Download exercise 6: measures of the welfare effect of a price change

MICROECONOMICS 2
TUTORIAL 6: MEASURES OF THE WELFARE EFFECT OF A PRICE
CHANGE
Timing of Tutorial This tutorial uses the material contained in chapter 19 and a part
of that in chapter 17.
Purpose of Tutorial To apply the material of chapter 19 to two particular types of
preferences and to show that the results asserted there are true for these particular
preferences.
Prior Preparation You should do as much as you can before the tutorial. You should
certainly have done parts (1) to (5) for both the case of perfect substitutes and the case
of perfect complements.
Written Work after Tutorial You should write up all the tutorial and hand it in to
your tutor.
Relevance to Examination It is possible that you will get a similar kind of exercise in
the examination.
We are going to consider various measures of the welfare effect of a price
change on two individuals who are alike in all respects except their preferences. To
avoid maths, we shall do all the analysis graphically and you are advised to draw very
precise graphs – following the instructions below.
The two individuals are A and B. We consider two ‘goods’ – the good whose
price will change and ‘all other goods’. We shall put the price of ‘all other goods’
equal to 1 and let p denote the price of the good in which we are interested. We are
going to get measures of the effect on the welfare of the two individuals of a rise in
the price p. Specifically we shall start with p = 0.8 and consider a rise to p = 1.2.
Both individuals have an income of 120.
(1)
We start with A. His or her preferences over the two ‘goods’, the good of
interest and all other goods, are perfect 1:1 substitutes. In other words, to this
individual the good of interest is effectively the same as all other goods. We will
analyse his or her behaviour graphically. Take a sheet of graph paper and draw on a
set of axes – we will measure the quantity of the good of interest along the horizontal
axis and we will measure ‘all other goods’ up the vertical axis. Calibrate both axes
from 0 to 150. Draw in the initial budget constraint – that given by an income of 120,
a price of the good of interest – henceforth just ‘the good’ – equal to 0.8, and the price
of ‘all other goods’ equal to 1. If you have drawn this correctly it goes from 150 on
the horizontal axis to 120 on the vertical axis.
(2)
Given A’s preferences argue that the optimal point on this budget line is
(150,0) – that is, A spends all his or her money on the good. Mark this point on your
graph. Draw in also the indifference curve passing through this point – which is
obviously the highest indifference curve that A can reach given the budget constraint.
(3)
Now draw in the new budget line – that with the higher price 1.2 for the good.
Argue that this goes from 100 on the horizontal axis to 120 on the vertical axis. Argue
that the optimal point for A on this new budget line is the point (0, 120) – that is, A
spends all his or her income on all other goods. Mark this point on your graph. Draw
in also the indifference curve passing through this point – which is obviously the
highest indifference curve that A can reach given the budget constraint. Note that this
indifference curves is lower than the originally highest attainable indifference curve.
Why?
(4)
The price rise of the good – from 0.8 to 1.2 – has made the individual worse
off. What we want to do is to get a monetary measure of how much worse off he or
she is. One possible measure is the compensating variation. This is the amount of
money we need to give to the individual to restore him or her to the same level of
happiness (the same indifference curve) as before the price rise. Show that in this case
the compensating variation is 30. Why is this less than the increased cost of the
originally purchased bundle of goods? (Originally the individual bought 150 units of
the good – costing 120 at the original price and 180 at the new price.)
(5)
Now calculate the equivalent variation – this is the amount of money that we
need to take away from Individual A at the original prices to have the same impact on
his or her welfare as the price rise. Show that the equivalent variation in this case is
24.
(6)
Now let us approximate the change in the surplus of the individual caused by
the price rise. As the surplus before the price rise was the area between the demand
curve of the individual and the price of 1.2, and the surplus after the price rise was the
area between the demand curve of the individual and the price of 0.8, it follows that
the loss in consumer surplus is the area between the two prices (1.2 and 0.8) and the
demand curve of Individual A. You should plot this demand curve carefully in a
separate graph – with the quantity of the good from 0 to say 250 along the horizontal
axis and the price of the good from 0 to say 1.5 up the vertical axis. What is the
demand curve of the individual? Well – if p is greater than 1 the individual does not
buy any of the good and so the demand is zero – and if p is less than 1 the individual
spends all his or her income on the good so the demand is q = 120/p (notice that if q =
120/p then it follows that pq = 120). Plot this carefully – and then approximate the
area between the two prices and the demand curve. It is around 27. (To calculate it
analytically requires the use of calculus.)
(7)
So we have shown that the compensating variation is 30, the change in the
consumer surplus is approximately 27 and the equivalent variation is 24. Why are
they not all equal? Are Perfect Substitutes not an example of quasi-linear preferences?
(Be careful – the condition for quasi-linearity is that the indifference curves are
parallel everywhere.)
(8)
Now let us repeat all the above for Individual B who has perfect 1 with 1
complements preferences over the two goods – the good of interest and ‘all other
goods’. Argue that at the original price (p = 0.8) B will buy 66.66666…. units of
each of the two goods and at the new price (p = 1.2) B will buy 54.545454…. units of
the two goods. Using the same methods as before show that the compensating
variation is 26.666… and the equivalent variation is 21.83838… Note that the
compensating variation in this case is exactly equal to the increased cost of the
originally purchased bundle of goods. (0.4 times 66.6666… equals 26.666….) Why?
(9)
Argue that the demand curve for the good of interest is given by the
intersection of the budget constraint (pq1 + q2 = 120) and the line that joins up the
corners of the indifference curves (q2 = q1) and hence is given by q1 = 120/(1 + p).
Draw this in a separate graph with q1 (the quantity of the good of interest) along the
horizontal axis (from 0 to 120) and with its price p up the vertical axis – from 0 to 2.
Argue as before that the change in consumer surplus caused by the price rise is the
area bounded by the two prices (0.8 and 1.2) and this demand curves. Find this area –
approximately. What is its relationship with the compensating and equivalent
variations found earlier?
(10) We note something interesting: for each measure of the loss in welfare – the
compensating variation, the change in surplus, and the equivalent variation – the
measure is always greater for Individual A than for Individual B. Why?
(11) Is the result we found in (10) above always true (that is, for any price rise)?
Why or why not?
(12) Just to pick up a point from Chapter 17 – if you are keen – can you draw the
aggregate demand curve of the two individuals and calculate the aggregate loss of
consumer surplus? Is it true that this equals A’s loss plus B’s loss?
Comments as to what you should take away from this tutorial Confidence in your
ability to calculate compensating and equivalent variations and to compute the
change in consumer surplus is the prime thing. You should also be beginning to get
some feel for what these measures of welfare change mean – what determines their
sizes and relative magnitudes.