Question

Exercise 14:

Assume preferences can be represented by the following utility function: u(x1,x2)=−x12 +100x1 +20x2

a. Is the utility function monotonic? Justify.

b. Set up the consumer’s utility maximization problem for prices p1, p2 and income m (the general case)

c. Solve the problem. You will obtain demand functions x∗1 (p1 , p2 , m) and x∗2 (p1, p2, m) in terms of the parameters (p1, p2, m) .

d. Graph the demand function for good 1 when the price of good 2 is p2 = 2 and income is m = 200.

e. Obtain the change in consumer surplus when the price of good 1 goes from p1 = 2 to p,1 = (B + 7) /2, where B is 0.

f. Again, assuming the price of good 1 increases to p,1 = (B + 7) /2. Find the Compensating and the Equivalent Variations

g. For the same price increase, obtain the income and substitution effects on good 1, both with Slutsky and Hicks compensations.

Need help with letter e

Answer 1

Solution.

(E) The consumer surpules at P=2, is the area of ABC

So ,The area of

ABC = 1/2 (40) (8)

= 160

Consumer Surplus at P=4 , the area of  AEF

The area of

AEF = 1/2 (30)(6)

=90

So the change in the consumer surplus is (-70) i.e (90-160)

(F) The compensating Variation will be as follows

When X1 = 40, then with M = 200

From the budget Constraint

P1X1 + P2X2 = M

X2 = (200-2*40)/2

= 60

Initial bundle ( X1,X2) = (40,60)

So U1 ( initial utility ) = -402 +100*40 +20*60

= -1600+4000+1200

= 3600

When the price rise to 4, then X1 = 30

X2 = (200-30*4)/2 = 60

New bundle ( X1, X2) = (30, 60)

New Utility U2 = -302+100*30 + 20*60

= -900 + 3000 + 1200

= 3300

So with new prices & original Utility level,

let new income level = M`

So , X1` = 30,

With new BC: P1`X1` + P2X2 = M`

X2 = (M` - 4*30)/2

= (M` - 120)/2

So from utility level

-(30)2 + 100*30 + 20*(M`-120)/2 = 3600

So, -900+3000 + 10M` - 1200 = 3600

10M` = (3600+900-3000+1200)

10M` = 2700

M` = $270

So the amount paid = $ 70 i.e(270-200)

(G). the income effect is positive. Therefore, when price of a normal good falls and results in increase in the purchasing power, income effect will act in the same direction as the substitution effect, that is, both will work towards increasing the quantity demanded of the good whose price has fallen.

The income effect expresses the impact of increased purchasing power on consumption, while the substitution effect describes how consumption is impacted by changing relative income and prices. ... Some products, called inferior goods, generally decrease in the consumption whenever incomes increase.

The income effect is the change in consumption patterns due to a change in purchasing power. This occurs with income increases, price changes, and even currency fluctuations. Since income is not a good in and of itself (it can only be exchanged for goods and services), price decreases increase purchasing power

Slutsky and Hicks ... The Slutsky Equation shows the relative changes between the Marshallian demand and the Hicksian demand functions. ... The demand changes based on the consumer's preferences, their income, and the price of goods. Hicks Demand Function is otherwise known as the Compensated Demand Function.

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