Microeconomics 3: Income-Substitution Effect (Slutsky), Revealed Preferences and Edgeworth box Flashcards
Introduction 1.1 Consumer choice building blocks: budget constraint, preferences, and utility function; 1.2 Consumer’s optimal choice; 1.3 Consumer demand Income effect; Price effect; Slutsky income and substitution effect.
You’ll need the demand function
Describe and interpret what the inverse demand function is
For a utility function 𝑈(𝑥, 𝑦) = 𝑥^𝑐 x 𝑦^𝑑, the solution for optimal consumption of
good 𝑥 is:
𝑥 = 𝑐 / 𝑐+𝑑 x 𝑚/𝑝bottom right𝑥
We can transform it and then:
𝑝bottom right𝑥 = 𝑚𝑐/𝑥(𝑐 + 𝑑)
The first representation is a demand function, the second – is the inverse
demand function.
Interpretation: the downward sloping inverse demand function shows also the
willingness to pay more when the amount of good x is little and less as x grows
larger.
Describe the inverse demand function graphically
If the demand curve is viewed measuring price as a function of quantity, we
have the inverse demand function
Graph with “Pbottom rightx” y-axis and “X” x-axis and a negative curve that’s decreasing less & less. Curve labelled “Inverse demand curve Pbottom rightx(x)
Illustrate & describe the overall effect from a change in price of a good. Then illustrate & describe the substitution effect. Then illustrate & describe the income effect
Graph with “y”-axis and “x”-axis and 2 straight budget constraint lines. Both start at same point on y-axis but one has steeper gradient than another. Each budget constraint has a tangent to an IC. The ICs are parallel to each other. The tangent at the steeper budget constraint is “A” and the other is “B”. The difference between A and B is “Total increase in x”.
Suppose the consumer is
maximising utility at point A.
If the price of good x falls, the
consumer will maximise utility
at point B.
On previous graph, there’s now a purple dotted budget constraint line, parallel to the previous budget constraint with less steep slope, going through A (imaginary budget constraint). There’s a new IC that matches for this budget constraint as the lowest point of this IC touches this imaginary line and is point “C”. The difference between A and C, labelled at x-axis is “Substitution effect”.
To isolate the substitution effect, we
hold purchasing power constant but allow
the relative price of good x to change:
➢The substitution effect is the
movement from point A to point C;
➢The individual substitutes good x for
good y because x is now relatively
cheaper
There’s a difference showing between point C and B, at x-axis it’s labelled “Income effect”
The income effect occurs because the
individual’s “real” income changes
when the price of good x changes:
- The income effect is the
movement from point C to
point B;
If x is a normal good, the
individual will buy more
because “real” income
increased
Describe the price changes for normal goods and inferior goods
Also, compare in what direction substitution effect operates vs income effect
If a good is normal, substitution and income effects reinforce one another
when price falls, both effects lead to a rise in quantity demanded;
when price rises, both effects lead to a drop in quantity demanded.
If a good is inferior, substitution and income effects move in opposite directions
and the combined effect is indeterminate:
when price rises, the substitution effect leads to a drop in quantity demanded, but the
income effect is opposite;
when price falls, the substitution effect leads to a rise in quantity demanded, but the
income effect is opposite
Substitution effect always operates in the same direction - we’re substituting towards the relatively cheaper good. Income effect can work in either direction
Describe & explain Slutsky’s Effects for Normal Goods. Include illustration afterwards
- Most goods are normal (i.e. demand increases with income).
The substitution and income effects reinforce each other when a normal
good’s own price changes:
Since both the substitution and income effects increase demand when own-price falls,
a normal good ordinary demand curve slopes down;
The Law of Downward-Sloping Demand therefore always applies to normal goods - Now for a model that describes the Slutsky Effect: Model: “y”-axis and “x”-axis. There’s already a consumption bundle set up with an optimal consumption bundle with “x1” and “y1” signified at their respective axes. The whole thing is about decomposing income and substitution effect. Explaining substitution effect first, purchasing power needs to be held constant as substitution effect doesn’t change this. To show only substitution effect, a purple line is drawn, this is the new budget line for a lower price of x, with a less steep gradient than the original budget line. It goes through (x1, y1). Now hypothetical optimal consumption can be found by drawing an indifference curve that is tangent to the purple line and parallel to the other indifference curve. The point where this IC and the purple line touch is the optimal consumption, “x2” - how much of x2 one would consume if we’re taking the new price ratio but holding purchasing power constant. It’s signified with a purple dot. “y2” is also shown at the axis.The substitution effect is x2-x1 (shown by arrow from x1 to x2 along axis). How do we know? By construction - the purple line held the income constant but the price has changed, so more of x2 is being consumed because it’s relatively cheaper which is the difference between x2 and x1. Now for the income effect. Essentially, there’s just more purchasing power so basically the new budget line is parallel to purple budget line but higher and there’s a higher IC. This budget line must touch the y-axis where the original budget line (the one for x1, y1) touches the y-axis as it illustrates the fact that the price of good y hasn’t changed.There’s a green dot at the new optimal consumption point, “(x3, y3)”. The income effect is x3 - x2 (shown by arrow along axis) because that’s the difference between the green optimal point and the purple point. Add together the income effect and substitution effect: x3 - x2 + x2 - x1 = x3 - x1. This is Slutsky Effect which is that total effect is the same as the sum of income and substitution effect. Arrows also drawn along y-axis from y1 to y2 and y2 to y3 Good x is normal because higher income increases demand, so the income and substitution effects reinforce each other.
