Unformatted text preview: Example:
If MU1 = 10 and P1 = 5
Then, MU1/P1 = 10/5 = 2
This means that the last dollar spent on x1 gives the consumer a utility of 2 units.
WHAT IF THE CONSUMER IS NOT AT THE OPTIMAL CHOICE?
Imagine that you have the following: MU 1 MU 2
>
P1
P2 In English this means that:
The extra utility obtained from the last dollar spent on x1 > The extra utility obtained from the last dollar spent on x2 This is implies that since the consumer gets a bigger bang from the last dollar spent on x1. Therefore, the consumer should consumer more of x1 since it gets more satisfaction from the last dollar spent. Also, the consumer should consume less of x2 because she gets less satisfaction from the last dollar spent on x2. MU 1 MU 2
>
the consumer increases x1 and reduces x2:
P1
P2
When that happens we have that:
MU 1
x 1 ↑⇒ MU 1 ↓⇒
↓
MU 1 MU 2
P1
⇒ this process continues until
=
P1
P2
MU 2
x 2 ↓⇒ MU 2 ↑⇒
↑
P2 Thus when Similary when MU 1 MU 2
<
P1
P2 the consumer decreases x1 and increase x2 until MU 1 MU 2
=
P1
P2 OPTIMAL CHOICE FOR OTHER UTILITY FUNCTIONS
The optimal choice condition we obtained before works only when we have well‐behaved utility functions. When we have Perfect Complements or Perfect Substitutes we cannot use the condition so we have to find the optimal choice in a different form.
1) Well‐behaved Utility Function:
Optimal Choice: MRS = ‐ P1/P2
x
2 Budget line I
P2 C
IC2
I
P1 x1 2) Perfect Substitutes Utility Function
Remember that in this case the IC’s are straight lines:
x2
U
b a
Slope = − = MRS
b IC1
U
a x1 However, the Budget Line is also a straight line, so this makes things a little bit different. In fact we are going to have three different cases. CASE A: The IC is steeper than the Budget Line
This means that we have the following:
x2
IC2 IC3 1) Notice that the IC’s are steeper than the Budget Line. IC4 2) The IC’s increase to the right. IC1 I
P2 Points W and Z are affordable and they exhaust the income of the consumer. However, they are not the optimal choice because the consumer can get higher utility by choosing an IC farther to the right. W Budget Line Point E gives the consumer high utility but it is not affordable because it is above the budget line. Z T I
P1 E x1 Point T is the optimal choice because it is affordable, exhausts all of the consumer’s income AND there is not a higher IC that the consumer can reach and stay within budget. Thus, at point T the optimal quantities of x1 and x2 are the following: *
x1 = I
P1 and *
x2 = 0 CASE B: The IC is flatter than the Budget Line
This means that we have the following:
x2 I
P2 1) Notice that the IC’s are flatter than the Budget Line. L 2) The IC’s increase to the right. M Points H and R are affordable and they exhaust the income of the consumer. However, they are not the optimal choice because the consumer can get higher utility by choosing higher IC’s. IC4 IC3 H IC2
Budget Line R IC1 I
P1 x1 Point L gives the consumer high utility but it is not affordable because it is above the budget line.
Point M is the optimal choice because it is affordable, exhausts all of the consumer’s income AND there is not a higher IC that the consumer can reach and stay within budget. Thus, at point L the optimal quantities of x1 and x2 are the following: *
x1 = 0 and *
x2 = I
P2 CASE C: The IC and the Budget Line have the same slope
This means that we have the following: 1) Notice that the IC’s have the same slope as the Budget Line. x2 2) The IC’s increase to the right. I
P2 The curves IC2 and the Budget Line are on top of each other. The optimal bundle will be any bundle on the budget line IC3
Budget Line
IC1 IC2 I
P1 x1 Now, remember that the Slope of the IC = MRS and also in the case of perfect substitutes MRS = ‐a/b
So we can say that the absolute value of the MRS is equal to a/b. Thus, MRS=a/b
Slope of the Budget Line = ‐ P1/P2
Thus, IF MRS=a/b > P1/P2 THEN THE IC IS STEEPER THAN THE BUDGET LINE
IF MRS=a/b < P1/P2 THEN THE IC IS FLATTER THAN THE BUDGET LINE
IF MRS=a/b = P1/P2 THEN THE IC AND THE BUDGET LINE HAVE THE SAME SLOPE
In other words, to know what line is steeper we have to compare a/b (from the utility function) to P1/P2 . That is, just looking at which has the highest number without any negative sign.
