Topic 5.3
Topic 5.3
I. Monotonic Transformations of a Cobb-Douglas
II. Utility Maximization
III. Using Tangency Conditions to Derive Demand
a. Complements (Perfect and Imperfect)
b. Substitutes
c. Cobb Douglas
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Slides on Monotonic Transformation of a Cobb Douglas | ECN 328, Study notes of Economics
Material Type: Notes; Class: Int Microeconomic Analys; Subject: Economics; University: Marshall ; Term: Unknown 1989;
Typology: Study notes
Pre 2010
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Topic 5. I. Monotonic Transformations of a Cobb-Douglas II. Utility Maximization III. Using Tangency Conditions to Derive Demand a. Complements (Perfect and Imperfect) b. Substitutes c. Cobb Douglas
A Special monotonic transformation of a Cobb-Douglas (C-D) Utility Function EX 1: V (x 1 x 2 ) = c ln x 1 + d ln x 2 Is this function a monotonic transformation of a C-D Utility function? Recall the rules for natural logarithms:
- b ln a = ln a b
- ln a + ln b = ln (a * b) Then, V (x 1 x 2 ) = c ln x 1 + d ln x 2 = ln x (^1) c + ln x 2 d (Using rule 1 above) = ln ( x (^1) c * x 2 d ) (Using rule 2 above) Note: If c + d = 1, then V (x 1 x 2 ) = ln ( x (^1) c * x 2 d ) is a monotonic transformation of a C-D Utility function.
Utility Maximization Max U(x 1 , x 2 ) s. t. y = p 1 x 1 + p 2 x 2 Given the budget constraint, you want to obtain the highest U curve possible. This tangency represents the First Order Conditions (FOC) -At the point of tangency: slope of U curve = slope of b.c. MRS = -p 1 p 2 slope of b.c. Slope of Indifference Curve
Note: Since monotonic transformations represent the same preferences, they will have the same FOC’s (tangency condition). We solve this tangency condition to find this consumer’s demand for good 1 (denoted x 1
) and demand for good 2 (denoted x 2
). The general form for the demand is: x 1
= x 1 (p 1 p 2 y) and x 2
= x 2 (p 1 p 2 y)
Imperfect Complements example: Little feminist Frances enjoys playing with her dolls. However, she only enjoys it if she has the combo of 2 Ken dolls for every Barbie doll. A) Draw some of her indifference curves. B) Write down 4 possible U curves. U=min {2B,K} U=min {4B,2K} U=min {B,1/2K} U=min {1/2 B, ¼ K} C) Give expressions for Frances demand for B and K. Find K *: Find B
PB B+ Pk K= Y PB B+PK K= Y PB * (½ K) + Pk K= Y PBB+PK*( 2B)= Y K(½PB + PK )= Y B(PB+2PK)= Y K = ___Y____ B
= Y ½ PB + PK PB +2PK
D) If PK= $1.00 and PB= $1.00 and Y=$6. K
= 6 = 6 = 6 x 2/3= 4 ½ +1 3/ B
= 6 =6 = 2 1+2 3 Ken intercept → Y/ PK = 6/1 = 6 Barbie intercept → Y/ PB = 6/1 = 6
Ex: Remember Joe who likes milk but doesn’t care about the carton size. x 1 = 8 oz. milk x 2 = 16 oz. milk Since Joe would need to consume 2 units of x 1 or 1 unit of x 2 we found his utility can be represented U = x 1 + 2x 2 If p 1 = $1.00 y = $ p 2 = $2. Note: The consumer would choose to consume only good 1, since: ap 1 < bp 2 2 ($1) < 1 ($2.05). To find demand: x 1
= y = 20 = 20 p 1 1 x 2
= 0
If we wanted to graph this scenario: Budget constraint in slope intercept form p 1 x 1 + p 2 x 2 = m x 1 + 2.05x 2 = 20 2.05x 2 = 20- x 1 x 2 = 20 – x 1 2.05 2. x 2 = 9.7561 – x 1
Form for the family of utility curves: 2x 2 = U – x 1 x 2 = U/2 – ½ x 1 →slope = – ½
To find the x 1 intercept (x 2 =0):
- 0 = 11.43 – x - 1.
- 11.43 = x
- x 1 = 19.
3. Cobb-Douglas Graph: Rule for finding demand: (To get derivation of demand, see appendix) x 2 * = y d x 1 * = y c p 2 c + d p 1 c + d
Max c ln x 1 + d ln x 2 st. p 1 x 1 + p 2 x 2 = y where c + d ≠ 1 = c ln x 1 + d ln x 2 + (y – p 1 x 1 – p 2 x 2 ) FOC 1. dL = c - p 1 = 0 dx 1 x 1
- dL = d - p 2 = 0 dx 2 x 2
- dL = y – p 1 x 1 – p 2 x 2 = 0 d Steps: A) Solve FOCs 1 and 2 for .
- c = p 1 2. d = p 2 x 1 x 2 c = d = p 1 x 1 x 2 p 2 B) Set the two equal to each other c = d c x 2 = p 1 p 1 x 1 x 2 p 2 x 1 d p 2 MRS = -MU 1 = -p 1 MU 1 = p 1 MU 2 p 2 MU 2 p 2 C) Solve for x 1 or x 2 & plug into 3 rd FOC. If you solve for x 1 , find x 2
. If you solve for x 2 , find x 1
.
x 1 = c x 2 p 2 x 1 = c p 2 x 2 p 1 d d p 1 y = p 1 ((c/d) (p 2 /p 1 ) x 2 ) + p 2 x 2 = x 2 ((c/d) p 2 + p 2 ) Take MRS = p 1 / p 2 → Tangency condition c x 2 = p 1 d x 1 p 2 Solve for x 1 , and put into b.c. to find x 2
. x 2
= y d Before we had x 1
= y c p 2 c + d p 1 c + d Demand eqs for C/D type.