Describe & explain Slutsky’s Effects for Income-Inferior Goods. Include illustration afterwards
Some goods are income-inferior (i.e. demand is reduced by higher income).
The substitution and income effects oppose each other when an income-
inferior good’s own price changes.
Model: “y”-axis and “x”-axis. There’s already a consumption bundle set up with an optimal consumption bundle with “x1” and “y1” signified at their respective axes. The whole thing is about decomposing income and substitution effect. Explaining substitution effect first, purchasing power needs to be held constant as substitution effect doesn’t change this. To show only substitution effect, a purple line is drawn, this is the new budget line for a lower price of x, with a less steep gradient than the original budget line (the good is income-inferior not price inferior so lower price still has same effect as it does for normal good). It goes through (x1, y1). Now hypothetical optimal consumption can be found by drawing an indifference curve that is tangent to the purple line and parallel to the other indifference curve. The point where this IC and the purple line touch is the optimal consumption, “x2” - how much of x2 one would consume if we’re taking the new price ratio but holding purchasing power constant. It’s signified with a purple dot. “y2” is also shown at the axis. The substitution effect is x2-x1 (shown by a rightwards arrow from x1 to x2 along axis labelled “substitution effect”). How do we know? By construction - the purple line held the income constant but the price has changed, so more of x2 is being consumed because it’s relatively cheaper which is the difference between x2 and x1. Essentially, there’s just more purchasing power so basically the new budget line is parallel to purple budget line but higher. This budget line must touch the y-axis where the original budget line (the one for x1, y1) touches the y-axis as it illustrates the fact that the price of good y hasn’t changed. The IC, however, is not parallel to the other ones. Good x is an income-inferior good, so now that income has risen, its demand has dropped. This new IC will have a more shallow gradient and its tangency point with the budget line will be more leftwards of x2. There’s a green dot at the new optimal consumption point, “(x3, y3)”. The income effect is x3 - x2 (shown by leftwards arrow along axis labelled “income effect”) because that’s the difference between the green optimal point and the purple point. If x3 - x2 is still smaller than x2-x1, the overall effect will still be x3>x1 so it’s still an ordinary good.
The pure substitution effect is as for a normal
good. But, the income effect is in the opposite
direction. Good x is income-inferior because an
increase to income causes demand to fall.
The overall changes to demand are
the sums of the substitution and
income effects.
Describe & explain Giffen Goods and Slutsky’s Effects on them. Include illustration afterwards
- In rare cases of extreme income-inferiority, the income effect may be larger in
size than the substitution effect, causing quantity demanded to fall as own-
price falls.
Such goods are Giffen goods.
Slutsky’s break down of the effect of a price change into a pure substitution effect and
an income effect thus explains why the Law of Downward-Sloping Demand is violated
for extremely income-inferior goods - Model: “y”-axis and “x”-axis. There’s already a consumption bundle set up with an optimal consumption bundle with “x1” and “y1” signified at their respective axes. The whole thing is about decomposing income and substitution effect. Explaining substitution effect first, purchasing power needs to be held constant as substitution effect doesn’t change this. To show only substitution effect, a purple line is drawn, this is the new budget line for a lower price of x, with a less steep gradient than the original budget line (the good is a Giffen good which is still income-inferior not price inferior so lower price still has same effect as it does for normal good). It goes through (x1, y1). Now hypothetical optimal consumption can be found by drawing an indifference curve that is tangent to the purple line and parallel to the other indifference curve. The point where this IC and the purple line touch is the optimal consumption, “x2” - how much of x2 one would consume if we’re taking the new price ratio but holding purchasing power constant. It’s signified with a purple dot. “y2” is also shown at the axis. The substitution effect is x2-x1 (shown by a rightwards arrow from x1 to x2 along axis labelled “substitution effect”). How do we know? By construction - the purple line held the income constant but the price has changed, so more of x2 is being consumed because it’s relatively cheaper which is the difference between x2 and x1. Essentially, there’s just more purchasing power so basically the new budget line is parallel to purple budget line but higher. This budget line must touch the y-axis where the original budget line (the one for x1, y1) touches the y-axis as it illustrates the fact that the price of good y hasn’t changed. The IC, however, is not parallel to the other ones. Good x is a Giffen good which is an income-inferior good, so now that income has risen, its demand has dropped. This new IC will have a more shallow gradient and its tangency point with the budget line will be more leftwards of x2 and even x1. There’s a green dot at the new optimal consumption point, “(x3, y3)”. The income effect is x3 - x2 (shown by leftwards arrow along axis labelled “income effect”) because that’s the difference between the green optimal point and the purple point. x3 - x2 is larger than x2-x1, the overall effect will be x3<x1, signifying that it’s a Giffen good
- In this case, X3 is smaller than X1. The overall effect is STILL X3 - X1 as in any case, it’s always the LAST consumption minus the first consumption, it’s juts that the overall effect is negative in this case
Describe the price changes algebraically for normal and inferior goods
What is this in words? What is this called?
