Microeconomics 2: Consumer's Constrained Optimisation Flashcards
1.1 Consumer choice building blocks: budget constraint, preferences, and utility function (continued). Consumer choice: utility function. 1.2 Consumer's optimal choice 1.3 Consumer demand (part 1)
Describe & explain the utility function
When do we use it? What is it? What are the 2 ways to measure how much a
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 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
Describe & explain ordinal vs cardinal utility
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.
Describe ‘neuroeconomics utility’
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)
Describe the use of indifference curves by the utility function
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.
Describe positive monotonic transformation in the ordinal utility function
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’)
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?
B is preferred to A but not by 10 times, in ordinal utility the quantity does not matter
Describe the algebraic expressions for different utility functions
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”
Questions on perfect substitutes have been the worst performing question by students in the last 11 years
Describe what ‘marginal utility’ is
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
Describe MRS with marginal utility
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 = − 𝑀Ubottom right𝑥/𝑀𝑈bottom right𝑦
What’s ‘microeconomics’ about?
Microeconomics is about using simple models to get some understanding of ‘real
world’ phenomena. Simplicity is gained through assumptions.
What does a consumer’s budget constraint do?
It identifies what they can afford to buy
What’s the general equation for the budget line?
𝑝bottom right𝑥 𝑥 + 𝑝bottom right𝑦 𝑦 = 𝑚
What’s the equation & interpretation for the budget line?
It’s the rate of substitution of one good for the other
Equation: ∆𝑦/∆𝑥 = − 𝑝bottom right𝑥 / pbottom righty
Describe when a budget line changes
If 𝑚, 𝑝bottom right𝑥 or 𝑝bottom right𝑦 change then we get a new budget constraint: changing 𝑚 changes the intercept
but not the slope; changing 𝑝bottom right𝑥 or 𝑝bottom right𝑦 changes the slope.
What’s the equation for ‘marginal rate of substitution’?
𝑅𝑆 = − 𝑀Ubottom right𝑥/𝑀𝑈bottom right𝑦
Show ‘Every consumer wishes to maximise their utility, but they are bound by
their budget constraint’ in notation form
Max 𝑈(𝑥, 𝑦)
subject to 𝑝bottom right𝑥 𝑥 + 𝑝bottom right𝑦 𝑦 = 𝑚
State the optimal consumption bundle of a consumer in notation form
When does it change?
(𝑥∗, 𝑦∗)
If prices and/or income change the optimum consumption bundle will chang
What’s ‘(𝑥∗, 𝑦∗)’ when it comes to consumption bundles?
The optimal consumption bundle
Describe & explain a consumer’s rational choice of a consumption bundle graphically
- “x”-axis and “y”-axis with a budget constraint and respective IC. At tangency point, there’s a point and dotted lines going to each axis. At each axis, there’s respective labels “x” and “y”
- Rational choice of the
representative consumer under
constraint:
Indifference curve is tangent to the
budget line,
(x,y) is the optimal consumption
bundle,
this is the optimal consumption
bundle for given prices and income:
𝑝bottom right𝑥, 𝑝bottom right𝑦, 𝑚
Describe & explain whether the tangency condition for finding a consumer’s optimal choice is true in all cases
Rational choice of the representative consumer under constraint: tangency
condition implies that indifference curve cannot cross the budget line and
does not take into account several possible cases…
1. Kinky tastes
- “x1”-axis and y-axis, “x2”, with a normal “budget constraint line”. But the indifference curve is not smooth, instead it has a ‘kink’ at the tangency point and the other “indifference curves” are parallel to this one. Dotted lines going to respective axis from the tangency point and at axes, there are the respective labels, “x1” and “x2”
- Here is an optimal consumption bundle where the indifference curve doesn’t technically have a tangent
2. Multiple tangencies
- “x1”-axis and y-axis, “x2”, with a normal “budget constraint line”. But the indifference curve is not smooth as usual, it’s smooth but it’s like a wave, going up and down from top left to bottom right of the graph and the other “indifference curves” are parallel to this one. 2 dips of the optimal IC are tangent to the budget line with label “optimal bundles” and the peak of the IC below the optimal IC is also tangent to the budget line, labelled “Nonoptimal bundle”
- Here there are 3 tangencies, but only 2 optimal points, so the tangency condition is necessary but not sufficient.
