Budget constraint, preferences, utility
Varian, Intermediate Microeconomics, 8e, chapters 2, 3, and 4
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In this lecture, you will learn
• what budget set and budget line are
• how their shape is influenced by taxes and food stamps
• what preferences are and how they are derived
• what the basic types of preferences are – why some indiference curves
are straight and some curved, or circle-shaped
• what we need a utility function for
• how to find out whether to reconstruct a stadium
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Budget constraint
We assume that the consumer chooses a bundle (x1, x2),
where x1 and x2 are quantities of goods 1 and 2.
Budget constraint is p1x1 + p2x2 ≤ m:
• p1 and p2 are prices of goods 1 and 2
• m is income
Budget set – bundles for which: p1x1 + p2x2 ≤ m.
Budget line (BL) – bundles for which: p1x1 + p2x2 = m.
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Budget set and budget line (graph)
Budget line: p1x1 + p2x2 = m
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Budget set and budget line (graph)
Budget line: p1x1 + p2x2 = m ⇐⇒ x2 = m/p2 − (p1/p2)x1
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Composite good
The theory works for more than two goods.
How to plot it in a 2D graph?
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Composite good
The theory works for more than two goods.
How to plot it in a 2D graph?
On the y axis we can plot the composite good
= money value of all other consumed goods.
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Change in income
A rise in income from m to m
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Change in income
A rise in income from m to m =⇒ parallel shift out
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Change in price
A rise in price from p1 to p1
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Change in price
A rise in price from p1 to p1 =⇒ pivot around the vertical intercept
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Change in more variables
Multiplying all prices and income by t...
tp1x1 + tp2x2 = tm
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Change in more variables
Multiplying all prices and income by t does not change BL:
tp1x1 + tp2x2 = tm ⇐⇒ p1x1 + p2x2 = m
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Change in more variables
Multiplying all prices and income by t does not change BL:
tp1x1 + tp2x2 = tm ⇐⇒ p1x1 + p2x2 = m
Multiplying all prices by t...
tp1x1 + tp2x2 = m
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Change in more variables
Multiplying all prices and income by t does not change BL:
tp1x1 + tp2x2 = tm ⇐⇒ p1x1 + p2x2 = m
Multiplying all prices by t has the same effect as dividing income by t:
tp1x1 + tp2x2 = m ⇐⇒ p1x1 + p2x2 =
m
t
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Numeraire
Any price or income can be normalized to 1 and adjust all variables so that
the BL stays the same.
Numeraire = an item with its value normalized to 1
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Numeraire
Any price or income can be normalized to 1 and adjust all variables so that
the BL stays the same.
Numeraire = an item with its value normalized to 1
Budget line p1x1 + p2x2 = m:
• Good 1 is numeraire – the same BL:
x1 +
p2
p1
x2 =
m
p1
• Good 2 is numeraire – the same BL:
p1
p2
x1 + x2 =
m
p2
• The income is numeraire – the same BL:
p1
m
x1 +
p2
m
x2 = 1
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Taxes and subsidies
Three types of taxes:
• Quantity tax – consumer pays the amount t for each unit.
→ Price of good 1 increases to p1 + t.
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Taxes and subsidies
Three types of taxes:
• Quantity tax – consumer pays the amount t for each unit.
→ Price of good 1 increases to p1 + t.
• Value tax (ad valorem) – consumer pays a share τ of price.
→ Price of good 1 increases to p1 + τp1 = (1 + τ)p1.
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Taxes and subsidies
Three types of taxes:
• Quantity tax – consumer pays the amount t for each unit.
→ Price of good 1 increases to p1 + t.
• Value tax (ad valorem) – consumer pays a share τ of price.
→ Price of good 1 increases to p1 + τp1 = (1 + τ)p1.
• Lump-sum tax – the value of the tax is independent from
consumer’s choice.
→ Consumer income decreases by the size of the tax.
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Taxes and subsidies
Three types of taxes:
• Quantity tax – consumer pays the amount t for each unit.
→ Price of good 1 increases to p1 + t.
• Value tax (ad valorem) – consumer pays a share τ of price.
→ Price of good 1 increases to p1 + τp1 = (1 + τ)p1.
• Lump-sum tax – the value of the tax is independent from
consumer’s choice.
→ Consumer income decreases by the size of the tax.
Subsidy = a tax with a negative sign
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Rationing
If there is rationing imposed on good 1, no consumer is allowed to buy a
higher quantity of good 1 than ¯x1.
