Slide 3 & 4 Asistensi Mikroekonomi-1

MICROECONOMICSTheory of Consumer BehaviorASISTENSI-3 & 4

Program Pascasarjana

Universitas Indonesia

Ilmu Ekonomi

Harry Patria

15 Maret 2016

3.1 Problem Exercises: Maximization Utility

3.2 Problem Exercises: Expenditure Minimization

3.3 Problem Exercises: Slutsky & Hicksian

3.4 Problem Exercises: Roy’s identity

3.5 Graphical Derivation (Marshall, Hicks & Slutsky)

UMP & EMP Schematic Diagram

Please explore:

a) Marshallian Demand

b) Indirect Utility

c) Hicksian Demand

d) Expenditure Function

e) Roy’s identity

f) Sheppard’ Lemma

g) Slutsky Equation

h) Substitution Effect

3.1 Problem Exercises: Maximum Utility

Jika fungsi utilitas untuk barang X dan Y adalah U = 7200X 0.75 Y 0.25

dengan harga $90 per unit X dan $ 30 per unit Y serta income total$ 1200, Tentukan:

a) berapa unit X dan Y yang memaksimalkan utilitas?

b) Utilitas maksimum?

c) Nilai dari lamda dan interpretasinya?

d) Intepretasi nilai lamda pada jawaban c?

3.1 Problem Exercises: Maximum Utility

Formulasikan soal sebelumnya dengan expenditure minimization,

Tentukan:

a) berapa unit X dan Y?

b) Minimum expenditure?

c) Nilai dari lamda dan interpretasinya?

d) Intepretasi nilai lamda pada jawaban c?

Assume a person has a utility function U = XY, and money income of$10,000, facing an initial price of X of $10 and price of Y of $15. Ifthe price of X increases to $15, answer the following questions:

a) What was the initial utility maximizing quantity of X and Y?

b) What is the new utility maximizing quantity of X and Yfollowing the increase in the price of X to $15?

c) What is the Hicks compensating variation in income that wouldleave this person equally well off following the price increase?What is the Slutsky compensating variation in income?

d) Calculate the pure substitution effect and the real incomeeffect on X of this increase in the price of X. Distinguishbetween the calculation of these effects using the Hicksiananalysis vs. the Slutsky analysis.

a. There are two related approaches. Both approaches requirethe simple derivation of the first order condition for maximizingutility subject to a budget constraint:

MU(x) /MU(y) = P(x)/(P(y)

In this simple case, this becomes Y / X = 10 / 15 = .67, or Y = .67 Xand X = 1.5 Y.

Thus, any “optimal” consumption combination must have thisrelationship between X and Y.

Approach 1 uses this information to solve for either X or Y bysimple substitution into the budget equation M = P(x) X + P(y) Y.For example, substituting X = 1.5 Y yields the following:

10,000 = $10 (1.5Y) + $15 Y

a. There are two related approaches. Both approaches requirethe simple derivation of the first order condition for maximizingutility subject to a budget constraint:

MU(x) /MU(y) = P(x)/(P(y)

In this simple case, this becomes Y / X = 10 / 15 = .67, or Y = .67 Xand X = 1.5 Y.

Approach 1 uses this information to solve for either X or Y bysimple substitution into the budget equation M = P(x) X + P(y) Y.For example, substituting X = 1.5 Y yields the following:

10,000 = $10 (1.5Y) + $15 Y

10,000 = 15 Y + 15 Y = 30Y so Y = 333.33, and X = 1.5Y = 500

Approach 2 is a bit more general, deriving the demand functionsfor both X and Y and then substituting the relevant parametervalues. From above, the first order condition was Y / X = P(x) /P(y), which can be manipulated to be P(y) Y = P(x) X, or “totalexpenditures on X = total expenditures on Y”

The next step is again to substitute back into the budgetequation to derive a more general demand function for X and Y.That is, M = P(x) X + P(y)Y, but since P(x)X = P(y)Y, this becomeseither M = 2 P(x) X or M = 2 P(y)Y, and finally:

X = M / 2 P(x) and Y = M / 2 P(y), which given M = 10,000,

P(x) = 10 and P(y) = 15 yields Y = 333.33 and X = 500 as withApproach 1.

b. It is then easy, especially with Approach 2 to determine howthe optimal consumption “bundle” changes as the price of Xincreases to $15. The quantity of Y does not change (essentiallythe pure substitution effect and real income effects for Y aretotally counterbalancing, so that Y does not change – anotherunique result of Cobb-Douglas utility functions). However, theoptimal X does change, dropping from 500 to 333.33, or by atotal of 166.67.

