quick reminder of the theory of consumer choice professor roberto chang rutgers university january...
Post on 20-Jan-2016
215 views
TRANSCRIPT
![Page 1: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/1.jpg)
Quick Reminder of the Theory of Consumer Choice
Professor Roberto Chang
Rutgers University
January 2007
![Page 2: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/2.jpg)
• Reminder of Theory of Consumer Choice, as given by Mankiw, Principles of Economics, chapter 21, and other elementary textbooks.
![Page 3: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/3.jpg)
A Canonical Problem
• Consider the problem of a consumer that may choose to buy apples (x) or bananas (y)
• Suppose the price of apples is px and the price of bananas is py.
• Finally, suppose that he has I dollars to spend.
![Page 4: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/4.jpg)
The Budget Set
• The budget set is the set of options (here, combinations of x and y) open to the consumer.
• Given our assumptions, the total expenditure on apples and bananas cannot exceed income, i.e.
px x + py y ≤ I
![Page 5: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/5.jpg)
• Rewrite
px x + py y = I
as
y = I/py – (px/py) x
This is the budget line
![Page 6: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/6.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
![Page 7: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/7.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
![Page 8: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/8.jpg)
Apples (x)
Bananas (y)
Budget Line:
px x + py y = I
(Slope = - px/py)
O I/px
I/py
![Page 9: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/9.jpg)
• If I increases, the new budget line is higher and parallel to the old one.
![Page 10: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/10.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
![Page 11: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/11.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
I’/px
I’/py
I’ > I
![Page 12: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/12.jpg)
• If px increases, the budget line retains the same vertical intercept, but the horizontal intercept shrinks
![Page 13: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/13.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
![Page 14: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/14.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
I/ px’
px’ > px
![Page 15: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/15.jpg)
Preferences
• Now that we have identified the options open to the consumer, which one will he choose?
• The choice will depend on his preferences, i.e. his relative taste for apples or bananas.
• In Economics, preferences are usually assumed to be given by a utility function.
![Page 16: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/16.jpg)
Utility Functions
• In this case, a utility function is a function U = U(x,y) , where U is the level of satisfaction derived from consumption of (x,y).
• For example, one may assume that
U = log x + log y
or that
U = xy
![Page 17: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/17.jpg)
Indifference Curves
• It is useful to identify indifference curves. An indifference curve is a set of pairs (x,y) that yield the same level of utility.
• For example, for U = xy, an indifference curve is given by setting U = 1, i.e.
1 = xy
• A different indifference curve is given by the pairs (x,y) such that U = 2, i.e. 2 = xy
![Page 18: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/18.jpg)
x
y
Utility = u0
Three Indifference Curves
![Page 19: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/19.jpg)
x
y
Utility = u0
Utility = u1
Three Indifference Curves
Here u1 > u0
![Page 20: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/20.jpg)
x
y
Utility = u0
Utility = u1
Utility = u2
Three Indifference Curves
Here u2 > u1 > u0
![Page 21: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/21.jpg)
Properties of Indifference Curves
• Higher indifference curves represent higher levels of utility
• Indifference curves slope down
• They do not cross
• They “bow inward”
![Page 22: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/22.jpg)
Optimal Consumption
• In Economics we assume that the consumer will pick the best feasible combination of apples and bananas.
• “Feasible” means that (x*,y*) must be in the budget set
• “Best” means that (x*,y*) must attain the highest possible indifference curve
![Page 23: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/23.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
Consumer Optimum
![Page 24: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/24.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
x*
y* C
Consumer Optimum
![Page 25: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/25.jpg)
Apples (x)
Bananas (y)
O I/px
I/py
x*
y* C
Consumer Optimum
![Page 26: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/26.jpg)
Key Optimality Condition
• Note that the optimal choice has the property that the indifference curve must be tangent to the budget line.
• In technical jargon, the slope of the indifference curve at the optimum must be equal to the slope of the budget line.
![Page 27: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/27.jpg)
The Marginal Rate of Substitution
• The slope of an indifference curve is called the marginal rate of substitution, and is given by the ratio of the marginal utilities of x and y:
MRSxy = MUx/ MUy
• Recall that the marginal utility of x is given by ∂U/∂x
![Page 28: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/28.jpg)
• Quick derivation: the set of all pairs (x,y) that give the same utility level z must satisfy U(x,y) = z, or U(x,y) – z = 0. This equation defines y implicitly as a function of x (the graph of such implicit function is the indifference curve). The Implicit Function Theorem then implies the rest.
![Page 29: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/29.jpg)
• Intuition: suppose that consumption of x increases by Δx and consumption of y falls by Δy. How are Δx and Δy to be related for utility to stay the same?
• Increase in utility due to higher x consumption is approx. Δx times MUx
• Fall in utility due to lower y consumption = -Δy times MUy
• Utility is the same if MUx Δx = - MUy Δy, i.e. Δy/ Δx = - MUx/ MUy
![Page 30: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/30.jpg)
• For example, with U = xy,
MUx = ∂U/∂x = y
MUy = ∂U/∂y = x
and
MRSxy = MUx/ MUy = y/x
• Exercise: Find marginal utilities and MRSxy
if U = log x + log y
![Page 31: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/31.jpg)
• Back to our consumer problem, we knew that the slope of the budget line is equal to the ratio of the prices of x and y, px/py. Hence the optimal choice of the consumer must satisfy:
MUx/ MUy = px/py
![Page 32: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/32.jpg)
Numerical Example
• Let U = xy again, and suppose px = 3, py = 3, and I = 12.
• The budget line is given by
3x + 3y = 12
• Optimal choice requires MRSxy = px/py, that is,
y/x = 3/3 = 1
• The solution is, naturally, x = y = 2.
![Page 33: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/33.jpg)
Changes in Income
• Suppose that income doubles, i.e. I = 24. Then the budget line becomes
3x + 3y = 24
• The MRS = px/py condition is the same, so now
x = y = 4
![Page 34: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/34.jpg)
x
y
O
C
I/px
I/py
![Page 35: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/35.jpg)
x
y
O
C
I/px
I/py
An increase in income
I’ > I
I’/px
I’/py
![Page 36: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/36.jpg)
x
y
O
C
I/px
I/py
An increase in income
I’ > I
I’/px
I’/py
C’
![Page 37: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/37.jpg)
• In the precious slide, both goods are normal. But it is possible that one of the goods be inferior.
![Page 38: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/38.jpg)
x
y
O
C
I/px
I/py
An increase in income, Good y inferior
I’ > I
I’/px
I’/py
C’
![Page 39: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/39.jpg)
Changes in Prices
• In the previous example, suppose that px falls to 1.
• The budget line and optimality conditions change to
x + 3 y = 12
y/x = 1/3
• Solution: x = 6, y = 2.
![Page 40: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/40.jpg)
x
y
O
C
I/px
I/py
![Page 41: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/41.jpg)
x
y
O
C
Effects of a fall in px
px > px’
I/px I/px’
I/py
![Page 42: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/42.jpg)
x
y
O
C’C
Effects of a fall in px
px > px’
I/px I/px’
I/py
![Page 43: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/43.jpg)
• If x is a normal good, a fall in its price will result in an increase in the quantity purchased (this is the Law of Demand)
• This is because the so called substitution and income effects reinforce each other.
![Page 44: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/44.jpg)
x
y
O
C’C
I/px I/px’
I/py
![Page 45: Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007](https://reader036.vdocuments.mx/reader036/viewer/2022062806/56649d4c5503460f94a2a6b6/html5/thumbnails/45.jpg)
x
y
O
C’C
Substitution vs Income Effects
I/px I/px’
I/py
C’’