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Rationing
If there is rationing imposed on good 1, no consumer is allowed to buy a
higher quantity of good 1 than ¯x1.
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Taxing consumption greater than ¯x1
If consumer pays a tax only on the consumption of good 1 that is in excess
of ¯x1...
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Taxing consumption greater than ¯x1
If consumer pays a tax only on the consumption of good 1 that is in excess
of ¯x1, budget line is steeper to the right of ¯x1.
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CASE: The food stamp program
Before 1979 (left graph):
• value subsidy – people pay a part of the value of the food stamp
• rationing – maximum value of stamps (e.g. 153 $)
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CASE: The food stamp program
Before 1979 (left graph):
• value subsidy – people pay a part of the value of the food stamp
• rationing – maximum value of stamps (e.g. 153 $)
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CASE: The food stamp program
Before 1979 (left graph):
• value subsidy – people pay a part of the value of the food stamp
• rationing – maximum value of stamps (e.g. 153 $)
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CASE: The food stamp program
Before 1979 (left graph):
• value subsidy – people pay a part of the value of the food stamp
• rationing – maximum value of stamps (e.g. 153 $)
After 1979 (right graph) – a specific number of food stamps for free
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CASE: The food stamp program
Before 1979 (left graph):
• value subsidy – people pay a part of the value of the food stamp
• rationing – maximum value of stamps (e.g. 153 $)
After 1979 (right graph) – a specific number of food stamps for free
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Preferences
Consumers compare bundles according to their preferences.
Preference relations – three symbols:
• bundle X is strictly preferred to bundle Y :
(x1, x2) (y1, y2)
• bundle X is weakly preferred to bundle Y
(bundle X is at least as good as bundle Y ):
(x1, x2) (y1, y2)
• consumer is indiferent between bundles X and Y :
(x1, x2) ∼ (y1, y2)
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Assumptions about preferences
Assumptions that allow ordering of bundles according to preferences:
• Completeness — any two bundles can be compared:
(x1, x2) (y1, y2), or (x1, x2) (y1, y2), or both
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Assumptions about preferences
Assumptions that allow ordering of bundles according to preferences:
• Completeness — any two bundles can be compared:
(x1, x2) (y1, y2), or (x1, x2) (y1, y2), or both
• Reflexivity — each bundle is at least as good itself: (x1, x2) (x1, x2)
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Assumptions about preferences
Assumptions that allow ordering of bundles according to preferences:
• Completeness — any two bundles can be compared:
(x1, x2) (y1, y2), or (x1, x2) (y1, y2), or both
• Reflexivity — each bundle is at least as good itself: (x1, x2) (x1, x2)
• Transitivity — if (x1, x2) (y1, y2) and (y1, y2) (z1, z2), then
(x1, x2) (z1, z2)
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Weakly preferred set and indifference curves
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Two indifference curves cannot cross
Two different IC such that X Y . Why cannot they cross?
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Two indifference curves cannot cross
Two different IC such that X Y . Why cannot they cross? It follows from
transitivity that if X ∼ Z and Z ∼ Y then X ∼ Y .
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Examples of preferences – perfect substitutes
Willingness to substitute one good for the other at a constant rate
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Examples of preferences – perfect substitutes
Willingness to substitute one good for the other at a constant rate =⇒
constant slope of the indifference curve (not necessarily −1).
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Examples of preferences – perfect complements
Consumption in fixed proportions (not necessarily 1:1).
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Examples of preferences – perfect complements
Consumption in fixed proportions (not necessarily 1:1).
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Examples of preferences – bads
The consumer likes pepperoni but does not like anchovies,
they are a bad for her.
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Examples of preferences – bads
The consumer likes pepperoni but does not like anchovies,
they are a bad for her.
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Examples of preferences – neutrals
The consumer likes pepperoni but is neutral about anchovies,
they are a neutral for her.
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Examples of preferences – neutrals
The consumer likes pepperoni but is neutral about anchovies,
they are a neutral for her.
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Examples of preferences – satiation point
Satiation point is the most preferred point (¯x1, ¯x2). When the consumer
has too much of one of the goods, it becomes a bad.
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Examples of preferences – satiation point
Satiation point is the most preferred point (¯x1, ¯x2). When the consumer
has too much of one of the goods, it becomes a bad.
