Theory of Consumer Choice and Theory of Firms
V. Hajko
2016
V. Hajko (FSS MU) Introduction to Economics 1 / 50
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 2 / 50
Consumer choice Consumer behavior
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 3 / 50
Consumer choice Consumer behavior
Key questions
How can we represent the consumer’s preferences?
How can we represent the choices a consumer can afford?
What determines how a consumer divides his or her resources between
two goods?
Consumer Optimization Problem
V. Hajko (FSS MU) Introduction to Economics 4 / 50
Consumer choice Consumer behavior
Rational behavior
The rational consumer behavior is driven by the effort to achieve the
highest level of satisfaction
In the economics, the consumers (sic!) are being satisfied by the
consumption of goods and services
The highest level of satisfaction is seen according to the consumer’s
preferences
With limited resources, people face tradeoffs
if you go swimming, you will have less time for jogging
if you buy more of one good, you will have less money to buy
something else
if you work more, you will earn more, but you will have less of the
leisure time
The consumers seek to consume the "best" bundle of goods they can
afford:
they maximize their utility functions subject to their budgetary
constrains
V. Hajko (FSS MU) Introduction to Economics 5 / 50
Consumer choice Consumer Preferences
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 6 / 50
Consumer choice Consumer Preferences
The preference relations
In order to represent the what is “better” or "worse", the consumer
has to be able to compare the choices.
This comparison can be represented by the preference relations.
This logic of comparison is used in the construction of the so-called
"utility function" - a representation of the consumer’s preferences.
The preference relation allows for the comparison of pairs of
alternatives, x and y
x y means x is at least as good as y; (x y means x is strictly
preffered to y)
x y means y is at least as good as x; (x y means y is strictly
preffered to x)
x ∼ y means x is indifferent to y
V. Hajko (FSS MU) Introduction to Economics 7 / 50
Consumer choice Consumer Preferences
The rational preference relations
The preference relation is rational, if it fulfills two basic assumptions:
completeness and transitivity
Completeness: for any pair of x, y ∈ X we have x y, x y or both
(i.e. x ∼ y).
consumer is able to compare any two pairs and decide whether x is
preferred to y, y is preferred to x, or whether he or she is indifferent
between x and y
Transitivity: for all alternatives x, y.z ∈ X , if x y and y z then
x z.
rational behavior rules out a preference cycle
V. Hajko (FSS MU) Introduction to Economics 8 / 50
Consumer choice Utility functions and indifference curves
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 9 / 50
Consumer choice Utility functions and indifference curves
Utility functions
A function u : X → R is a utility function representing preference
relation , if for all x, y ∈ X we have x y ⇔u (x) ≥ u (y)
The utility function is used to represent the preferences (the logical
relationships) with a numerical representation
The utility function is strictly individual
A preference relation can be represented by a utility function only if
it is rational
V. Hajko (FSS MU) Introduction to Economics 10 / 50
Figure: The visual representations, function f (x, y) = z
(a) We need 3-D graphs to visually capture
the value of the function with two inputs
0
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
0
1
2
3
4
5
6
7
x 10
−3
(b) 2-D contour plot for f (x, y) = z
5 10 15 20 25 30
5
10
15
20
25
30
Consumer choice Utility functions and indifference curves
Utility function
The contour lines of the utility function represent all points, where
the various combinations of amounts of x and y lead to the same
value of the utility function
Assume the utility function U = X2Y 2; for the given value of utility
function, e.g. U = 100, we can write down the specific contour line
(which is called the indifference curve) as 100 = X2Y 2, or Y = 100
X2
The indifference curve shows all combination of bundles of x and y
that bring the same utility. Therefore the consumer is indifferent
between all bundles lying on the indifference curve.
Properties of the well behaved utility function
monotonicity (or its weaker version, local nonsatiation)
in essence, the larger amount of consumed goods or services will not
decrease the utility (the goods are desirable)
convexity
a formal expression of a basic inclination of economic agents for
diversification (the assumption that consumers prefer variety)
V. Hajko (FSS MU) Introduction to Economics 12 / 50
Figure: The visual representation of the convex utility function with nonsatiation
(a) The utility function:
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
(b) The indifference map is the contour
plot of the utility function:
30 40 50 60 70 80 90 100 110
30
40
50
60
70
80
90
100
110
Consumer choice Utility functions and indifference curves
Indifference curves
For well behaved utility functions, the higher indifference curves show
the contours of higher values of the utility function
The higher indifference curves are preferred to lower ones
Indifference curves do not cross (this would violate the transitivity of
the preference relation)
Indifference curves are bowed inward (becase of the convexity
property)
V. Hajko (FSS MU) Introduction to Economics 14 / 50
Consumer choice Utility functions and indifference curves
Marginal rate of substitution in consumption
Marginal rate of substitution in consumption (MRSC ) is the rate at
which a consumer is willing to trade one good for another
It is represented by the slope of the indifference curve
MRSC = MUX
MUY
, also MRSC = −∆Y
∆X
C
B
A
X
Y
IC1
IC2
V. Hajko (FSS MU) Introduction to Economics 15 / 50
Consumer choice Consumer’s optimization problem
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 16 / 50
Consumer choice Consumer’s optimization problem
Budget constraint
Due to the limited resources, some of the consumption bundles are
unavailable to the consumer.
