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 curv
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 pr
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 (
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
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 win
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 ave
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
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 gam
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 den
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
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
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, stationa
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
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 Mar
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 applicatio
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 d
Unbundling Institutions
Daron Acemoglu and Simon Johnson
Massachusetts Institute of Technology
This paper evaluates the importance of property rights institutions,
which protect citizens against expro
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
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
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 i
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 s
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 whic
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
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, maximu
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
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 indi
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: imperf
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 i
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-ter
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 pers