Sign of the substitution effect:
If the price decreases, then the demand increases or the substitution effect is negative
Total change in demand:
Δx=x(Pbottom rightxtop right2,mtop right1) - x(Pbottom rightxtop right1, mtop right1) =
=[x(Pbottom rightxtop right2,m2)-x(Pbottom rightxtop right1, mtop right1)]+[x(Pbottom rightxtop right2, mtop right1)-x(Pbottom rightxtop right2,mtop right2)]
- It’s essentially saying X3 - X1 = X2 - X1 + X3 - X2.
Otherwise said, the total change in demand is equal to the sum of the substitution and the
income effect: Slutsky identity
Describe & explain the types of utility functions and whether they have income and
substitutions effects
- Cobb-Douglas utility functions can have both income & substitution effect
- Perfect complements have an income effect but don’t have substitution effect because they’re perfect complements, they’re consumed in fixed proportion so we can’t substitute one for another
Describe the idea of revealed preferences
We cannot observe consumers’ preferences directly: we can either ask about
the preferences or observe choices.
Revealed preferences analysis is based on observing the consumption choices
(demand) of consumers for different prices and incomes.
These observations enable us to test our modelling hypothesis and to
“discover” consumers’ preferences
What do we assume about revealed preferences to make our analysis of it simpler?
To make our analysis simpler we will assume that:
Preferences do not change over time;
Are monotonic and convex – this means that the preferred affordable bundle is
unique
There’s a simple budget line with point “(x1, x2)” on the line and point “(y1, y2)” beneath the line”. What can we say to compare the 2 points?
(𝑥1, 𝑥2) is ‘revealed preferred to’ or ‘chosen over’ (𝑦1, 𝑦2) or ‘x ≻underneathD(means ‘directly’) y’
On a budget line, when can we say that a certain point is ‘revealed preferred’ to another point?
Both graphically and algebraically
How can we write it as?
- Graphically: If it’s above the other point; if we were to draw parallel budget lines on both points, which budget line is higher
- Algebraically: If
- 𝑝bottom right1𝑥bottom right1 + 𝑝bottom right2𝑥bottom right2 ≥ 𝑝bottom right1𝑦bottom right1 + 𝑝bottom right2𝑦bottom right2
We can write is as ‘x ≻underneathD(means ‘directly’) y’
Describe ‘indirectly revealed preferences’
- Suppose X is directly revealed preferred to Y and Y is directly revealed preferred to Z.
By transitivity we then have that X is revealed indirectly preferred to Y - Graph with “x1”-axis and y-axis, “x2”. With 2 “budget constraint lines” with 2 different gradients. There’s: a point on the steeper gradient line, “(x1, x2)” , a point on the shallower gradient line, “(y1, y2)” and a point beneath both lines, “(z1, z2)”
- x ≻underneathD(means ‘directly’) y and y ≻underneathD z and x ≻underneathI (means ‘indirectly’) z
Describe the ‘principle of revealed preference’
Let (𝑥1, 𝑥2) be the chosen bundle when prices are (𝑝1, 𝑝2), and let (𝑦1, 𝑦2) be
some other bundle such that
𝑝1𝑥1 + 𝑝2𝑥2 ≥ 𝑝1𝑦1 + 𝑝2𝑦2.
Then, if the consumer is choosing the most preferred bundle they can afford,
we must have that (𝑥1, 𝑥2) ≻ (𝑦1, 𝑦2).
This may look obvious or circular – but we can reformulate this principle as:
“If a bundle X is chosen over Y, then X must be (is revealed) preferred to Y”
