EC204: Economics 2
Week 1 Topic 1: consumer theory
Monotonicity: more is better, e.g. If you assume monotonicity, you will choose the bundle on a
higher indifference curve (for well-behaved preferences) graph shows convexity
Convexity: averages are better than extremes e.g., preferring an average of both goods
instead having extreme amounts of a good in a bundle (for well-behaved preferences)
Z = tx1 + (1-t)y1, tx2 + (1-t)y2 where for convexity, t = 0.5 (z is point between x1y1,x2y2
Transitivity: if a person prefers option x to option y and they also prefer y to z, then they
would prefer x to z
~ means indifferent, > means strictly preferred over, ≥ means at least preferred
Exogenous: a variable that is not affected by other variables
Budget constraint: tells you the maximum amount that you can buy (If you
spend all your income); worked out by doing income/price of the good
represented by the y (if you only buy good y) and x intercept (if you only buy
good x).
Any point on the budget constraint shows that you are spending all of your
money. Any point inside the budget constraint is part of the feasible set, but
we don’t think about that as we always think about consming on our budget
constraint.
Combining budget constraint with indifference curve, you can find the optimum
consumption point when the slope of the budget line = slope of the indifference
curve (tangent); at this point you are on your highest possible IC so utility is
maximised but you are constrained by your budget
Completeness, continuity and transitivity required for rationality
Utility functions
Utility functions: describe preferences, assigning higher numbers to more preferred bundles. 2
goods that give the same level of utility lie on the same indifference curve,
The further from the origin, the higher the utility
Ordinal utility: when utility can only be ranked, and it can not be measured in numbers
Cardinal utility: when utility is observable and so can be given a number
Types of preferences: how to increase utility
Continuity: means that tiny changes in bundles will not change the ordering of preferences
Completeness: means that we can always compare or rank bundles. It is not possible to say that you
cannot compare 2 bundles
Perfect substitutes: if you had 2 perfect substitutes, you would not care
about which good you have (e.g., 3 Pepsi 0 Cola, or 1 Pepsi 2 Cola, you
would only care about the sum of the goods to maximise your utility
U(C,P) = aC +bP (where U represents units of happiness). Having one
extra can would always be betters as we assume monotonicity.
,EC204: Economics 2
Perfect complements: e.g., right shoes and left shoes. Having 1R and 0L shoes, U would be 0 and you
would have no satisfaction. 1R and 1L would mean U = 1. 2R and 1L would still mean U = 1.
The trade is not always 1 for 1, e.g., trading a £10 note would require 2 £5 notes.
The minimum of the 2 goods that you have is what is stopping you from achieving a higher utility. So,
U(SR, SL) = min{aSR, bSL} where the minimum of the 2 goods represents that maximum amount of
utility that you can achieve with the goods that you have.
Cobb-Douglas: a set of different utility functions that lie between the 2 extremes of perfect
substitutes and perfect complements. The format of the function explains the degree of
substitutability. The Cobb-Douglas utility
function is the simplest example of well-
behaved preferences as it gives us a nice
convex shaped indifference curve. The
powers sum to 1
You can take one Cobb-Douglas and transform it into another monotonic utility function for instance
by adding a Constance k
Monotonic transformations
Monotonic transformation: When you take a given utility function and apply a transformation to it
to create a new function, but with the same set of preferences; for this reason one indifference
curve can be described by an infinite number of utility functions.
It involves transforming a set of numbers into another set, but preserving the order (ordinal); it
doesn’t matter how much extra utility you get, it only matters that you get more utility
For example, if u = xy and you have values of x and y, you can work out the utility for each bundle by
multiplying x and y. The bundle with the highest u value has the highest utility. The values do not
matter, the only thing that matters is the order so you know which bundles are preferred to other
bundles.
You can transform the utility function to a different utility function (monotonic transformation).
For example, if v = 2xy (or v = 2u), there is no change in which
bundle is preferred to which, the order of them are still the same
so the new utility function still describes the same preferences
as the other. The same would apply for the utility function w = xy
+3
So you can take different transformations of different utility
functions to describe the exact same preferences and the order is
preserved.