The limitation of the consumption is called the budget constraint:
I ≥ px X + py Y , where I is consumers disposable income, px is the
price of one unit of X and X is the number of units of X consumed.
The slope of the budget line represents the so called Marginal rate of
substitution in exchange (MRSE = −pX
pY
)
this represents the ratio at which the goods are valued by the market
(e.g. “1 apple = 2 bananas” - this is called the relative price)
An increase in income shifts the budget constraint outward
A change in the prices causes the budget constraint to change its
slope ("rotate")
V. Hajko (FSS MU) Introduction to Economics 17 / 50
Consumer choice Consumer’s optimization problem
Budget constraint
D
C
B
A
X
Y
IC1
IC2
V. Hajko (FSS MU) Introduction to Economics 18 / 50
Consumer choice Consumer’s optimization problem
Consumer’s optimization problem
The consumer wants to attain maximum utility
The higher level of utility is equivalent with higher value of the utility
function
Therefore, the consumer seeks to find such a combination that will
maximize the value of the utility function AND satisfy the budget
constraint
max U (X, Y ) subject to: I ≥ px X + py Y
since we are assuming monotonicity, we can simplify the budget
constraint to I = px X + py Y
we can write the Lagrangian for this optimization problem as:
L (X, Y , λ) = U (X, Y ) + λ (I − px X − py Y )
By differentiating L with respect to x,y and λ and setting the
derivatives equal to zero, we get the first order conditions:
∂U
∂X − λpx = 0
∂U
∂Y − λpy = 0
I − px X − py Y = 0
V. Hajko (FSS MU) Introduction to Economics 19 / 50
Consumer choice Consumer’s optimization problem
The optimal consumption
The first order conditions imply the consumer will seek to consume
the goods in the ratio that fulfills MRSC = MRSE , i.e. MUX
MUY
= pX
pY
or
MUX
pX
= MUY
pY
The marginal utility MUX is the change in the total utility caused by
infinetisimaly small change in the consumption of X, i.e.
MUX = ∂U
∂X
This implies the optimum will occur at the point where the indifference
curve and the budget constraint are tangent
The the attainable consumption, hence also the attainable utility, is
limited by the budget constraint
Combined with the previous condition, this implies the optimum will
occur at the point where the highest indifference curve and the budget
constraint are tangent
V. Hajko (FSS MU) Introduction to Economics 20 / 50
Consumer choice Consumer’s optimization problem
Income and substitution effects
The income effect is the change in consumption that results when a
price change moves the consumer to a higher or lower indifference
curve.
the price change increases (decreases) the purchasing power of the
consumer (his relative wealth), which encourages (discourages) the
consumer to buy the goods in question
The substitution effect is the change in consumption that results
when a price change moves the consumer along an indifference curve
to a point with a different marginal rate of substitution.
the price change encourages greater (lower) consumption of the good,
since it has become relatively cheaper (more expensive)
V. Hajko (FSS MU) Introduction to Economics 21 / 50
Consumer choice Consumer’s optimization problem
Normal goods vs. inferior goods
Based on the income elasticity of demand (εID =
∂QD
Q
∂I
I
) we can
distinguish normal and inferior goods:
With the increase in income, the consumer buys more of a normal good.
With the increase in income, the consumer buys less of an inferior good.
V. Hajko (FSS MU) Introduction to Economics 22 / 50
Consumer choice Consumer’s optimization problem
Individual demand
The demand function QD = f (p, ·) represents the quantity of the
given goods demanded, dependent on the price (and possibly other
factors, e.g. QD
X = f (px , py , I)).
Consumer’s demand curve summarizes the solutions of the
optimization problems for the various levels of price (given no changes
in the other factors, e.g. for the given level of income and prices of
other goods).
Typically a demand curve slopes downward (with a decrease in price,
the consumer would buy more of the goods).
There is an exception (but very rare): the so-called Giffen goods - an
increase in the price raises the quantity demanded (the income effect
overwhelmes the substitution effect).
V. Hajko (FSS MU) Introduction to Economics 23 / 50
Theory of Firms Firm behavior
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 24 / 50
Theory of Firms Firm behavior
Key questions
What is the goal of the firm?
How do firms make decisions?