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Examples of preferences – discrete goods
A discrete good is not divisible – consumption in integer amounts:
• indiference
”
curves“
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Examples of preferences – discrete goods
A discrete good is not divisible – consumption in integer amounts:
• indiference
”
curves“ – a set of discrete points
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Examples of preferences – discrete goods
A discrete good is not divisible – consumption in integer amounts:
• indiference
”
curves“ – a set of discrete points
• a weakly preferred set
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Examples of preferences – discrete goods
A discrete good is not divisible – consumption in integer amounts:
• indiference
”
curves“ – a set of discrete points
• a weakly preferred set – a set of line segments
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Well-behaved preferences
Assumptions of well-behaved preferences: monotonicity and convexity
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Well-behaved preferences
Assumptions of well-behaved preferences: monotonicity and convexity
Monotonicity – more is better (it excludes bads)
=⇒ indifference curves have negative slope.
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Well-behaved preferences (cont’d)
Convexity – if (x1, x2) ∼ (y1, y2), then it holds for all 0 ≤ t ≤ 1
that (tx1 + (1 − t)y1, tx2 + (1 − t)y2) (x1, x2).
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Well-behaved preferences (cont’d)
Convexity – if (x1, x2) ∼ (y1, y2), then it holds for all 0 ≤ t ≤ 1
that (tx1 + (1 − t)y1, tx2 + (1 − t)y2) (x1, x2).
Strict convexity – if (x1, x2) ∼ (y1, y2), then it holds for all 0 ≤ t ≤ 1
that (tx1 + (1 − t)y1, tx2 + (1 − t)y2) (x1, x2).
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Marginal rate of substitution
Marginal rate of substitution
(MRS) = slope of the indifference
curve:
MRS =
∆x2
∆x1
=
dx2
dx1
Diminishing marginal rate of
substitution – absolute value of
MRS decreases as we increase x1.
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Interpretation of marginal rate of substitution
Interpretation of MRS:
• The amount of good 2 one is willing to pay for one unit of good 1.
• If good 2 is measured in money: MRS = marginal willingness to
pay = how many dollars you would just be willing to give up for an
additional unit of good 1.
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APPLICATION: Build a stadium for Minnesota Vikings?
The club does not like the stadium – considers leaving Minnesota.
Fenn a Crooker (SEJ, 2009) measure how much households are willing to
pay for Vikings staying in Minnesota = MRS between composite good and
Vikings in Minnesota.
MRS of an average household: 531 $
Value of the stadium: 531 $ × 1,323 million households = 702 mil. $
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APPLICATION: Build a stadium for Minnesota Vikings?
The club does not like the stadium – considers leaving Minnesota.
Fenn a Crooker (SEJ, 2009) measure how much households are willing to
pay for Vikings staying in Minnesota = MRS between composite good and
Vikings in Minnesota.
MRS of an average household: 531 $
Value of the stadium: 531 $ × 1,323 million households = 702 mil. $
Estimated costs are 1 billion $.
The new stadium opens in 2016
– the state provided 500 million $.
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Utility
Two concepts of utility:
Cardinal utility – attach a significance to the magnitude
of utility:
• difficult to assign the magnitude
• not needed to describe choice behavior
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Utility
Two concepts of utility:
Cardinal utility – attach a significance to the magnitude
of utility:
• difficult to assign the magnitude
• not needed to describe choice behavior
Ordinal utility – important is only the order
of preference:
• easy to set the utility – 1 rule: preferred bundle
has a higher utility
• we can derive a complete theory of demand
We will use the ordinal utility.
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Ordinal utility
Utility function is a way of assigning a number to every possible
consumption bundle such that more-preferred bundles get assigned larger
numbers than less-preferred bundles.
If (x1, x2) (y1, y2), then u(x1, x2) > u(y1, y2).
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Ordinal utility
Utility function is a way of assigning a number to every possible
consumption bundle such that more-preferred bundles get assigned larger
numbers than less-preferred bundles.
If (x1, x2) (y1, y2), then u(x1, x2) > u(y1, y2).
Different ways to assign utilities that describe the same preferences:
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Monotonic transformation
Positive monotonic transformation f (u) = any increasing function of u.
Describes the same preferences as the original utility function u.
Examples of the function f (u): f (u) = 3u, f (u) = u + 3, f (u) = u3
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Monotonic transformation
Positive monotonic transformation f (u) = any increasing function of u.
Describes the same preferences as the original utility function u.
Examples of the function f (u): f (u) = 3u, f (u) = u + 3, f (u) = u3
Example:
Two bundles X and Y , preferences: X Y
We assign utility so that u(X) > u(Y ), e.g. u(X) = 1, u(Y ) = −1
Do monotonic transformations f1(u) = 3u a f2(u) = u + 3 represent the
same preferences as the original utility function u?
