(Requires Appendix Material and Calculus)The Logarithm of the Likelihood Function

question 15

Essay

(Requires Appendix material and Calculus)The logarithm of the likelihood function (L)
for estimating the population mean and variance for an i.i.d.normal sample is as follows
(note that taking the logarithm of the likelihood function simplifies maximization.It is a
monotonic transformation of the likelihood function, meaning that this transformation
does not affect the choice of maximum): $$L = - \frac { n } { 2 } \log \left( 2 \pi \sigma ^ { 2 } \right) - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \mu _ { Y } \right) ^ { 2 }$$
Derive the maximum likelihood estimator for the mean and the variance.How do they
differ, if at all, from the OLS estimator? Given that the OLS estimators are unbiased,
what can you say about the maximum likelihood estimators here? Is the estimator for the
variance consistent?

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Describe the relationship between the price of goods and consumer income.

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Definitions:

Elastic

Describes a situation where the demand for a good or service significantly changes in response to price changes, indicating sensitivity to price.

Midpoint Method

is a technique in economics used to calculate the elasticity of demand or supply, providing a more accurate measure by averaging the starting and ending points.

Elasticity of Demand

A measure of how sensitive the demand for a good or service is to changes in its price.

Quantity Demanded

The complete amount of a product or service that purchasers have the willingness and capacity to obtain at a set price.