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Citeas: Chia-HuiChen,coursematerialsfor14.01PrinciplesofMicroeconomics,Fall2007. MITOpenCourseWare(http://ocw.mit.edu),Massachusetts InstituteofTechnology. Downloadedon [DDMonthYYYY].

11CornerSolutionofOptimization

14.01 Principles of Microeconomics, Fall 2007ChiaHui Chen

September 17, 2007

Lecture6Optimization,RevealedPreference,andDeriving IndividualDemand

Outline1. Chap 3: CornerSolutionofOptimization2. Chap 3: RevealedPreference3. Chap 4: Deriving IndividualDemand,EngleCurve1 CornerSolutionofOptimizationWhen we have an interior solution,

Px Ux=

Py Uymust be satisfied. However, sometimes a consumer gets highest utility levelwhen x= 0 or y= 0. If thats the case, we have corner solutions, and

Px Ux= ,Py Uy

as shown in Figure 1.In Figure 1, because people cannot consume negative amounts of goods

(bundle A), their best choice is to consume bundle B, so the quantity of yconsumed is zero. Conditions for corner solutions:

Ux Px

M RS= > when y= 0.Uy Py

Ux Px

M RS=

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21CornerSolutionofOptimization

Figure 1: Corner Solution to Consumers Problem.

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32RevealedPreference

I= 2.The utility function is U(x, y) = x+ 2y.The budget constraint is

x+ y= 2.According to the condition for an interior solution:

Px Ux= .

Py Uy=

1 1= .

1 1y

= y= 1 = x= 1.

If the price ychanges to 1:Py = 1,

then the solution isy= 4 = x= 3

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43Deriving IndividualDemand,EngleCurve

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

X: 1.478Y: 3.761

y

X: 1.929Y: 5.142

X: 4.751Y: 2.124

X: 3.949Y: 1.101

B

CA

D

x

Figure 2: A Contradiction of Preference. A and B are the Choices.

Example (Revealed preference). In Figure 2, two budget constraint lines inter-sect. Assume one persons choices are A and B respectively. Then we have

AC,BD.

And Figure 2 obviously shows that

C B,D A.

Thus,AC BD A,

which is a contradiction, which means utility does not optimized and the choiceis not rational.

3 Deriving IndividualDemand,EngleCurveUse the following utility function again:

U(x, y) = x+ 2y,with a budget constraint:

Pxx+ Pyy= I.

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WhenP2

I x

,Pywe have an interior solution. M RS= Px/Py. Thus,

I Pxx= ,

Px Py 2

Pxy= .

PyWhen

P2I x,

Pywe have a corner solution.

x= 0,I

y= .Py

Figure 3 shows a demand function ofyand Py as an example. (Assumethat I, xand Px are held constant.)

Engle Curve describes the relation between quantity and income. Figure4 shows the relation between x and income, and Figure 5 shows thatbetween yand income.Normalgood. Quantity demanded of good increases with income.Inferiorgood. Quantity demanded of good decreases with income.Substitutes. Increase in price of one leads to an increase in quantity

demanded of the other.

Complements. Increase in price of one leads to an decrease in quantitydemanded of the other.

For this problem,

P2x ifI

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Figure 3: Demand Function for Goods y.

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9

10

0 1 2 3 4

8

7

6

0

1

2

3

4

5x

Px

2/P

y

5 6 7 8 9 10I


Figure 4: The Relation between Income and Quantity Demanded of x. Englecurve ofx.

0

1

2

3

4

5

6

7

8

9

10

y

Px

2/P

y

0 1 2 3 4 5 6 7 8 9 10

I

Figure 5: The Relation between Income and Quantity Demanded of y. Englecurve ofy.


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