You can’t work backwards from the optimal demand to find the
original utility function because every indifference curve has an
infinite number of utility functions
If you differentiate the utility function, and make that = 0, you
will know the x value in which the function is at its maximum and utility is maximised. You can use
the x value to work out the value on the y axis. 2u will give you the same x value as function u, and
the value on the y axis would double in this case
IC curves can not cross otherwise this violates rationality and transitivity.
The marginal rate of substitution
,EC204: Economics 2
MRS: if we cut the consumption of a good, you increase the consumption of the other good to put
you back on the same indifference curve, so the same level of utility is maintained. MRS tells you the
rate of substitution between the 2 goods.
Calculated by ΔX2/ ΔX1. You can also calculate MRS by calculating the ratio of the marginal utilities
(differentiate). Marginal utility tells us how valuable the good is to the consumer, and so how many
units of each good you are willing to trade to get more of the other good.
MRS and perfect substitutes
U(C,P) = aC +bP
Perfect substitutes have linear downward sloping indifference curves
because all of the bundles of goods give the exact same satisfaction
MRS = muC/muP = -a/b
To work out muC (marginal utility of coca cola) you have to find the
derivative of the function with respect to C. Do the same to work out mu P
As you move down the indifference curve, MRS remains constant as our ICs
are linear; this means the rate at which you are willing to substitute coca cola for Pepsi will always
remain constant
MRS and perfect compliments
U(SR, SL) = min{aSR, bSL}
IC is L shaped because you could have 1R and 1L, but if you have 2R your
utility doesn’t increase if you still have 1L. Same if you have 1R and 2L, the
extra shoe is pointless so your utility still doesn’t go up.
The only way utility can go up is if you have the same number of each shoe.
MRS is not calculated the same way because it not a linear graph.
MRS = ΔY/ ΔX, this would always be ∞ along the vertical proportion as ΔY/ 0
=∞
Along the horizontal proportion MRS would be 0/ ΔX = 0
And at the kink MRS is undefined
MRS and Cobb-Douglas
MRS = muX1 / muX2
u = x1∝ x21- ∝
differentiate with respect to x1 to get mux1 = ∝x1∝-1 x21-∝
mux2 = (1-∝)x1∝ x21-∝-1
divide the 2 and you get ∝/1-∝ X x2/x1 = MRS
A slope tangent to the IC would give us the MRS. The MRS
diminishes as you go down the IC
If you take a monotonic transformation, you would get the same MRS as
the first utility function. This means that you would have 2 different utility
functions that are describing the exact same preferences
Revealed preferences
, EC204: Economics 2
Offering consumers a choice between 2 bundles of goods and observing what they select, and based
we can make inferences about their preferences
if an
optimising consumer chooses (X1,X2) over (Y1,Y2), when the bundles are different this means that
they prefer bundle X. Given bundle X lies on their budget constraint, this means that the amount
that the consumer is spending on bundle X at prices p1 and p2 is exactly = to their income (as it is on
their budget constraint). Whereas if they buy bundle Y at the same prices p1 and p2, then they will
be spending less than their income as it is within the budget constraint. Both bundles can be
afforded by the consumer.
You are left with the inequality because bundle x is preferred over y and bundle x is on a higher
indifference curve
A consumer would buy bundle Y if X is no longer affordable; this is the only reason that makes the
most sense in terms of rationality.
The expenditure from buying bundle X at the original price must be at least as high as the
expenditure form buying bundle Y. At the new prices, the expenditure at prices Q1,Q2 can not be
greater than the expenditure at price X1,X2.
Indifference maps
Homothetic tastes: a situation where the consumer’s preferences
depend solely on the ratio of good 1 to good 2. When you draw a line
through the origin of a curve, through the ICs, the MRS on each IC on
that one ray is the same. So if you were to increase your income, the
demand for the 2 goods would increase by the same proportion. E.g. if
you double your income, you will double the number of the 2 goods
that you buy.
