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## Finance 510: Microeconomic Analysis

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### Finance 510: Microeconomic Analysis

The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more of X

Consumer Demand Analysis

Suppose that you observed the following consumer behavior

P(Bananas) = $4/lb.

P(Apples) = $2/Lb.

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

Choice A

P(Bananas) = $3/lb.

P(Apples) = $3/Lb.

Q(Bananas) = 15lbs

Q(Apples) = 15lbs

Choice B

What can you say about this consumer?

Is strictly preferred to

Choice B

Choice A

How do we know this?

Consumers reveal their preferences through their observed choices!

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

Q(Bananas) = 15lbs

Q(Apples) = 15lbs

P(Bananas) = $4/lb.

P(Apples) = $2/Lb.

Cost = $80

Cost = $90

P(Bananas) = $3/lb.

P(Apples) = $3/Lb.

Cost = $90

Cost = $90

B Was chosen even though A was the same price!

Choice C

Cost = $90

P(Bananas) = $2/lb.

P(Apples) = $4/Lb.

Q(Bananas) = 25lbs

Q(Apples) = 10lbs

Q(Bananas) = 15lbs

Q(Apples) = 15lbs

Cost = $90

Choice B

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

Cost = $100

Choice A

Is strictly preferred to

Is choice C preferred to choice A?

Choice C

Choice B

Choice B

Choice A

Is strictly preferred to

Choice C

Choice B

C > B > A

Is strictly preferred to

Choice C

Choice A

Rational preferences exhibit transitivity

Consumer theory begins with the assumption that every consumer has preferences over various consumer goods. Its usually convenient to represent these preferences with a utility function

Set of possible choices

“Utility Value”

Using the previous example (Recall, C > B > A)

Choice A

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

Choice B

Q(Bananas) = 15lbs

Q(Apples) = 15lbs

Choice C

Q(Bananas) = 25lbs

Q(Apples) = 10lbs

We only require a couple restrictions on Utility functions

- For any two choices (X and Y), either U(X) > (Y), U(Y) > U(X), or U(X) = U(Y) (i.e. any two choices can be compared)
- For choices X, Y, and Z, if U(X) > U(Y), and U(Y) > U(Z), then U(X) > U(Z) (i.e., the is a definitive ranking of choices)

- However, we usually add a couple additional restrictions to insure “nice” results
- If X > Y, then U(X) > U(Y) (More is always better)
- If U(X) = U(Y) then any combination of X and Y is preferred to either X or Y (People prefer moderation to extremes)

Suppose we have the following utility function

U = 20

Imagine taking a “cross section” at some utility level.

The “cross section” is called an indifference curve (various combinations of X and Y that provide the same level of utility)

Any two choices can be compared

There is a definite ranking of all choices

A

C

B

The “cross section” is called an indifference curve (various combinations of X and Y that provide the same level of utility)

More is always better!

C

A

B

The “cross section” is called an indifference curve (various combinations of X and Y that provide the same level of utility)

People Prefer Moderation!

A

C

B

The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more of X

+

= 0

Suppose you are given a little extra of good X. How much Y is needed to return to the original indifference curve?

The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more of X

+

= 0

Now, let the change in X become arbitrarily small

The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more of X

Marginal Utility of X

Marginal Utility of Y

If you have a lot of X relative to Y, then X is much less valuable than Y MRS is low)!

The elasticity of substitution measures the curvature of the indifference curve

Consumers solve a constrained maximization – maximize utility subject to an income constraint.

As before, set up the lagrangian…

Suppose that we raise the price of X

Can we be sure that demand for x will fall?

Suppose that we raise the price of X, but at the same time, increase your income just enough so that your utility is unchanged

Substitution effect

Now, take that extra income away…

Income effect

Suppose that we have the following demand curve

$100

A demand curve tells you the maximum a consumer was willing to pay for every quantity purchased.

$50

D

100

For the 100th sale of this product, the maximum anyone was willing to pay was $50

Suppose that we have the following demand curve

$100

$75

$50

D

50

100

For the 50th sale of this product, the maximum anyone was willing to pay was $75

Consumer surplus measures the difference between willingness to pay and actual price paid

$100

$75

Whoever purchased the 50th unit of this product earned a consumer surplus of $25

$50

D

50

100

For the 50th sale of this product, the maximum anyone was willing to pay was $75

Consumer surplus measures the difference between willingness to pay and actual price paid

$100

If we add up that surplus over all consumers, we get:

CS = (1/2)($100-$50)(100-0)=$2500

$2500

$50

Total Willingness to Pay ($7500)

$5000

- Actual Amount Paid ($5000)

D

Consumer Surplus ($2500)

100

In economics, we are often interested in elasticity as a measure of responsiveness (price, income, etc.)

Given our model of demand as a function of income, and prices, we could specify a demand curve as follows:

We could, instead, use a semi-log equation:

We could, instead, use a semi-log equation:

The most common is a log-linear demand curve:

Log linear demand curves are not straight lines, but have constant elasticities!

If we assumed that this was the maximization problem underlying a demand curve, what form would we use to estimate it?

Suppose you observed the following data points. Could you estimate the demand curve?

D

A bigger problem with estimating demand curves is the simultaneity problem.

S

Market prices are the result of the interaction between demand and supply!!

D

Case #1: Both supply and demand shifts!!

Case #2: All the points are due to supply shifts

S

S

S’

S’

S’’

S’’

D

D’

D’’

D

Suppose you get a random shock to demand

Demand

The shock effects quantity demanded which (due to the equilibrium condition influences price!

Supply

Therefore, price and the error term are correlated! A big problem !!

Equilibrium

Suppose we solved for price and quantity by using the equilibrium condition

We could estimate the following equations

The original parameters are related as follows:

We can solve for the supply parameter, but not demand. Why?

By including a demand shifter (Income), we are able to identify demand shifts and, hence, trace out the supply curve!!

S

D

D

D

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