Lecture 1: The market and consumer theory

Intermediate microeconomics

Jonas Vlachos

Stockholms universitet

1

The market

• Demand

• Supply

• Equilibrium

• Comparative statics

• Elasticities

2

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

3

Demand curve: own price change

4

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)

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

5

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

6

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

Supply curve: increase in input price

Shifts the supply curve

At any given pork price, less is supplied

8

Market equilibrium: demand=supply

Qd = 286-20p = QS = 88+40p => Q = 220

Since Q = 220, p = 3.3

9

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

10

55.3$

407320286

p

pp

QQsd

21555.320286 d

Q

21555.34073 s

Q

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:

12

<0

>0 <0

>0 (equilibrium prices increase

when a increases)

Elasticities

How responsive is a variable to a change in another variable?

Price elasticity of demand:

Price elasticity of supply:

13

Elasticity: example with linear demand

14

Linear demand: Qd = a-bp => dQ/dp=-b

휀 =

𝑑𝑄

𝑑𝑝

𝑝

𝑄= −𝑏

𝑝

𝑄

Constant elasticity demand curves

(Q = Apε)

15

Constant elasticity supply curves

(Q = Bpη)

16

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

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

23

/

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

Compare to temperatures

Celsius Fahrenheit Kelvin

50 122 323

100 212 373

25

Describe the same temperature relation, but units differ

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

27

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

28

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.

32

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

35

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)

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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