Before we move on we will summarize the perfect substitutes information in one table. SUMMARY OF CASES FOR PERFECT SUBSTITUTES CASE SLOPE SOLUTION GRAPH
x2 IC steeper than BL IC flatter than BL a/b>P1/P2
or
MRS>P1/P2
a/b<P1/P2
or
MRS<P1/P2 IC
BL X1 *= I/P1
X2* = 0 x1
x2 X1 *= 0
X2* = I/P2 IC
BL
x1 IC and BL have the same slope a/b=P1/P2
or
MRS=P1/P2 x2 Any point on the Budget Line BL
IC
x1 Example: Consider the following information
U = 4 x1 + 24 x2 P1 = 10 P2= 15 I = 2,800
Question 1: Graph the optimal solution
Question 2: Obtain the optimal choice of x1 and x2
The first thing to do is to figure out the slope of the IC and the budget line so that we can compare them and figure out which line is steeper. Step 1: Obtain a/b From before (see slide 14 of the utility topic) we know that for perfect substitutes the MRS is given by the following:
MRS = ‐ a/b (where U = a x1 + b x2)
Thus, in this case: MRS = ‐ a/b=‐ 4 / 24 = ‐ 0.17, so MRS=a/b =0.17
Step 2: Obtain P1/P2
P1/P2 = 10/15 = 0.67 Step 3: Compare a/b and P1/P2 to determine which line is steeper:
a/b= 0.17 < 0.67 = P1/P2 ‐> the Budget Line is steeper than the IC
Step 4: Graph it (try to do this first always to avoid confusion)
x2
I
2,800
=
= 186.7
P2
15 K IC Budget Line I
2,800
=
= 280
P1
10 x1 Step 5: Point K is the optimal solution. At K: X1* = 0 and X2*=186.7 3) Perfect Complements Utility Function
We have covered the optimal choice for the well‐behaved and the perfect complements utility functions. We will now go over the optimal choice when the utility function is the perfect complements:
Utility Function: U = min {ax1, bx2}
The corresponding IC’s are the following:
x2 IC2
IC1
x1 Putting together the budget line with the IC’s:
1) The IC’s increase to the right.
x2 H V Points V and R are affordable and they exhaust the income of the consumer. However, they are not the optimal choice because the consumer can get higher utility by choosing higher IC’s. IC3 T Point H gives the consumer high utility but it is not affordable because it is above the budget line. IC2
R IC1
x1 Point T is the optimal choice because it is affordable, exhausts all of the consumer’s income AND there is not a higher IC that the consumer can reach and stay within budget. Also, remember that at the corner of the IC Now, lets obtain the actual values for the optimal x1 and x2.
Step 1: Remember that at the corner of the IC we have that: ax1 = bx2
(see slide 13 of the utility power points) x2 ax1 = bx2 T IC2 x1 b
a Step 2: Solve for x1: ax 1 = bx 2 ; x 1 = x 2
Step 3: Use the budget equation, substitute x1 and solve for x2
b
⎛b
⎞
⎛b
⎞
P1 x 1 + P2 x 2 = I ; P1 ⎜ x 2 ⎟ + P2 x 2 = I ;
P1 x 2 + P2 x 2 = I ; x 2 ⎜ P1 + P2 ⎟ = I
a
⎝a
⎠
⎝a
⎠
*
x2 = I
b
P1 + P2
a
b
a x1 = x 2
Step 4: Substitute x2* into ⎛
b⎜
I
* b*
x1 = x 2 = ⎜
a
a⎜ b
⎜ P1 + P2
⎝a
⎛
I
* b⎜
x1 = ⎜
a⎜ b
⎜ P1 + P2
⎝a ⎞
⎟
⎟
⎟
⎟
⎠ ⎞
⎟
⎟
⎟
⎟
⎠ Example: Consider the following information
U =min{ 10 x1, 40 x2} P1 = 3 P2= 8 I = 2,000
Question 1:Obtain the optimal choice of x1 and x2
Question 2: Graph the optimal solution
Step 1: Obtain the expression for the corner: 10 x 1 = 40 x 2
Step 2: Solve for x1: 10 x 1 = 40 x 2 ; x 1 = 40
x 2 ; x1 = 4x 2
10 Step 3: Use the budget equation and substitute x1 and solve for x2
3 x 1 + 8 x 2 = 2000; 3 ( 4 x 2 ) + 8 x 2 = 2000; 12x 2 + 8 x 2 = 2000; 20x 2 = 2000
*
x 2 = 100 Step 4: Substitute x2* into x1: *
x 1 = 4 x 2 ; x 1 = 4(100) = 400; x 1 = 400 ...
View
Full
Document
This note was uploaded on 06/19/2011 for the course ECON 3357 taught by Professor Fidel during the Summer '11 term at Sam Houston State University.
 Summer '11
 Fidel
 Microeconomics, Utility

Click to edit the document details