Microeconomics 2: Consumer's Constrained Optimisation

Question 1

Describe & explain the utility function

When do we use it? What is it? What are the 2 ways to measure how much a

Answer 1

To represent preferences when there are a great many outcomes, we use
utility functions.
 Utility functions are a mathematical expression that translates the full range of
possible outcomes into a person’s valuation of the outcome – their payoffs.
 A utility function is assigns a number 𝑈(𝑥, 𝑦) to every bundle 𝑥, 𝑦 , representing a
person’s valuation of that bundle.
 For example, given the choice between two bundles (𝑥, 𝑦) and (𝑥bottom right0, 𝑦bottom right0), an individual
will choose (𝑥, 𝑦) if 𝑈(𝑥, 𝑦) > 𝑈(𝑥bottom right0, (𝑦bottom right0).
 We measure how much a person values various outcomes in two ways, either:
1. By indicating how valuable each is on some absolute scale, or
2. By simply ranking them in order
A utility function U assigns a number to each consumption bundle such that:
 A ≻ B implies that U(A) > U(B) and
 A ~ B implies that U(A) = U(B).
 Definition of utility function: a numerical score representing the satisfaction
that a consumer derives from a given consumption bundle.
Utility can be used for states of nature or of elections for presidential candidate’s - it’s not just for consumption bundles, it’s an agent’s attitude towards a state of nature

Question 2

Describe & explain ordinal vs cardinal utility

Answer 2

Note that we consider ordinal utility functions and so all that matters is
whether U(A) is bigger than or smaller than U(B).
 This means the difference 𝑈(𝐴) – 𝑈(𝐵) is meaningless and we cannot make
interpersonal comparisons.
 Cardinal utility attaches a significance to the magnitude of utility. The
size of the utility difference between two bundles has some sort of
significance.

Question 3

Describe ‘neuroeconomics utility’

Answer 3

The economics utility we are used to: measure of how desirable is the outcome of a choice.
 How do we actually measure that?
 Neuroeconomics utility: the average firing rate of a population of neurons that
encodes the subjective value of the object.
 This is a real number, which can be measured.
 It predicts choices!
 Subjective value is the averaged firing rate of a population of neurons coding
behavioural preferences.
 From neurobiological perspective, an object has a subjective value if it is a
reward or a punishment.
 How to measure subjective value?!
 Idea: a being (human, animal) will work for a reward or work to avoid punishment.
 In other words, a being will work to obtain activation of certain neurons (e.g. nucleus
accumbens and orbitofrontal cortex)

Question 4

Describe the use of indifference curves by the utility function

Answer 4

Utility function, in general form for two goods, 𝑈(𝑥, 𝑦), can be
constructed using the indifference curves:
 Each curve is labelled and due to our assumption of monotonic preferences, all
bundles that are on a higher indifference curve have to have bigger labels
 Indifference curves can also be drawn from the utility function:
 All bundles that have the same utility follow 𝑈(𝑥, 𝑦) = 𝑐𝑜𝑛𝑠𝑡,
 Each value of constant will give a different indifference curve.
 Example: 𝑈 (𝑥,𝑦) = 𝑥𝑦
 Let’s assume that the constant is 𝑘 or 𝑈 = 𝑘.
 Solving for 𝑦, we have: 𝑦 = 𝑘/𝑥.
 We can draw indifference curves for different values of 𝑘.
Graph with y-axis labelled “xbottom right2” and x-axis labelled “xbottom right1” with multiple reciprocal curves, closest one to origin labelled “k=1”, next one labelled “k=2” and so on
 Note! For a utility function 𝑈1 𝑥, 𝑦 = 𝑥2𝑦2indifference
curves look the same:
 observe the transformation: 𝑈1 𝑥, 𝑦 = 𝑥2𝑦2 = (𝑥𝑦)2= 𝑈2(𝑥, 𝑦),
 as we are reasoning with ordinal utility.

Question 5

Describe positive monotonic transformation in the ordinal utility function

Answer 5

Positive monotonic transformation preserves the order of preferences’ ranking
*2 graphs - both labelled “V” for y-axis and “u” for x-axis. Graph on left, “A”, has a curve with varying degrees of positive gradient but no inflection points. Titled “yes” (as in, ‘this does follow positive monotonic transformation’) Graph on right, “B”, is overall upward sloping but has points of inflections with negative gradients at times. Titled “no” (as in, ‘this does not follow positive monotonic transformation’)

Question 6

Reasoning with ordinal utility, consider two bundles A and B. For a consumer
with well-behaved preferences, bundle A gives utility of 10 and B gives utility of
100. Can we say that B is preferred ten times to A? Can we say that B is much
better than A? Can we say that B is preferred to A?