Production choices maximizing the firm’s profit
What is a production function, marginal product, fixed and variable
costs, marginal costs and marginal revenue?
What is a perfectly competitive market and what is imperfet
competition?
What is the link to the demand function?
How the firm determines its supply function?
V. Hajko (FSS MU) Introduction to Economics 25 / 50
Theory of Firms Firm behavior
The goal of the firm
The producers of the goods and services are called firms (to
distinguish the specific production units and their behavior)
The business operation of the firm is the transformation of (typically
multiple) inputs into (one or more) outputs.
The formal description of this transformation is called the production
function
This transformation faces economic constraints - there are costs of
production
The firm sells its production - it has to be aware of the demand in
order to calculate its revenues
The goal of the firm is to maximize profits (profit = total revenues total
costs, Π = TR − TC)
Can you provide the reason why?
V. Hajko (FSS MU) Introduction to Economics 26 / 50
Theory of Firms Firm behavior
Firm’s profit maximization problem
In general, the firm aims to maximize profits Π = TR − TC
the solution to the problem is found by setting the optimal Q (since
Π (Q) = TR (Q) − TC (Q)
the optimal Q can be produced using various combinations of inputs
(typically, K and L) (i.e. Q (K, L)), hence the firm can find the
solution by setting optimal K and L (i.e.
Π (Q (K, L)) = TR (Q (K, L)) − TC (Q (K, L)))
alternative method is to formulate the cost minimization problem to
produce a given quantity ¯Q
V. Hajko (FSS MU) Introduction to Economics 27 / 50
Theory of Firms Production function
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 28 / 50
Theory of Firms Production function
The production function
The production function (Q = f (F1, F2, . . . FN )) describes the
technological process of the transformation of inputs to outputs.
The typical production factors (F) considered in the production of the
firm are: capital (K) and labor (L).
Q = f (K, L)
V. Hajko (FSS MU) Introduction to Economics 29 / 50
Theory of Firms Production function
Short run and long run
Due to the technological limitations, it is usually time consuming to
change the amount of capital used in production.
This assumption is reflected in the distinction between the the short
run and the long run
in the short run, we consider some of the production factors (typically
capital (K)) as fixed and only some as variable (typically labor (L))
i.e. Q = f ( ¯K, L), or for the sake of simple notation Q = f (L)
in the long run, we consider all the production factors as variable
V. Hajko (FSS MU) Introduction to Economics 30 / 50
Theory of Firms Production function
Marginal product
The marginal product of factor Fi (MPFi = ∂Q
∂F ; note it also shows
the slope of the production function!) is the increase in output caused
by the additional unit of the input (while other inputs remain
constant)
The marginal product is not constant, but is (typically) diminishing
In contrast, if you hire additional worker, or buy additional machine, the
costs of the aditional unit are often the same as with the previous unit
You can compare the marginal benefit (the increase in production) with
the marginal costs (the increase in costs) to determine whether or not
you want to add the additional unit of the input.
Try to provide an example
V. Hajko (FSS MU) Introduction to Economics 31 / 50
Figure: Production function in the long run
8 10 12 14 16 1820
40
60
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10
6
Figure: Production function in the short run
1 2 3 4 5 6 7 8 9
−5000
−4000
−3000
−2000
−1000
0
1000
2000
3000
4000
Figure: Production function in the long run, no diminishing returns
(non-decreasing function)
0
5
10
15
20
0
20
40
60
0
1000
2000
3000
4000
5000
Theory of Firms Production function
Isoquants
Isoquants are the contour lines of the production function.
Isoquants represent all points, where the various combinations of
amounts of inputs (K and L) lead to the same values of the
production function.
they show the one given level of output that can be produced using
various combinations of inputs
In a similar fashion as the indifference curves, the isoquants are the 2-D
representation of the 3-D figure
the slope of an isoquant is called the marginal rate of technical
substitution - it shows the ratio in which we can substitute the
production factors while keeping the output constant
V. Hajko (FSS MU) Introduction to Economics 35 / 50
Figure: Isoquants of the production function in the long run, no diminishing
returns (non-decreasing function)
2 4 6 8 10 12 14 16
5
10
15
20
25
30
35
40
45
Theory of Firms Costs
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 37 / 50
Theory of Firms Costs
Costs
In order to produce something, the firm has to pay for the inputs.
We need to consider both implicit and explicit costs
Explicit costs: everything that will show up in the accounting (e.g.
the electricity bill, the wages of hired workers)
Implicit costs: the opportunity costs (the value of foregone alternative
use of resources)
Illustration:you want to buy a machine, the price of the machine is $10,000, you can both
lend or borrow money, with the fixed interest rate 3%.
you can take a loan for $10,000 and pay 3% interest to the bank → explicit cost = $300
you can take your own $10,000 and pay 0% interest to the bank → explicit cost = $0
But you could also lend $10,000 to someone else and collect the 3% interest!
