31/01/11
E.g. Supply and Demand of Automobiles:
People want to drive more, taste in preference shift
Demand falls, supply increases, the price will fall, quantity effects are indeterminate
Demand curve is downward sloping higher price, people are less wil
IEOR 6711: Conditional expectation
Here we review some basic properties of conditional expectation that are useful for doing computations and give several examples to help the reader memorize these properties. (A more rigorous account can be found, for ex
Copyright c 2009 by Karl Sigman
1
IEOR 6711: Continuous-Time Markov Chains
A Markov chain in discrete time, cfw_Xn : n 0, remains in any state for exactly one unit of time before making a transition (change of state). We proceed now to relax this restrict
Copyright c 2009 by Karl Sigman
1
Discrete-time renewal processes
Imagine busloads of passengers, where the ith bus contains Hi passengers, and the cfw_Hi are iid with pmf p(k ) = P (H = k ), k 1. (1) Imagine further that the seats on a bus are labeled 1
Copyright c 2009 by Karl Sigman
1
Gamblers Ruin Problem
Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an
initial fortune of $i and then on each successive gamble either wins $1 or loses $1 independent
of the past with probabili
Copyright c 2009 by Karl Sigman
1
Notes on Littles Law (l = w)
We consider here a famous and very useful law in queueing theory called Littles Law, also known as l = w, which asserts that the time average number of customers in a queueing system, l, is eq
Copyright c 2009 by Karl Sigman
1
1.1
Discrete-time Markov chains
Stochastic processes in discrete time
A stochastic process in discrete time n I = cfw_0, 1, 2, . . . is a sequence of random variables N (rvs) X0 , X1 , X2 , . . . denoted by X = cfw_Xn : n
Copyright c 2009 by Karl Sigman
1
IEOR 6711: Introduction to Martingales in discrete time
Martingales are stochastic processes that are meant to capture the notion of a fair game in the context of gambling. In a fair game, each gamble on average, regardle
Copyright c 2009 by Karl Sigman
1
IEOR 6711: Notes on the Poisson Process
We present here the essentials of the Poisson point process with its many interesting properties. As preliminaries, we rst dene what a point process is, dene the renewal point proce
1
Regenerative Processes
Copyright c 2009 by Karl Sigman Given a stochastic process X = cfw_X (t) : t 0, suppose that there exists a (proper) random time = 1 such that cfw_X ( + t) : t 0 has the same distribution as X and is independent of the past, C1 =
Copyright c 2009 by Karl Sigman
1
IEOR 6711: Introduction to Renewal Theory
Here, we will present some basic results in renewal theory such as the elementary renewal theorem and the inspection paradox (Section 1), and the renewal reward theorem (Section 2
Copyright c 2009 by Karl Sigman
IEOR 6711: Introduction to Renewal Theory II
Here we will present some deeper results in renewal theory such as a central limit theorem for counting processes, stationary versions of renewal processes, renewal equations, th
Copyright c 2009 by Karl Sigman
1
1.1
Stopping Times
Stopping Times: Denition
Given a stochastic process X = cfw_Xn : n 0, a random time is a discrete random variable on the same probability space as X, taking values in the time set I = cfw_0, 1, 2, . . .
Copyright c 2009 by Karl Sigman
1
Time-reversible Markov chains
In these notes we study positive recurrent Markov chains cfw_Xn : n 0 for which, when in steady-state (stationarity), yield the same Markov chain (in distribution) if time is reversed. The fu
Copyright c 2009 by Karl Sigman
1
Uniform integrability
Given a sequence of rvs cfw_Xn for which it is known apriori that Xn X, n , wp1. for some r.v. X , it is of great importance in many applications to determine conditions ensuring that E (Xn ) E (X )
Vol. 48 No. 2 (Nov., 2011), 157179
A unified growth model
for independent chile*
J. rodrigo fuentes*
This article analyzes long-term patterns of growth of the Chilean economy.
Examining 200 years of data, it shows evidence in favor of using a
neoclassical
Unbundling Institutions
Daron Acemoglu and Simon Johnson
Massachusetts Institute of Technology
This paper evaluates the importance of property rights institutions,
which protect citizens against expropriation by the government and
powerful elites, and con
Journal of Economic PerspectivesVolume 22, Number 1Winter 2008 Pages 25 44
The Productivity Gap between Europe
and the United States: Trends and
Causes
Bart van Ark, Mary OMahony, and
Marcel P. Timmer
T
he benefits of the modern knowledge economy differ g
Copyright c 2009 by Karl Sigman
Borel-Cantelli Lemmas
Suppose that cfw_An : n 1 is a sequence of events in a probability space. Then the event A(i.o.) = cfw_An ocurrs for innitely many n is given by A(i.o.) = k An , k=1 n= Lemma 1 Suppose that cfw_An : n
26/01/11
Comparative Statics:
Start at equilibrium, stationary situation, change a factor and compare a situation
If tastes and preferences change, demand curve will shift
If demand increases, there is increase in the price and an increase in the equilibr
02/02/11
Theory behind demand curve:
Utility function is a function of consumption of various goods.
Good A (x)
0
1
2
3
4
Consumer behavior: how we as individuals make decisions about how and why we spend our money. As
rational individuals, what is our go
07/02/11
Consumer Behavior:
X
A
1
B
2
C
3
D
4
Y
20
19
16
11
4 choices of good x that can be consumed, with options of good y combinations, where utility is
constant
The consumer is indifferent to which combination they receive
There are several possible i
09/02/11
Consumption in time: two periods of life, working life (C1) and retired life (C2)
A worker can earn 100K over his lifetime
Change in interest rates: interest rates increase, the graph rotates on the x-axis, point on C2 axis can be
higher opportun
14/02/01
Upward sloping labor supply curve (A)
o The higher the wages someone is willing to pay you, the more willing you are to work
Downward sloping labor supply curve (B)
In a one day model, maximum about of leisure or work is 24
o Leisure by consumpti
16/02/11
Given a standard market demand supply curve
in a perfectly competitive market, a firm takes an individual demand curve that is horizontal where P =
MC = MR
Any number of firms in the industry have the same curve that combine to make up the indust
21/02/11
Firm operating in perfect competition establishes a market equilibrium at a quantity produced where P = MR = MC =
ACmin
P = MC is a socially desirable result
Perfect competition: lots of individual sellers with no control over market price
Monopo
28/02/11
P = MC A positive outcome (Adam Smith)
Perfectly competitive industry horizontal price line, producing at P = MR = MC.
o With production, the firm pollutes its environment
Distortions: imperfect competition
Externalities: unintended consequences,
Macroeconomics:
21/03/11
Ideal macroeconomic factors: high employment, high price stability/low inflation, high growth in GDP
High inflation and high unemployment: stagflation
o 1973-1978: oil price issues
Inflation, unemployment axis
Is there a trade-off
28/03/11
Macroeconomics: Employment, Inflation and Growth
How much should the Government intervene in these macroeconomic issues
1946: Employment Act
Unemployment:
o Frictional unemployment: short-term unemployment, people searching for jobs, time period
30/03/11
Long-Run Growth:
Policies that will develop a long term growth
Output growth: the growth rate of the output of the entire economy
Per-capital output growth: the growth rate of output per person in the economy
Labor productivity growth: the growth