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Lecture 1: The market and consumer theory
Intermediate microeconomics
Jonas Vlachos
Stockholms universitet
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The market
• Demand
• Supply
• Equilibrium
• Comparative statics
• Elasticities
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Demand
• Demand function. Mathematical relation between quanitity
demanded (Qd), own price (p) and other factors
• For example
• Where p is own price, pb and pc are prices of substitutes,
and Y is income
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Demand curve: own price change
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Holding everything but own price equal
Movement along the demand curve
dQ/dp=-20 (million kg/$)
Since p is on y-axis, slope=1/(dQ/dp)=dp/dQ=-0.05
($/million kg)
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Demand curve: changes in other
prices or income
Increases in substitute goods or income shifts the
demand curve
At any given price of pork, demand for pork increases
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Supply
• Supply function: mathematical relation between
quantity supplied (QS), own price (p) and other factors
(for example input prices)
• For example
Where p is own price and ph is input price
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Supply curve: own price change
Holding everything but own price equal
Movement along the supply curve
dQ/dp=40 (million kg/$)
Since p is on y-axis, slope=1/(dQ/dp)=dp/dQ=0.025
($/million kg) 7
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Supply curve: increase in input price
Shifts the supply curve
At any given pork price, less is supplied
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Market equilibrium: demand=supply
Qd = 286-20p = QS = 88+40p => Q = 220
Since Q = 220, p = 3.3
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Comparative statics: large shocks
What happens when some ”shock” affects the demand or
supply of a good?
For example: large increase in input price, from 1.5 to 1.75
Previous: QS = 88+40p. Now: QS = 73+40p
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55.3$
407320286
p
pp
QQsd
21555.320286 d
Q
21555.34073 s
Q
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Comparative statics: small shocks
Demand is a function of price, holding everything else
constant: Q = D(p)
Supply is a function of price and some exogenous factor a.
(exogenous here means ”outside the control of firms”)
Q = S(p,a)
Prices will indirectly depend on a, so in equilibrium:
D(p(a)) = S(p(a),a)
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Comparative statics: small shocks
To analyze the impact of a small change in a (for example
input price), use the chain rule when differentiating the
equilibrium condition with respect to a
Rearrange:
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<0
>0 <0
>0 (equilibrium prices increase
when a increases)
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Elasticities
How responsive is a variable to a change in another variable?
Price elasticity of demand:
Price elasticity of supply:
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Elasticity: example with linear demand
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Linear demand: Qd = a-bp => dQ/dp=-b
휀 =
𝑑𝑄
𝑑𝑝
𝑝
𝑄= −𝑏
𝑝
𝑄
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Constant elasticity demand curves
(Q = Apε)
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Constant elasticity supply curves
(Q = Bpη)
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Consumer theory
• The demand function builds on assumptions concerning
consumer preferences
• Consumers also face constraints
• Also assume that consumers try to do as well as they
can, that is maximize their well-being
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Preferences: notation
• Consumers have preferences over ”bundles” of
goods and services
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Strict preference. E.g. a b (a is better than b)
Weak preference. Bundle a is at least as good as b
~ Indifference. Bundle a exactly as good as b
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Preferences: basic assumptions
• Completeness: all bundles can be ranked
a consumer can rank them so that either a b,
b a, or a ~ b
• Transitivity:
Consumers’ rankings are logically consistent in the
sense that if a b and b c, then a c.
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Well behaved preferences
• Non-satiation
(monotonicity):
– More is better
• Convexity:
– Averages are prefered to
extremes
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From preferences to indifference curves
Consider combinations of two possible goods. The bundles that
make a consumer equally happy make up indifference curves
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Indifference curves
• The preference assumptions imply certain properties for
indifference curves
– Every bundle is on an indifference curve (completeness)
– Indifference curves further from the origin are better (non-
satiation)
– Indifference curves cannot cross (transitivity)
– Indifference curves cannot slope upwards (non-satiation)
– Indifference curves cannot be thick (non-satiation)
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Indifferences curves cannot cross
Y ~ Z
Z ~ X
But
Y ~ X
Transitivity violated
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From preferences to utility and utility
functions
• Utility is an analytical tool to describe preferences
• Ordinal: utility only describes how bundles are ranked
relative to each other
– Not a cardinal measure that tells us by how much bundles are
preferred
• A utility function assignes a larger value to the more
prefered bundle, but units to not matter
– If x y, then u( ) is a utility function if u(x) > u(y)
• A utility function can be transformed into another utility
function such that the preference ordering is maintained
– Positive monotonic transformation 24
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Compare to temperatures
Celsius Fahrenheit Kelvin
50 122 323
100 212 373
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Describe the same temperature relation, but units differ
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Marginal rate of substitution (MRS)
MRS is the amount of one good that an individual is willing to give
up for another good for any given utility level: MRS = dq2/dq1
Marginal utility is the increase in utility that a consumer gets from
consuming the last unit of a good, holding other consumption
constant: 𝛿𝑈
𝛿𝑞1= 𝑈1
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Different preferences => different utility
functions => different MRS:s
We often assume that goods are imperfect substitues, but
that is not necessary
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Indifference curves
Perfect subsitutues and perfect complements are ”extremes”,
many different standard, convex, utility functions
Cobb-Douglas never hit the axis
Quasi-linear hit one of the axis
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Some commonly used utility functions
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The budget constraint: basics
Consumers maximize utility, subject to constraints
Given prices p1, p2, and income Y, the budget line is
If p1 = $1 p2 = $2 and Y = $50, the budget line is:
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The budget constraint: MRT
The marginal rate of transformation, MRT, tells how the market
allows a consumer to trade (”transform”) one good for another
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Constrained consumer choice
Maximize utility subject to the budget constraint
Given ”standard” indifference curves, there is an interior
optimum. The highest feasible indifference curve.