By using your own money your foregone interest is $300 = implicit costs
V. Hajko (FSS MU) Introduction to Economics 38 / 50
Theory of Firms Costs
Accounting profit vs. economic profit
Accounting can only record actual events, not the alternate foregone
opportunities.
BUT economic costs are not meaningless → they influence the
decisions! (it would not be rational / economical to pursue a certain
activity, if there is a more profitable alternative)
Accounting profit = total revenue minus total explicit costs
Economic profit = total revenue minus (total explicit and total
implicit costs)
V. Hajko (FSS MU) Introduction to Economics 39 / 50
Theory of Firms Costs
Fixed costs vs. variable costs
We have already differentiated between the short run and the long run
In the short run, we can not change the quantity of some production
factors
BUT we still have to pay for them
These costs are fixed by the inability to change the amount of certain
production factors in the short run
If we can change the quantity of the inputs we include into the
production, we are dealing with the variable costs
In the short run, there are fixed costs (typically the cost of capital)
and variable costs (cost of labor)
In the long run, all inputs and hence all costs are variable
V. Hajko (FSS MU) Introduction to Economics 40 / 50
Theory of Firms Costs
Marginal costs
Marginal costs (MC = ∂TC
∂Q ; note it also shows the slope of the
function of total costs!):
the change in total costs caused by producing the additional unit of the
output
The marginal costs are not constant, but are (typically) increasing
(due to the diminishing returns of the production function!).
V. Hajko (FSS MU) Introduction to Economics 41 / 50
Figure: Total and marginal costs
ATC
AVC
AFC
MC
Q
FC
VC
TCTC
VC
FC
Q
MC
ATC
AVC
AFC
Theory of Firms Revenues and demand
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 43 / 50
Theory of Firms Revenues and demand
Revenues
Firms generate revenues by selling their output, i.e. TR = p.Q (K, L)
we have seen the firm is able to influence the quantity of the product it
produces by using various amounts of inputs
Depending on the market structure (influencing the demand for their
products), firms either take the price from the market (p is constant),
or they are able to influence the market price by changing the
quantity produced (p is a function of Q)
perfect competition: TR = p.Q (K, L)
imperfect competition: TR = p (Q (K, L)) .Q (K, L)
V. Hajko (FSS MU) Introduction to Economics 44 / 50
Figure: Total, marginal and average revenue
TR
TR
Q
TR
Q
TR
Q
MR
Q
MR
MR=AR
MR
AR
Theory of Firms Revenues and demand
Marginal revenue
The marginal revenue (MR = ∂TR
∂Q ; note it also shows the slope of the
function of total revenues)
the change in the total revenue caused by producing the additional unit
of the output
The demand function is different in perfect competition and in
imperfect competition: the demand function shows the quantity
demanded for the given price
The perfectly competetive firm can sell as many units as it can
produce for the market price
The demand function is a horizontal line, AR=MR
The imperfect competitive firm: in order to sell more units of Q, it has
to lower the price on all units → it faces downward sloping demand
It has downward sloping marginal revenues
V. Hajko (FSS MU) Introduction to Economics 46 / 50
Theory of Firms Profit maximization problem
1 Consumer choice
Consumer behavior
Consumer Preferences
Utility functions and indifference curves
Consumer’s optimization problem
2 Theory of Firms
Firm behavior
Production function
Costs
Revenues and demand
Profit maximization problem
V. Hajko (FSS MU) Introduction to Economics 47 / 50
Theory of Firms Profit maximization problem
Profit maximization problem
To maximize the profits Π = TR − TC, the firm will set the quantity
produced so that
∂Π
∂Q =
∂TR
∂Q
MR
−
∂TC
∂Q
MC
= 0 → MR − MC = 0 → MR = MC
In other words, the firm will produce additional unit of production, as
long as the marginal revenue of this additional unit is larger or equal
to what this additional unit of production costs
Holds true for both perfectly and imperfectly competitive firms
V. Hajko (FSS MU) Introduction to Economics 48 / 50
Figure: Total and marginal revenues
MC
ATC
π
π
TR
ATC
AVC
AFC
MC
Q
TCTC
VC
FC
Q
MR=AR
Theory of Firms Profit maximization problem
Supply function
The supply function of the firm is the upward sloping part of the MC
curve above the AVC (or ATC in the long run)
The profit maximization problem explains the supply of the firm - a
firm will supply as long as the last unit of the product that is still
profitable
In the short run (with existence of fixed costs), a firm might choose to
produce, even if the production ends in red numbers (in loss)
The alternative option is to produce nothing, yet still pay fixed costs
If a firm covers at least variable costs for producing given amount of
product, it will engage in production
V. Hajko (FSS MU) Introduction to Economics 50 / 50