1. Frank Cowell: Microeconomics Distributions MICROECONOMICS Principles and Analysis Frank Cowell August 2006

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

3. Frank Cowell: Microeconomics Ingredients of a probability model The variate 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 support of the distribution  The smallest closed set  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 Distribution function  Captures the probability concept in a convenient and general way  Encompass both discrete and continuous distributions.

4. Frank Cowell: Microeconomics Discrete and continuous Discrete distributions Discrete distributions   is usually a finite collection of points   could also be countably infinite Continuous distributions 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  then define the density function f as the derivative of F Take some particular cases: Take some particular cases: a collection of examples

5. Frank Cowell: Microeconomics Some examples Begin with two cases of discrete distributions 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 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: Finally a standard example of a continuous distribution with unbounded support:  Lognormal distribution

6. Frank Cowell: Microeconomics Examples 1 & 2 Discrete distributions Discrete distributions  is the set { x 0, x 1, …, x  - 1 }  is the set { x 0, x 1, …, x  - 1 }   is number of elements in the support  assumed positive, finite Example 1 Example 1   = 2.  Probability  of value x 0 ; probability 1–  of value x 1. Example 2 Example 2   = 5  Probability  i of value x i, i = 0,...,4.

7. Frank Cowell: Microeconomics F: Example 1 x   Below x 0 probability is 0.   Probability of x ≤ x 0 is . x1x1 x0x0 1    Probability of x ≤ x 1 is .   Suppose of x 0 and x 1 are the only possible values. F(x)   Probability of x ≥ x 0 but less than x 1 is .

8. Frank Cowell: Microeconomics F: Example 2 x   Below x 0 probability is 0.   Probability of x ≤ x 0 is  . x1x1 x0x0 1 00   Probability of x ≤ x 1 is   +  .   There are five possible values: x 0,…, x 4. F(x)  0  1  0  1  2  3   Probability of x ≤ x 2 is   +   +  . x4x4 x2x2 x3x3  0  1  2   Probability of x ≤ x 3 is   +   +   +  .   Probability of x ≤ x 4 is 1.     +   +   +   +   = 1

9. Frank Cowell: Microeconomics Examples 3 & 4 Continuous distributions Continuous distributions In both cases  is the interval [ x 0, x 1 ] In both cases  is the interval [ x 0, x 1 ]  where x 0 is non-negative…  … and x 1 is finite  F is differentiable, so…  …density f is defined Example 3 Example 3  rectangular distribution  f(x) = 1 / [ x 1 − x 0 ] Example 4 Example 4  Beta distribution  [ x 0, x 1 ] = [0,1]

10. Frank Cowell: Microeconomics Example 3 – rectangular distribution x   Below x 0 probability is 0. x1x1 x0x0   Suppose values are uniformly distributed between x 0 and x 1. f(x)

11. Frank Cowell: Microeconomics F: Example 3 (cont) x   Below x 0 probability is 0.   Probability of x ≥ x 0 but less than x 1 is [x  x 0 ] / [x 1  x 0 ]. x1x1 x0x0 1   Probability of x ≤ x 1 is .   Values are uniformly distributed over the interval [x 0, x 1 ]. F(x)

12. Frank Cowell: Microeconomics Example 4: Beta distribution 01 x   Support is bounded (above and below)   The density function with parameters a=0.5, b=0.5

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

14. Frank Cowell: Microeconomics Example 5 Lognormal distribution Lognormal distribution  Continuous Unbounded support Unbounded support   is the interval [0, ∞) F is differentiable, so… F is differentiable, so…  …density f is defined

15. Frank Cowell: Microeconomics Example 5: Lognormal distribution x 012345678910   Support is unbounded above.   The density function with parameters  =1,  =0.5   The mean

16. Frank Cowell: Microeconomics Lognormal distribution function x 012345678910 1