If the consumer prefers (X1,X2) > (y1,y2) then they will also prefer
(tx1,tx2) > (ty1,ty2)
Where MRS = -2, the blue line represents a lower income where less is
consumed, when income increases, the budget constraint shifts out
and you are on a higher IC in which the ratio of consumption of each
good stays the same, but you consume more so the ratio of trousers
to tops remains constant as income changes
Quasilinear tastes: tastes are linear in one good, but may not be in
another good – DOES NOT depend on the ratio so MRS depends on the
quantity of the quasilinear good and is independent on how much of
the other good that you have
Week 1 Topic 1: consumer theory
Monotonicity: more is better, e.g. If you assume monotonicity, you will choose the bundle on a
higher indifference curve (for well-behaved preferences) graph shows convexity
Convexity: averages are better than extremes e.g., preferring an average of both goods
instead having extreme amounts of a good in a bundle (for well-behaved preferences)
Z = tx1 + (1-t)y1, tx2 + (1-t)y2 where for convexity, t = 0.5 (z is point between x1y1,x2y2
Transitivity: if a person prefers option x to option y and they also prefer y to z, then they
would prefer x to z
~ means indifferent, > means strictly preferred over, ≥ means at least preferred
Exogenous: a variable that is not affected by other variables
Budget constraint: tells you the maximum amount that you can buy (If you
spend all your income); worked out by doing income/price of the good
represented by the y (if you only buy good y) and x intercept (if you only buy
good x).
Any point on the budget constraint shows that you are spending all of your
money. Any point inside the budget constraint is part of the feasible set, but
we don’t think about that as we always think about consming on our budget
constraint.
Combining budget constraint with indifference curve, you can find the optimum
consumption point when the slope of the budget line = slope of the indifference
curve (tangent); at this point you are on your highest possible IC so utility is
maximised but you are constrained by your budget
Completeness, continuity and transitivity required for rationality
Utility functions
Utility functions: describe preferences, assigning higher numbers to more preferred bundles. 2
goods that give the same level of utility lie on the same indifference curve,
The further from the origin, the higher the utility
Ordinal utility: when utility can only be ranked, and it can not be measured in numbers
Cardinal utility: when utility is observable and so can be given a number
Types of preferences: how to increase utility
Continuity: means that tiny changes in bundles will not change the ordering of preferences
Completeness: means that we can always compare or rank bundles. It is not possible to say that you
cannot compare 2 bundles
Perfect substitutes: if you had 2 perfect substitutes, you would not care
about which good you have (e.g., 3 Pepsi 0 Cola, or 1 Pepsi 2 Cola, you
would only care about the sum of the goods to maximise your utility
U(C,P) = aC +bP (where U represents units of happiness). Having one
extra can would always be betters as we assume monotonicity.
,EC204: Economics 2
Perfect complements: e.g., right shoes and left shoes. Having 1R and 0L shoes, U would be 0 and you
would have no satisfaction. 1R and 1L would mean U = 1. 2R and 1L would still mean U = 1.
The trade is not always 1 for 1, e.g., trading a £10 note would require 2 £5 notes.
The minimum of the 2 goods that you have is what is stopping you from achieving a higher utility. So,
U(SR, SL) = min{aSR, bSL} where the minimum of the 2 goods represents that maximum amount of
utility that you can achieve with the goods that you have.
Cobb-Douglas: a set of different utility functions that lie between the 2 extremes of perfect
substitutes and perfect complements. The format of the function explains the degree of
substitutability. The Cobb-Douglas utility
function is the simplest example of well-
behaved preferences as it gives us a nice
convex shaped indifference curve. The
powers sum to 1
You can take one Cobb-Douglas and transform it into another monotonic utility function for instance
by adding a Constance k
Monotonic transformations
Monotonic transformation: When you take a given utility function and apply a transformation to it
to create a new function, but with the same set of preferences; for this reason one indifference
curve can be described by an infinite number of utility functions.
It involves transforming a set of numbers into another set, but preserving the order (ordinal); it
doesn’t matter how much extra utility you get, it only matters that you get more utility
For example, if u = xy and you have values of x and y, you can work out the utility for each bundle by
multiplying x and y. The bundle with the highest u value has the highest utility. The values do not
matter, the only thing that matters is the order so you know which bundles are preferred to other
bundles.
You can transform the utility function to a different utility function (monotonic transformation).
For example, if v = 2xy (or v = 2u), there is no change in which
bundle is preferred to which, the order of them are still the same
so the new utility function still describes the same preferences
as the other. The same would apply for the utility function w = xy
+3
So you can take different transformations of different utility
functions to describe the exact same preferences and the order is
preserved.
