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Statistical Distributions in Microeconomics

Explore key concepts, probability models, discrete and continuous distributions, and examples like Beta and Lognormal distributions in microeconomics. Get insights into income modeling and empirical estimation methods.

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Statistical Distributions in Microeconomics

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  1. Distributions MICROECONOMICS Principles and Analysis Frank Cowell August 2006

  2. Purpose • This presentation concerns statistical distributions in microeconomics • a brief introduction • it does not pretend to generality • Distributions make regular appearances in • models involving uncertainty • representation of aggregates • strategic behaviour • empirical estimation methods • Certain concepts and functional forms appear regularly • We will introduce basic concepts and some key examples

  3. Ingredients of a probability model • The variate • could be a scalar – income, family size… • Could be a vector – basket of consumption, list of inputs • The support of the distribution • The smallest closed set W whose complement has probability zero • A convenient way of specifying clearly what is logically feasible (points in the support) and infeasible (other points). • Distribution function • Captures the probability concept in a convenient and general way • Encompass both discrete and continuous distributions.

  4. Discrete and continuous • Discrete distributions • Wis usually a finite collection of points • Wcould also be countably infinite • Continuous distributions • for univariate distributions S is usually an interval on the real line… • … could be bounded or unbounded • for multivariate distributions usually a connected subset of real space • if F is differentiable on W then define the density functionf as the derivative of F • Take some particular cases: a collection of examples

  5. Some examples • Begin with two cases of discrete distributions • where the variate can take just one of two values • where it can take one of five values. • Then two simple examples of continuous distributions with bounded support • The rectangular distribution – uniform density over an interval. • Beta distribution – a single-peaked distribution. • Finally a standard example of a continuous distribution with unbounded support: • Lognormal distribution

  6. Examples 1 & 2 • Discrete distributions • W is the set {x0, x1, …, x-1} •  is number of elements in the support • assumed positive, finite • Example 1 •  = 2. • Probability p of value x0; probability 1–p of value x1. • Example 2 •  = 5 • Probability pi of value xi, i = 0,...,4.

  7. F: Example 1 • Suppose of x0 and x1 are the only possible values. • Below x0 probability is 0. • Probability of x ≤ x0is p. 1 • Probability of x ≥ x0 but less than x1 is p. F(x) • Probability of x ≤ x1 is 1. p x x0 x1

  8. F: Example 2 • There are five possible values: x0 ,…, x4. • Below x0 probability is 0. • Probability of x ≤ x0is p0. 1 • Probability of x ≤ x1 is p0+p1. • Probability of x ≤ x2 is p0+p1 +p2. p0+p1+p2+p3 • Probability of x ≤ x3 is p0+p1+p2+p3. F(x) p0+p1+p2 • Probability of x ≤ x4 is 1. p0+p1 • p0 + p1+ p2+ p3+ p4 = 1 p0 x x2 x3 x4 x0 x1

  9. Examples 3 & 4 • Continuous distributions • In both cases W is the interval [x0, x1] • where x0is non-negative… • … and x1 is finite • F is differentiable, so… • …density f is defined • Example 3 • rectangular distribution • f(x) = 1 / [ x1− x0 ] • Example 4 • Beta distribution • [x0, x1] = [0,1]

  10. Example 3 – rectangular distribution • Suppose values are uniformly distributed between x0 and x1. • Below x0 probability is 0. f(x) x x0 x1

  11. F: Example 3 (cont) • Values are uniformly distributed over the interval[x0,x1]. • Below x0 probability is 0. 1 • Probability of x ≥ x0 but less than x1 is [x x0] / [x1 x0] . • Probability of x ≤ x1 is 1. F(x) x x0 x1

  12. Example 4: Beta distribution • Support is bounded (above and below) • The density function with parameters a=0.5, b=0.5 x 0 1

  13. Beta distribution (cont) • Support is bounded in [0,1] • The distribution function with parameters a=0.5, b=0.5 1 0 0 1

  14. Example 5 • Lognormal distribution • Continuous • Unbounded support • W is the interval [0, ∞) • F is differentiable, so… • …density f is defined

  15. Example 5: Lognormal distribution • Support is unbounded above. • The density function with parameters m=1, s=0.5 • The mean x 0 1 2 3 4 5 6 7 8 9 10

  16. x 0 1 2 3 4 5 6 7 8 9 10 Lognormal distribution function 1

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