So the answer to b. is Y = no change; X = 333.33, a reduction of167.67.

c. The Hicksian compensating variation in income is that amount of M,holding the price of X constant at its higher level of $15, that will allow theperson to as well off as they were before the price increase. In Hicksian termsof course, being equally well off means having the same level of utility.

Original utility was U = XY = (333.33)(500) = 166,665.

Methodological footnote: Note that any “monotonic transformation” of thisutility function such as U = X 2 Y 2, or U = X . 5 Y .5 will only change the totalnumber of “utils” (which is irrelevant, since utility in this context is only anordinal concept (ordering from “less to more’ rather than a “cardinal” concepthaving a specific quantity). Thus, when X and Y are especially large numbers,it is easier to use the “square root” transformation above to generate lowerand easier to work with numbers.

Back to the problem: When the price ratio of X to Y had been 10/15, weobserved above that Y = .67 X and X = 1.5 Y. Now, with both prices equal to$15, the price ratio is 1:1, and Y = X, meaning that the utility maximizingbundle at the new relative price ratio on the original indifference curve musthave the same quantity of X and Y. Substitution allows the following result:

U = XY, but if X=Y, U = X 2 or original utility 166,665 = X 2 or X = 408.25 and Y= 408.25. Note that on a graph, these would be the quantities at thetangency point of the shifted budget line (with the higher relative price of X)and the original indifference curve.

Final step is to calculate the amount of M that must be spent to achieve X = Y= 408.25, which is M = $15 (408.25) + $15 (408.25) = $12,247.50. Original Mwas $10,000, so the Hicksian compensating variation of income (HCV) =$12,247.50 - $10,000 = $2,247.50.

The Slutsky compensating variation (SCV) is much easier to calculate: At thenew prices the money income required to consume the original X,Y bundle ofX = 500, Y = 333.33 is simply: M = $15 (500) + $15 (333.33) = $12,499.95.This is the money income required to allow a budget line at the new slope(with higher price of X) to go through the original consumption point. Since$12,499.95 - $10,000 = $2,499.95, that is the SCV. Note that in this case, withthe price of X having gone up, the SCV > HCV by $252.45.

So, to summarize the answer to this part: HCV = $2,247.50 and SCV =$2,499.95

d. Finally, if no compensating variation is actually paid, the full reduction inthe consumption of X is from 500 to 333.33 or 167.67. How much of this167.67 reduction is due to a pure substitution effect and how much is due toa real income effect. We can rely on the analysis in c to derive the results forboth the Slutsky and the Hicksian analysis.

Hicksian analysis: We found above that if the real income effect is eliminatedby “hypothetically” (in this case) giving the person another $2,247.50, thenew point on the original indifference curve is X = 408.25, Y = 408.25.Therefore, the movement along that original indifference curve representingthe pure substitution effect is 500 – 408.25 = 91.75. Then, the remainingchange in X of 408.25 – 333.33 = 74.92 is the real income effect (the result ofnow taking that $2,247.50 away from the person, so there is a parallel shift tothe left to the lower indifference curve at X = 333.33 and Y = 333.33).

Slutsky derivation of substitution and income effects: We found above thatthe elimination of the real income effect as defined by Slutsky would requirea “hypothetical” increase in M of $2,499.95 to M = $12,499.95. We alsoknow as stated earlier that with P(x) = P(y), the first order condition requiresthat Y = X. Thus, we can calculate the point on the higher utility indifferencecurve that can be achieved with $12,499.95 (and the more steeply slopedbudget line incorporating the higher price of X) as $12,499.95 = $15 X + $15 Y,or since X = Y, $12,499.95 = 15 X + 15 X, so X = 416.67, and Y = 416.67.

Therefore, the pure substitution effect related to X is 500 – 416.67 = 83.33and the real income effect is then the “residual” of 416.67 – 333.33 = 83.34(essentially equal, just a rounding difference).