Answer 6

B is preferred to A but not by 10 times, in ordinal utility the quantity does not matter

Question 7

Describe the algebraic expressions for different utility functions

Answer 7

 For goods which are perfect substitutes the utility function will be of the
following form:
𝑈 = 𝑎𝑥 + 𝑏𝑦:
 Example: 𝑈(𝑥,y) = 𝑥 + 3𝑦
 For goods which are perfect complements the utility function will be of
the following form:
𝑈(𝑥, 𝑦) = 𝑚𝑖𝑛(𝑎𝑥,𝑏𝑦):
 Example: 𝑈(𝑥, 𝑦) = 𝑚𝑖𝑛 (𝑥, 3𝑦)
 If one of the considered goods (second good 𝑦) is a neutral good, the
utility function will be of the following form:
𝑈(𝑥, 𝑦) = 𝑎𝑥
 Example: 𝑈(𝑥, 𝑦) = 𝑥
 If one of the considered goods (second good 𝑦) is a bad good the utility
function can be of the following form:
𝑈(𝑥, 𝑦) = 𝑎𝑥 − 𝑏𝑦
 Note: this a particular example of a bad good and perfect substitutes.
- Also:
1. Quasi-linear preferences: Graph with “ x2” y-axis and “x1” x-axis and 3 parallel “indifference curves” with different heights all shaped as negative at a decreasing rate. Starts with touching vertical axis. For example: 𝑢(𝑋1, 𝑋2) = ln( 𝑋1) + 𝑋2
2. Cobb-Douglas utility functions: 2 graphs both with “x2” y-axis and “x1” x-axis and have multiple curves shaped as reciprocal (asymptotic with both axis but each curve becomes more linear as it’s farther from the origin. Graph on left has steep asymptote towards both axis and label underneath graph is “A: c=1/2 d=1/2”. Graph on right also has steep (close) asymptote towards x-axis but only curves a slight little towards vertical axis before the curve stops. Label underneath graph is “B: c=1/5 d=4/5”. Label underneath both graphs - “𝑢(𝑋1, 𝑋2) = 𝑋1^c𝑋2^𝑑”

Question 8

Describe what ‘marginal utility’ is

Answer 8

Consider a consumer who is consuming a bundle of goods 𝑥, 𝑦 . How does
this consumer’s utility change as we give him or her a little more of good 1?
This rate of change is called the marginal utility with respect to good 1.
 We write it as MUx and think of it as being a ratio:
𝑴𝑼bottom right𝒙 = ∆𝑼/∆𝒙 = (𝑈(𝑥 + ∆𝑥, 𝑦) − 𝑈(𝑥, 𝑦))/∆𝑥
 Marginal utility measures the rate of change in utility due to a small change in
the amount of good 1 with the amount of good 2 is held constant, 𝑴𝑼𝒙 = 𝒅𝑼/𝒅𝒙.
 Note: we can re-write ∆𝑈 = 𝑀𝑈bottom right𝑥∆not bottom right𝑥

Question 9

Describe marginal utility with MRS

Answer 9

MRS is the rate of substitution of a small amount of good 2 for good 1,
𝑀𝑅𝑆 =∆𝑦/∆𝑥.
 Let’s consider a change in both goods that keeps utility level the same –
moving along an indifference curve:
𝑀𝑈bottom right𝑥∆𝑥 + 𝑀𝑈bottom right𝑦∆𝑦 = ∆𝑈 = 0
 we can rearrange:
𝑀𝑅𝑆 = ∆𝑦/∆𝑥 = − 𝑀𝑈bottom right𝑥/𝑀𝑈bottom righty¬

Or, with a different notation:
 We consider a utility function 𝑈 = 𝑢(𝑥, 𝑦) ; the total differential of the utility
function is:
𝑑𝑈 = 𝑑𝑈/𝑑𝑥 𝑑𝑥 + 𝑑𝑈/𝑑𝑦 𝑑𝑦
 Moving along one indifference curve means that 𝑑𝑈 = 0 (i.e. utility doesn’t change) and rearranging the above gives the slope of an indifference curve:
𝑑𝑦/𝑑𝑥 = - 𝑑𝑈/𝑑𝑥 / 𝑑𝑈/𝑑𝑦 = 𝑀𝑅𝑆
 Or:
𝑀𝑅𝑆 = − 𝑀bottom right𝑥/𝑀𝑈bottom right𝑦