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Monotonic transformation
Positive monotonic transformation f (u) = any increasing function of u.
Describes the same preferences as the original utility function u.
Examples of the function f (u): f (u) = 3u, f (u) = u + 3, f (u) = u3
Example:
Two bundles X and Y , preferences: X Y
We assign utility so that u(X) > u(Y ), e.g. u(X) = 1, u(Y ) = −1
Do monotonic transformations f1(u) = 3u a f2(u) = u + 3 represent the
same preferences as the original utility function u?
Yes:
• f1(u) = 3u: f1(u(X)) = 3 > −3 = f1(u(Y ))
• f2(u) = u + 3: f2(u(X)) = 4 > 2 = f2(u(Y ))
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Construction of indifference curves from utility function
Utility function u(x1, x2) = x1x2
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Construction of indifference curves from utility function
Utility function u(x1, x2) = x1x2 =⇒ indifference curves x2 = k
x1
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PROBLEM: The slope of indifference curves
The slope of indifference curves for two utility functions:
1. What is the slope of IC x2 = 4/x1 v point (x1, x2) = (2, 2)?
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PROBLEM: The slope of indifference curves
The slope of indifference curves for two utility functions:
1. What is the slope of IC x2 = 4/x1 v point (x1, x2) = (2, 2)?
Slope of indifference curves = MRS =
dx2
dx1
=
−4
x2
1
= −1
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PROBLEM: The slope of indifference curves
The slope of indifference curves for two utility functions:
1. What is the slope of IC x2 = 4/x1 v point (x1, x2) = (2, 2)?
Slope of indifference curves = MRS =
dx2
dx1
=
−4
x2
1
= −1
2. What is the slope of IC x2 = 10 − 6
√
x1 v point (4, 5)?
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PROBLEM: The slope of indifference curves
The slope of indifference curves for two utility functions:
1. What is the slope of IC x2 = 4/x1 v point (x1, x2) = (2, 2)?
Slope of indifference curves = MRS =
dx2
dx1
=
−4
x2
1
= −1
2. What is the slope of IC x2 = 10 − 6
√
x1 v point (4, 5)?
Slope of indifference curves = MRS =
dx2
dx1
=
−3
√
x1
=
−3
2
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Examples of utility functions – perfect substitutes
The consumer is willing to exchange
• coke and pepsi at a ratio 1:1
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Examples of utility functions – perfect substitutes
The consumer is willing to exchange
• coke and pepsi at a ratio 1:1
important is the total number: e.g. u(K, P) = K + P
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Examples of utility functions – perfect substitutes
The consumer is willing to exchange
• coke and pepsi at a ratio 1:1
important is the total number: e.g. u(K, P) = K + P
• 2 buns for 1 baguette
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Examples of utility functions – perfect substitutes
The consumer is willing to exchange
• coke and pepsi at a ratio 1:1
important is the total number: e.g. u(K, P) = K + P
• 2 buns for 1 baguette
baguette has a double weight: e.g. u(R, H) = R + 2H
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Examples of utility functions – perfect complements
The consumer demands
• left and right shoes at a fixed ratio 1:1
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Examples of utility functions – perfect complements
The consumer demands
• left and right shoes at a fixed ratio 1:1
lower quantity matters: e.g. u(L, P) = min{L, P}
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Examples of utility functions – perfect complements
The consumer demands
• left and right shoes at a fixed ratio 1:1
lower quantity matters: e.g. u(L, P) = min{L, P}
• rum and coke at a fixed ratio 1:5
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Examples of utility functions – perfect complements
The consumer demands
• left and right shoes at a fixed ratio 1:1
lower quantity matters: e.g. u(L, P) = min{L, P}
• rum and coke at a fixed ratio 1:5
goal: same numbers in the bracket – we need only 1/5 of coke:
e.g. u(R, K) = min{5R, K}
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Examples of utility functions – quasilinear preferences
Indifference curves are vertically parallel (a practical property)
Utility function u(x1, x2) = v(x1) + x2, e.g. u(x1, x2) =
√
x1 + x2
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Examples of utility functions – Cobb-Douglas preferences
• A simple utility function representing well-behaved preferences.
• Utility function of the form u(x1, x2) = xc
1 xd
2 .
• More convenient to use the transformation f (u) = u
1
c+d
and write xa
1 x1−a
2 , where a = c/(c + d).
