
As we i
ECON 505: Introduction to Game Theory
Jorge Lemus
August 28, 2017
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Welcome to ECON 505 - Introduction to Game Theory
My Contact Information:
Office:
213 David Kinley Hall
Office Phone: (217) 244 7468
E-mail:
[email protected]
Office H
ECON 505: Introduction to Game Theory
Jorge Lemus
August 30, 2017
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Last Class
Single agent decision making.
Static decision with no uncertainty:
Actions are deterministically mapped into payoffs.
Static decision with uncertainty:
Actions
ECON 505: Introduction to Game Theory
Jorge Lemus
September 11, 2017
Jorge Lemus
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Announcement
We will not have a lecture on
Wednesday, September 13
Jorge Lemus
2 / 14
Last Class
How to solve a game? Different solution concepts.
Rational Players, wh
ECON 505: Introduction to Game Theory
Jorge Lemus
September 17, 2017
Jorge Lemus
1 / 25
Last class
Different notions of equilibrium:
Strictly dominant strategy equilibrium
Iterated elimination of strictly dominated strategies
Rationalizable Strategies
(It
ECON 505: Introduction to Game Theory
Jorge Lemus
September 6, 2017
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Announcement
Next Monday
September 11
First Quiz (35 minutes).
Jorge Lemus
2 / 32
Last Class: Static Games of Complete Information
Si : Player is strategy set.
si Si a
ECON 505: Introduction to Game Theory
Jorge Lemus
September 25, 2017
Jorge Lemus
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Equilibrium Concepts
So far, we have seen different equilibrium concepts for static games
of complete information.
SDS, IESDS, IENBR, Pure Strategy Nash.
Different con
Comparative Statics
Comparative Statics
We look at the following cases:
1) Temporary changes in income: today and tomorrow
2) Permanent changes in income
3) Changes in interest rate
Comparative Statics: Change in Income
0
Increase in todays income (recall
Example: Jacks father
Life-time utility and income:
log(C1) + 0.9 log(C2 )
Y1 = 40000
and Y2 = 10500
Interest rate: r = 5%
Optimality conditions
C1 = 50000 = 26316
1.9
Solution:
C2
= 0.9 50000 = 24868
1.9
Example: Compare the two
Jack and his dad have
Two Period Model: Optimality
Problem of the household:
0
max u(c) + u(c )
c,c0
(1 + r)c + c0 = we(1 + r)
Subject to
From first order conditions:
u0 (c) = (1 + r)u0 (c0 )
Optimality:
0
u (c)
u0cfw_z 0 )
| (c
= (1 + r)
0
c,c
Two Period Model: Optimality
To
What is novel about producitivity?
Why do we get growth in per capita terms with technological
progress?
Answer: No diminishing returns in per capita terms to growth in
productivity
How to see this? Write output per worker:
N
Y
=F
K
K
,HN
and in the long
What is Labor-Augmenting productivity Growth?
Solow does not explain why we have growth:
just assumes g or n
Does not tell us what to do to improve long run growth
Obvious Questions:
How do we improve long-run growth? (esp. if it has slowed down
recently)
The Golden Rule: Introduction
Different values of s lead to different steady states.
How do we know which is the best steady state?
The best steady state has the highest possible consumption per
person:
c = (1 s)f (k )
An increase in s
Leads to higher k a
T
The Golden Rule capital stock
c = f(k*) - !k*
is biggest where the
slope of the production
function
equals
the slope of the
depreciation line:
*
! k*
f(k*)
*
gold
sgold f(k*)
MPK = !
*
k
steady-state
capital per
*
worker, k
gold
How to find. . .
Golden
Population growth
Assume that the population (and labor force) grow at rate n
Nt+1
N = 1+ n
t
n is exogenous
Now, a steady state equilibrium satisfies
K
0
K
N0
=
constant, or
K
N
Break-even investment
(d + n)kt = break-even investment, the amount of
inves
Depreciation
E&$'&5.-6)7%$&'%
()'*&'+%!"%
!%B%#2&%'-#&%)8%4&$'&5.-6)7%
%B%#2&%8'-56)7%)8%#2&%5-$.#-/%3#)5*%#2#% (&-'3%)"#%&-52%$&'.)4%
!"'
!
A%
,-$.#-/%$&'%
()'*&'+%"%
The equation of motion for k
0
k k = sf (k) dk
The Solow models central equation
Determ
Starting with too much capital
If k *
gold
*
y
then increasing c
requires a fall in s.
In the transition to
the Golden Rule, c
consumption is
i
higher at all points
in time.
t
0
!me
t
!me
Starting with too little capital
If k *
gold *
then increasing c
re