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2012年5月17日木曜日

Convex Function ⇔ Positive Semidefinite etc.

Let f (x) be a twice differentiable function in n variables defined on an open convex set S. Then we have:

1. f''(x) is positive semidefinite for all x ∈ S ⇔ f is convex in S
2. f''(x) is negative semidefinite for all x ∈ S ⇔ f is concave in S
3. f''(x) is positive definite for all x ∈ S ⇔ f is strictly convex in S
4. f''(x) is negative definite for all x ∈ S ⇔ f is strictly concave in S

http://home.bi.no/a0710194/Teaching/BI-Mathematics/GRA-6035/2010/lecture5-hand.pdf

Extremum Value Theorem

If an objective function is continuous and its domain is compact, there exists the global max and min points.

http://en.wikipedia.org/wiki/Extreme_value_theorem

2012年5月15日火曜日

Stone-Geary Utility Function/ Linear Expenditure System

Stone-Geary Utility Function

$U = \prod_{i} (q_i-\gamma_i)^{\beta_{i}}$

Linear Expenditure System (Demand function derived from Stone-Geary utility function)

$q_i = \gamma_i + \frac{\beta_i}{p_i} (y - \sum_j \gamma_j p_j)$

http://en.wikipedia.org/wiki/Stone%E2%80%93Geary_utility_function

http://www.ier.hit-u.ac.jp/~kitamura/lecture/Hit/02Statsys4_c.pdf

Cournot/ Engel Aggregation

While these results may look like meaningless mathematics exercises, they are in fact useful for empirical work.

http://faculty.apec.umn.edu/pglewwe/documents/Ap840301.pdf

Comparative statics

In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter.

http://en.wikipedia.org/wiki/Comparative_statics

2012年5月14日月曜日

Special Markov Chain (IID Case)

Let P be the transition matrix of the two state Markov chain, where
p11=1-a, p12=a, p21=b, p22=1-b. In other words,

If a=1-b, then the state X1, X2, ... are independently identically distributed random variables with P{Xn=0}=b and P{Xn=1}=a.

http://www.rslabntu.net/Random_Processes/Chapter_2.pdf

2012年5月8日火曜日

Negishi Method to Compute Competitive Equilibria

1. Solve the social planners problem for Pareto efficient allocations indexed
by Pareto weight α

2. Compute transfers, indexed by α , necessary to make the efficient allocation
affordable. As prices use Lagrange multipliers on the resource constraints
in the planner's ’problem.

3. Find the Pareto weight(s) α^ that makes the transfer functions 0.

4. The Pareto efficient allocations corresponding to α^ are equilibrium allocations; the supporting equilibrium prices are (multiples of) the Lagrange multipliers from the planning problem.

Dirk Krueger "Macroeconomic Theory" p.17
http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.201.7768&rep=rep1&type=pdf