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Examples of utility functions – Cobb-Douglas preferences
• A simple utility function representing well-behaved preferences.
• Utility function of the form u(x1, x2) = x1
1 x2
2 .
• More convenient to use the transformation f (u) = u1/3
and write x
1/3
1 x
2/3
2 .
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Marginal utility
Marginal utility (MU) is the change in utility from an increase in
consumption of one good, while the quantities of other goods are constant.
Partial derivatives of u(x1, x2) with respect to x1 or x2.
Pˇr´ıklady:
• u(x1, x2) = x1 + x2 → MU1 = ∂u/∂x1 = 1
• u(x1, x2) = xa
1 x1−a
2 → MU2 = ∂u/∂x2 = (1 − a)xa
1 x−a
2
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Marginal utility
Marginal utility (MU) is the change in utility from an increase in
consumption of one good, while the quantities of other goods are constant.
Partial derivatives of u(x1, x2) with respect to x1 or x2.
Pˇr´ıklady:
• u(x1, x2) = x1 + x2 → MU1 = ∂u/∂x1 = 1
• u(x1, x2) = xa
1 x1−a
2 → MU2 = ∂u/∂x2 = (1 − a)xa
1 x−a
2
The value of MU changes with a monotonic transformation of the utility
function. If we multiply utility times 2, MU increases times 2.
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Relationship between MU and MRS
We want to measure MRS = slope of IC u(x1, x2) = k,
where k is a constant.
We are interested in (∆x1, ∆x2), for which the utility is constant:
MU1∆x1 + MU2∆x2 = 0
MRS =
∆x2
∆x1
= −
MU1
MU2
We can calculate MRS from the utility function. E.g. for u =
√
x1x2:
MRS = −
∂u/∂x1
∂u/∂x2
= −
0,5x−0,5
1 x0,5
2
0,5x0,5
1 x−0,5
2
= −
x2
x1
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Relationship between MU and MRS
We want to measure MRS = slope of IC u(x1, x2) = k,
where k is a constant.
We are interested in (∆x1, ∆x2), for which the utility is constant:
MU1∆x1 + MU2∆x2 = 0
MRS =
∆x2
∆x1
= −
MU1
MU2
We can calculate MRS from the utility function. E.g. for u =
√
x1x2:
MRS = −
∂u/∂x1
∂u/∂x2
= −
0,5x−0,5
1 x0,5
2
0,5x0,5
1 x−0,5
2
= −
x2
x1
The value of MRS does not change with monotonic transformation.
If we multiply utility function times 2, MRS= −2MU1
2MU2
= −MU1
MU2
.
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APPLICATION: Utility from commuting
People decide whether to take bus or car.
Each type of transport represents a bundle with different characteristics,
e.g.:
• x1 is walking time
• x2 is time taking a bus or car
• x3 is the total cost of commuting
• ...
Assume that the utility function has a linear form
U(x1, ..., xn) = β1x1 + ... + βnxn.
Then we use statistical techniques to estimate the
parameters βi that best describe choices.
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APPLICATION: Utility from commuting (cont’d)
Domenich and McFadden (1975) estimated the following utility function:
U(TW , TT, C) = −0,147TW − 0,0411TT − 2,24C
• TW = total walking time in minutes
• TT = total driving time in minutes
• C = total cost in dollars
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APPLICATION: Utility from commuting (cont’d)
Domenich and McFadden (1975) estimated the following utility function:
U(TW , TT, C) = −0,147TW − 0,0411TT − 2,24C
• TW = total walking time in minutes
• TT = total driving time in minutes
• C = total cost in dollars
The parameters can be used for different purposes.
For instance, we can:
• calculate the marginal rate of substitution between two characteristics
• forecast consumer response to proposed changes
• estimate whether a change is worthwhile in a benefit-cost sense
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What should you know?
• Budget set = consumption bundles available at
given prices and income
• Budget line are bundles for which the entire
income is spent.
• If the preference relation is complete, reflexive
and transitive, consumer can order bundles
according to preferences.
• Monotonicity and convexity are reasonable
assumptions – easier to find the optimum
bundle.
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What should you know? (cont’d)
• Utility function assigns numbers to different
bundles so that the bundles are ordered
according to preferences.
• The numbers have no meaning in itself.
Monotonic transformation of u represents the
same preferences..
• MRS measures the slope of IC.
• The slope of IC measures the willingness to pay
for good 1 (in units of good 2)
• The slope of BL measures the opportunity cost
of good 1(in units of good 2)
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