Optimal Choice

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Optimal Choice in the Theory of Consumer Behavior

Important Ideas and Vocabulary
Optimal choice, x1* and x2*, is where satisfaction is maximized given the budget constraint.
Two Approaches to Finding the Optimal Solution:
1) Analytical
2) Numerical
At the optimal solution, MRS = p1/p2.

Finding the Initial Optimal Solution
c

d

max u( x1 , x 2 )=x 1 x 2
x1 , x 2

s . t . m≥ p1 x 1 + p2 x 2
Goal
max Utility

200

40
35
30

Endogenous Variables
x1
20
x2
10

x2

25
20
15

Exogenous Variables
p1
2
p2
3
m
100
c
1
d
1
Constraint

price of x1
price of x2
income
exponent for x1
exponent for x2

10

9
9
9

5
0
0

10

20

30
x1

30 income left over
-p1/p2
-0.667

MRS at x1, x2
-0.500

40

50

60

40

50

60
Axes locked down.
Double-click an axis to unlock the scale.

U
step

200 fixed level of utility
3 amount of the change in x1

x1
8
11
14
17
20
23
26
29
32

U = 200
200
x2
25
18.18182
14.28571
11.76471
10
8.695652
7.692308
6.896552
6.25

x2/x1
-2.272727
-1.298701
-0.840336
-0.588235
-0.434783
-0.334448
-0.265252
-0.215517

dx2/dx1
x1 budget c
-3.125
0
-1.652893
5
-1.020408
10
-0.692042
15
-0.5
20
-0.378072
25
-0.295858
30
-0.237812
35
-0.195313
40
45
50

change U

x2 budget constraint
33.333333
30
26.666667
23.333333
20
16.666667
13.333333
10
6.6666667
3.3333333
0

x1
8
11
14
17
20
23
26
29
32

100

U = 300
300
x2
37.5
27.27273
21.42857
17.64706
15
13.04348
11.53846
10.34483
9.375

x2/x1
-3.409091
-1.948052
-1.260504
-0.882353
-0.652174
-0.501672
-0.397878
-0.323276

dx2/dx1
-4.6875
-2.479339
-1.530612
-1.038062
-0.75
-0.567108
-0.443787
-0.356718
-0.292969

x1
8
11
14
17
20
23
26
29
32

U = 100
100
x2
12.5
9.090909
7.142857
5.882353
5
4.347826
3.846154
3.448276
3.125

x2/x1
-1.136364
-0.649351
-0.420168
-0.294118
-0.217391
-0.167224
-0.132626
-0.107759

dx2/dx1
-1.5625
-0.826446
-0.510204
-0.346021
-0.25
-0.189036
-0.147929
-0.118906
-0.097656

Corner

Questions
Open Word and answer the following questions. Save the document and print it when you are done.
1) Find the optimal solution, using Solver, for the problem in the OptimalChoice sheet.

2) Insert a new sheet in the workbook (execute Insert: Worksheet) and solve the same problem as in the OptimalChoice sheet
except that the utility function is ln U = c ln x1 + d ln x2.
Take a picture of the cells in your worksheet that includes the goal, endogenous variables, exogenous variables, and constrain

WARNING: Do not simply copy the cells from the OptimalChoice sheet and paste them in your new sheet because there will b
Excel will continue to use the named cells from the OptimalChoice sheet instead of the new sheet.
It is recommended that you set up the optimization problem on your new sheet "by hand"—while a bit tedious, this is good pra
3) Compare the x1*, x2*, and U* from the OptimalChoice and ln U problems. What effect does changing the utility to natural
log have on the solution? Why does the natural log transformation have this effect?

4) Return to your natural log version of the utility function sheet and change c to 2. Solve the problem via Solver and mathema
Hint: Remember that dlnx/dx = 1/x dx so, e.g., dlnU/dx1 = c/x1.
Take a picture of Solver's answer and use Word's Equation Editor as needed to show your answer via the analytical approach.
5) Compare the x1* and x2* from problems 3 and 4. Do the different decisions make common sense? Explain.

6) Since U* is higher for the consumer in question 4 than in question 3, can we conclude that the Q4 consumer is better off tha

blem as in the OptimalChoice sheet,
exogenous variables, and constraint.

your new sheet because there will be a serious Named cells confusion.

—while a bit tedious, this is good practice.

does changing the utility to natural

he problem via Solver and mathematically.
answer via the analytical approach.

mon sense? Explain.

hat the Q4 consumer is better off than the Q3 consumer? Explain.

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