To summarize the analysis:

Hicks or Hicksian compensating variation in income = $2,247.50

Slutsky compensating variation in income = $2,499.95

Hicksian derived pure substitution effect on X = 91.75

Hicksian derived real income effect on X = 74.92

Slutsky derived pure substitution effect on X = 83.33

Slutsky derived real income effect on X = 83.34

In this case, following a price increase for X, the Slutsky approach understatesthe pure substitution effect by giving “too much” compensating variation ofincome to the person (and thus attributing part of the “real” substitutioneffect erroneously to an income effect). Thus, the Slutsky real income effectis “too large” and its pure substitution effect is “too small” relative to themore theoretically correct, but more difficult to measure, Hicksian analysis.

3.4 Problem Exercises: Roy’s identity

Exercise

The Marshall, Hicks and Slutsky Demand Curves

Graphical Derivation

In this part of the diagram we have drawn the choice between x on the horizontal axis and y on the vertical axis. Soon we will draw an indifference curve in here.

Down below we have drawn the relationship between x and its price Px.

This is effectively the space in which we draw the demand curve.

We start with the following diagram:

Next we draw in the indifference curves showing the consumers’ tastes for x and y.

Then we draw in the budget constraint and find the initial equilibrium.

Recall the slope of the budget constraint is:

From the initial equilibrium we can find the first point on the demand

Projecting x0 into the diagram below, we

map the demand for xat px

Next consider a rise in the price of x, to px

1. This causes the budget constraint to swing in as – px

1/py0 is

greater.

To find the demand for xat the new price we

locate the new equilibrium quantity of x

demanded.

Then we drop a line down from this point to

the lower diagram.

This shows us the new level of demand at p1

We are now in a position to draw the ordinary demand curve.

First we highlight the px and x combinations we have found in the lower diagram and then connect them with a line.

This is the Marshalliandemand curve for x.

Our next exercise involves giving the consumer enough income so that they can reach their original level of utility U2.

To do this we take the new budget constraint and gradually increase the agent’s income, moving the budget constraint out until we reach the indifference curve U2

The new point of tangency tells us the

demand for x when the consumer had been

compensated so they can still achieve utility

level U2, but the relative price of x and y has

risen to px1/py

The level of demand for x represents the pure substitution effect of the increase in the price of x.

This is called the Hicksian demand for xand we will label it xH.

We derive the Hicksiandemand curve by projecting the demand for x downwards into the demand curve diagram.

Notice this is the compensateddemand for x when the price is px

To get the Hicksian demand curve we connect the new point to the original demand x0px

Notice that the Hicksiandemand curve is steeper than the Marshalliandemand curve when the good is a normal good.

We label the curve Hx

Notice that an alternative compensation scheme would be to give the consumer enough income to buy their original bundle of goods x0yo

In this case the budget constraint has to move out even further until it goes through the point x0y0Hx

But now the consumer doesn’t have to consume x0y0

So they will choose a new equilibrium point on a higher indifference curve.

Once again we find the demand for x at this new higher level of income by dropping a line down from the new equilibrium point to the x axis.

We call this xs . It is the Slutsky demand.

Once again this income compensated demand is measured at the price px

Finally, once again we can draw the Slutskycompensated demand curve through this new point xspx

1 and the original x0px

The new demand curve Sx is steeper than either the Marshallianor the Hicksian curvewhen the good is normal.

We can derive three demand curves on the basis of our indifference curve analysis.

Summary

1. The normal Marshalliandemand curve2. The Hicksian compensated demand curve where agents are given sufficient income to maintain them on their original utility curve.

3. The Slutsky income compensated demand curve where agents have sufficient income to purchase their original bundle.

Finally, for a normal good the Marshallian demand curve is flatter than the Hicksian, which in turn is flatter than the Slutskydemand curve.

Problems to consider

1. Consider the shape of the curves if X is an inferior good.

2. Consider the shape of each of the curves if X is a Giffen good.

3. Will it matter if Y is a Giffen or an inferior good?

Please watch this video!• A.10 Marshallian and Hicksian demand curves | Consumption -

Microeconomics

Policonomics

https://www.youtube.com/watch?v=T4GAFG8TVt0

A.9 Income and substitution effects | Consumption - Microeconomics

https://www.youtube.com/watch?v=w9z_I-5QZ0o

• Example Income and Substitution Effects For Normal and Inferior Goods Economicsfun

https://www.youtube.com/watch?v=pLhh_D5b_Lg&ebc=ANyPxKqwG8yJUL72V2SwEbht8KNHh6Ipdu7qx6upHYo50yTRUgfMm27-DivPovsb200-Gm8-S63l

Best Motivational Video For Students - Don't Count The Cost

https://www.youtube.com/watch?v=vVJJj9hshCM

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