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Maximizing utility using calculus: I
One way to solve consumer maximization is to directly
assume an interior solution and thus MRS=MRT (here
Cobb-Douglas utility)
max𝑋,𝑍
𝑈(𝑋, 𝑍) = 𝑋𝛼𝑍1−𝛼 subject to 𝑌 = 𝑃𝑋𝑋 + 𝑃𝑍𝑍
𝑀𝑈𝑋 =𝛿𝑈(𝑋, 𝑍)
𝛿𝑋= 𝛼𝑋𝛼−1𝑍1−𝛼
𝑀𝑈𝑍 =𝛿𝑈(𝑋, 𝑍)
𝛿𝑍= (1 − 𝛼)𝑋𝛼𝑍−𝛼
𝑀𝑅𝑆𝑋𝑍 =
𝑀𝑈𝑋
𝑀𝑈𝑍=
𝛼𝑋𝛼−1𝑍1−𝛼
(1 − 𝛼)𝑋𝛼𝑍−𝛼=
𝛼
1 − 𝛼
𝑍
𝑋=
𝑃𝑋
𝑃𝑍= 𝑀𝑅𝑇
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Maximizing utility using calculus: I (cont)
From this we see that
𝑍 =
(1 − 𝛼)𝑃𝑋
𝛼 𝑃𝑍𝑋, where
(1 − 𝛼)𝑃𝑋
𝛼 𝑃𝑍 is a constant.
We now have the optimal allocation between Z and X, but how much we
consume also depends on income Y. Insert expression for Z in the budget
constraint
𝑌 = 𝑃𝑋𝑋 + 𝑃𝑍
(1 − 𝛼)𝑃𝑋
𝛼 𝑃𝑍𝑋 = 𝑃𝑋𝑋 1 +
1 − 𝛼
𝛼= 𝑃𝑋𝑋
𝛼
𝛼+
1 − 𝛼
𝛼=
𝑃𝑋
𝛼𝑋 ⇒
𝑋∗ =𝛼𝑌
𝑃𝑋 (optimal quantity of X). Continuing for Z, we get
𝑍∗ =(1 − 𝛼)𝑃𝑋
𝛼 𝑃𝑍𝑋∗ =
(1 − 𝛼)𝑃𝑋
𝛼 𝑃𝑍
𝛼𝑌
𝑃𝑋=
1 − 𝛼 𝑌
𝑃𝑍 (optimum quatity of Z)
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Maximizing utility using calculus: II
The Lagrangian method: optimization with constraints
General: maximize f(x) + λg(x)
Maximize 𝑈 𝑞1, 𝑞2 subject to 𝑌 = 𝑝1𝑞1 + 𝑝2𝑞2
Chose optimal values of q1, q2, and λ
Think of λ as the cost of violating the constraint
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Maximizing utility using calculus: II
Find first order conditions:
Equating (1) and (2) conditions yields
λ is the marginal utility of income (”shadow value” of income)
The value of relaxing the constraint
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(1)
(2)
(3)
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Special case 1: Perfect complements
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Special case 2: Perfect substitutes
If the marginal rate of substitution is not equal to
the marginal rate of transformation, you will only
chose the relatively cheap good. Corner solution
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Special case 3: Quasilinear preferences
Under quasilinear preferences, the price of one good may be
so high that you do not consume any of it. Corner solution
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Expenditure minimization
An alternative approach to consumer optimization is to
consider the lowest cost at which you can achieve a certain
level of utility. I.e. to mimimize expenditures
The solution gives you the expenditure function, the
minimum expendtures needed to achive a certain utility
level
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Are the basic assumptions correct?
• Behavioural economics analyses the basic rationality
assumptions. Numerous deviations recorded
– Endowment effects: what people actually own affects their
preferences
– Salience: people do not (fully) incorporate costs such as
taxes when making economic decisions
– Transitivity: basically seems to hold
• But, we are working with models
– The economic importance of such deviations is not fully
understood
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