You can’t work backwards from the optimal demand to find the
original utility function because every indifference curve has an
infinite number of utility functions
If you differentiate the utility function, and make that = 0, you
will know the x value in which the function is at its maximum and utility is maximised. You can use
the x value to work out the value on the y axis. 2u will give you the same x value as function u, and
the value on the y axis would double in this case
IC curves can not cross otherwise this violates rationality and transitivity.
The marginal rate of substitution
,EC204: Economics 2
MRS: if we cut the consumption of a good, you increase the consumption of the other good to put
you back on the same indifference curve, so the same level of utility is maintained. MRS tells you the
rate of substitution between the 2 goods.
Calculated by ΔX2/ ΔX1. You can also calculate MRS by calculating the ratio of the marginal utilities
(differentiate). Marginal utility tells us how valuable the good is to the consumer, and so how many
units of each good you are willing to trade to get more of the other good.
MRS and perfect substitutes
U(C,P) = aC +bP
Perfect substitutes have linear downward sloping indifference curves
because all of the bundles of goods give the exact same satisfaction
MRS = muC/muP = -a/b
To work out muC (marginal utility of coca cola) you have to find the
derivative of the function with respect to C. Do the same to work out mu P
As you move down the indifference curve, MRS remains constant as our ICs
are linear; this means the rate at which you are willing to substitute coca cola for Pepsi will always
remain constant
MRS and perfect compliments
U(SR, SL) = min{aSR, bSL}
IC is L shaped because you could have 1R and 1L, but if you have 2R your
utility doesn’t increase if you still have 1L. Same if you have 1R and 2L, the
extra shoe is pointless so your utility still doesn’t go up.
The only way utility can go up is if you have the same number of each shoe.
MRS is not calculated the same way because it not a linear graph.
MRS = ΔY/ ΔX, this would always be ∞ along the vertical proportion as ΔY/ 0
=∞
Along the horizontal proportion MRS would be 0/ ΔX = 0
And at the kink MRS is undefined
MRS and Cobb-Douglas
MRS = muX1 / muX2
u = x1∝ x21- ∝
differentiate with respect to x1 to get mux1 = ∝x1∝-1 x21-∝
mux2 = (1-∝)x1∝ x21-∝-1
divide the 2 and you get ∝/1-∝ X x2/x1 = MRS
A slope tangent to the IC would give us the MRS. The MRS
diminishes as you go down the IC
If you take a monotonic transformation, you would get the same MRS as
the first utility function. This means that you would have 2 different utility
functions that are describing the exact same preferences
Revealed preferences
, EC204: Economics 2
Offering consumers a choice between 2 bundles of goods and observing what they select, and based
we can make inferences about their preferences
if an
optimising consumer chooses (X1,X2) over (Y1,Y2), when the bundles are different this means that
they prefer bundle X. Given bundle X lies on their budget constraint, this means that the amount
that the consumer is spending on bundle X at prices p1 and p2 is exactly = to their income (as it is on
their budget constraint). Whereas if they buy bundle Y at the same prices p1 and p2, then they will
be spending less than their income as it is within the budget constraint. Both bundles can be
afforded by the consumer.
You are left with the inequality because bundle x is preferred over y and bundle x is on a higher
indifference curve
A consumer would buy bundle Y if X is no longer affordable; this is the only reason that makes the
most sense in terms of rationality.
The expenditure from buying bundle X at the original price must be at least as high as the
expenditure form buying bundle Y. At the new prices, the expenditure at prices Q1,Q2 can not be
greater than the expenditure at price X1,X2.
Indifference maps
Homothetic tastes: a situation where the consumer’s preferences
depend solely on the ratio of good 1 to good 2. When you draw a line
through the origin of a curve, through the ICs, the MRS on each IC on
that one ray is the same. So if you were to increase your income, the
demand for the 2 goods would increase by the same proportion. E.g. if
you double your income, you will double the number of the 2 goods
that you buy.
If the consumer prefers (X1,X2) > (y1,y2) then they will also prefer
(tx1,tx2) > (ty1,ty2)
Where MRS = -2, the blue line represents a lower income where less is
consumed, when income increases, the budget constraint shifts out
and you are on a higher IC in which the ratio of consumption of each
good stays the same, but you consume more so the ratio of trousers
to tops remains constant as income changes
Quasilinear tastes: tastes are linear in one good, but may not be in
another good – DOES NOT depend on the ratio so MRS depends on the
quantity of the quasilinear good and is independent on how much of
the other good that you have