3. Boundary solution
- “x1”-axis and y-axis, “x2”. The “indifference curves” are a normal shape and parallel but they decay into the x1-axis and touch it. The “budget line” is only tangent to the most leftward IC and it touches it where the IC touches the x1-axis
- The optimal consumption involves consuming 0 units of good 2. The IC is not tangent to the budget line.
As you can see, tangency is not a sufficient condition; it’s only a sufficient condition for interior tangency solutions
Describe the relationship between the optimum consumption bundle and the demand function
The optimum consumption bundle is the demanded bundle
If prices and/or income change then the demanded bundle will change as well.
The demand function explains the relationship between different demanded
consumption bundles and different prices and incomes.
Demand functions can be denoted by 𝑥(𝑝bottom right𝑥, 𝑝bottom right𝑦, 𝑚) and y(𝑝bottom right𝑥, 𝑝bottom right𝑦, 𝑚).
We will mostly be working with Cobb-Douglas preferences:
They have very “nice” properties and we can demonstrate that consumer always
spends a fixed fraction of her income on each good.
Describe graphically & explain optimal choice / consumer demand for 1-1 perfect substitutes
1-1 perfect substitutes:
- Graph with “x”-axis and “y”-axis and a “budget constraint” line. Multiple “indifference curves; slope - -1” (so they’re completely straight lines). The graph is so that the budget constraint touches one of the higher ICs at y=0 and the IC just below it at x=0). Where the budget line meets the higher IC at y=0 is the “Optimal choice”.
- Different cases are
possible, depending
on price of goods
- Demand function is:
1. 0 when p2< p1
2. m/p1 when p2> p1
3. Any number between 0 and m/p1 when p1=p2 (the slope of the budget line would be the same as the slope of the IC, and so there would be infinite solutions)
Describe graphically optimal choice / consumer demand for concave preferences
- *Graph with “x1”-axis and y-axis, “x2”, with normal “budget line”. The “indifference curves” are concave (think of it like a baseball field where the origin is where the batsman is and each IC is shaped how a baseball field is, from one axis to the other). One of the ICs is tangent to the budget line at the x-axis, point “Z”, the “optimal choice”. The IC below this one is also tangent to budget line somewhere in the middle of this IC. This is point “X”, the “nonoptimal choice”.
In consumer preferences, what will a consumer’s income be spent on between ‘good x’ and ‘good y’ if good y is “bad”?
“Good” good x: m/pbottom rightx
In consumer preferences, what will a consumer’s income be spent on between ‘good x’ and ‘good y’ if good y is “neutral”?
“Good” good x: m/pbottom rightx
Describe graphically optimal choice / consumer demand for perfect complements
Graph with “x1”-axis and y-axis, “x2”. Normal “budget line” but “indifference curves” are ‘L-shaped’, the tangent between the budget line and one of the ICs is at the angle of the L-shaped IC, the “optimal choice” point. From this point, there are dotted lines to the respective axes with respective labels, “x1” and “x2”. …
Describe graphically optimal choice / consumer demand for perfect complements
2 graphs, both with “x1”-axis and y-axis, “x2”. x-axis has a discrete scale (1, 2, 3 etc.):
First graph, titled “A: Zero units demanded”, has parallel ICs that are very uniquely structured: instead of a curve, the ICs have points, aligned with each number on the x-axis, at varying levels of “x2” , and for each IC, the points are joined up by dotted lines. The “budget line” for this graph has a tangent where it meets one of the ICs at its point on the y-axis. This point is the “optimal choice”.
The second graph, “B: 1 unit demanded”, has the same type of ICs. This time, the “budget line” goes through the point of one of the ICs that aligns with 1 on the x-axis. This is the “optimal choice”.
- We compare the utility of
consumption bundles:
(1, m-p1), (2, m-2p1), etc
Describe the ways we can find optimal choice for well-behaved preferences
- Graphically: draw the consumers budget constraint and indifference curves and see
which consumption bundle would maximise utility.
This is not a very precise option. - Mathematically:
Using Lagrangian or
Using tangency condition
Note! These methods are useful only for interior (both goods >0) tangency
solution
