CARDIFF BUSINESS SCHOOL
MACROECONOMICS (BS1652)
Spring Semester 2005-06
Maths Macro Lecture 1
Deriving Aggregate Supply in the 4-quadrant model
The key element here is understanding the labour market which, of course, was
something that you studied in Microeconomics in the Autumn Semester.
There are two key components to the labour market: demand and supply – let us
examine these in turn.
The demand for labour
Labour is not demanded for its own sake but rather because it can help a firm earn
profits, which it does by increasing output which the firm can sell. Assuming that the
firm is a profit-maximiser, and that labour is the only variable factor in the short-run,
then we have the following situation:
The firm faces a production function:
Q = f (L)
where Q is output and L is the amount of labour employed.
The firm’s cost (C) of production will be:
C = w.L + F
where w is the cost of labour and F is Fixed Costs.
The firm’s revenue (R) from selling the good is:
R = p. Q = p. f (L)
The firm’s profits () are given by:
 = R–C
= p. f (L) – ( w.L + F )
= p. f (L) - w.L - F
To maximise profits:
d / dL = 0
d / dL = p. (df / dL) – w = 0
or
p. (df / dL) = w
………(1)
df / dL = (w / p)
………(2)
Equation (1) can be interpreted as saying that a profit maximising firm will employ
labour up to the point where the value marginal product of labour is equal to the
money wage rate.
Equation (2) can be interpreted as saying that a profit maximising firm will employ
labour up to the point where the marginal product of labour is equal to the ‘real
wage’.
Note that, as the ‘real wage’ changes, so too will the demand for labour. Since the
marginal product of labour eventually diminishes (as more of the variable factor,
labour, is used with the fixed factor, capital), then the demand for labour schedule will
also diminish, i.e. it will be negatively-sloped.
The supply of labour
Most individuals/households do not work because they like it, rather it is a means to
an end, namely, gaining income with which to buy goods and services. How much
individuals/households will be willing to work will clearly depend upon the amount of
money being offered to them to give up their leisure time in order to work. Assuming
that individuals/households are utility maximisers, then their choice as to how much
to work will depend on their tastes and preferences and the wage being offered: this
can be analysed using indifference curve analysis, as was done in microeconomics in
the Autumn semester. From this we can examine the response of the
individual/household to differing wage rates. Clearly, though:
nS = g (w / p)
Where nS is the amount of labour supplied and (w / p) is the real wage. Under
normal circumstances we would expect this function to be a positive one (i.e. has a
positive slope) for at least part of its length, though the possibility that it then bends
‘backwards’ (i.e. the slope becomes negative) can not be ruled out (it will depend on
the relative sizes of the income and substitution effects, assuming that leisure is a
normal good).
The macroeconomic labour market
Having determined a firm’s demand for labour and the individuals/households supply
of labour, how can we derive the macroeconomic labour market? Clearly it is
necessary to aggregate across all firms and all individuals/households. In this way it
will be possible to determine both the aggregate demand for labour and the aggregate
supply of labour which, when brought together, will make it possible to determine the
equilibrium ‘real wage’ in the economy and the equilibrium level of employment.
The aggregate demand for labour
If we assume that all firms operate in perfect competition, then all firms will have an
identical demand for labour. If there are n such firms in the economy, and each of
them wishes to demand the same level of labour at each real wage rate, then the
aggregate demand at each wage rate will simply be n times that level of labour. So if
each firm should desire to employ L1 units of labour at a particular ‘real wage’, then
aggregate demand will be n. L1 . [Note that the amount a firm would wish to employ
in order to maximise profits will, as we have already seen, depend upon the ‘real
wage’.]
The aggregate supply of labour
The situation is not quite the same for labour supply as for labour demand, since all
individuals/households will not be identical (as were the firms). Nevertheless the
same underlying principle applies, namely, that the aggregate supply of labour at any
given ‘real wage’ will be the sum of the amounts each individual/household in the
economy is willing to supply at that ‘real wage’.
A key point to notice here is that both the aggregate demand for labour and the
aggregate supply of labour will depend upon the ‘real wage’. The macroeconomic or
‘aggregate’ labour market can therefore be depicted by two equations, as follows
Demand:
ND = h (w / p)
Supply:
NS = j (w / p)
CARDIFF BUSINESS SCHOOL
MACROECONOMICS (BS1652)
Spring Semester 2005-06
Lecture 1
Numerical examples
1.
A profit-maximising firm operating in perfect competition is able to sell its
product for £5 per unit and can purchase labour for £25 per unit. If the firm
faces a production function given by Q = 40. L½ , how many units of labour
will the firm employ and what will be its level of output?
Answer
To profit maximise, firm must set the marginal product of labour equal to the
‘real wage’.
The marginal product of labour is given by:
MPL = 20. L-½
The ‘real wage’ is equal to:
( w / p) = £25 / £5 = 5
Therefore the firm will demand labour where:
20. L-½ = 5
20 = 5. L½
L½ = 4
L = 16
At this level of employment, output is:
Q = 40. (16)½
= 40. (4)
= 160
2.
Assume that a perfectly competitive economy consists of 1000 identical firms,
all facing the production function given by Q = 17. L - (½). L2.
If the aggregate supply of labour curve is given by the equation
NS = 2000 ( w / p ) - 10000 , and the market wage rate of labour is £18 and
the price of the firm’s product is £2, show that the labour market is in
macroeconomic equilibrium. Find the level of aggregate supply in this
economy.
Answer
For an individual firm, profit maximisation occurs where
dQ / dL = ( w / p )
17 – L = ( 18 / 2 )
L = 8
Since there are 1000 firms, all identical, the aggregate demand for labour will
be 1000 x 8 = 8000.
From the aggregate supply of labour curve, we get that:
NS = 2000 ( w / p ) - 10000
= 2000 ( 9 ) - 10000
= 8000
Since ND = NS , the labour market is in macroeconomic equilibrium.
With each firm employing 8 units of labour, the firm will produce:
Q = 17. (8) - (½) (8)2
=
136 - 32
=
104
Aggregate supply will therefore be equal to 104 x 1000 = 104,000.
3.
Let us assume that the aggregate supply of labour schedule is given by:
NS = 1000 + 25 ( w / p)
and the aggregate labour demand schedule by:
ND = 2000 - 15 ( w / p )
(a)
(b)
Find the equilibrium levels of the ‘real wage’ and employment.
Assume that individuals/households now determine to work less at
every ‘real wage’. Write down a new aggregate supply of labour
schedule to represent this change, and show that the equilibrium levels
of both employment and output will fall.
Answer
(a)
NS = 1000 + 25 ( w / p)
ND = 2000 - 15 ( w / p )
For market to be in equilibrium, ND = NS , i.e.
1000 + 25 ( w / p ) = 2000 - 15 ( w / p )
40 ( w / p ) = 1000 ,
 ( w / p ) = 25
Given this wage rate:
ND = 2000 - 15 ( 25 ) = 2000 – 375 = 1625
(b)
If individuals/households decide to work less at each ‘real wage’, then
the labour supply schedule will shift upwards to the left. This can be
represented by a reduction in the intercept term of the labour supply
function. Hence, let the new labour supply function be:
NS = 800 + 25 ( w / p )
The new labour market equilibrium will therefore be at:
800 + 25 ( w / p ) = 2000 - 15 ( w / p )
40 ( w / p ) = 1200 ,
( w / p ) = 30
Given this wage rate:
ND = 2000 - 15 ( 30 ) = 2000 – 450 = 1550
Hence, the reduction in willingness of individuals/households to work,
raises the equilibrium ‘real wage’ (from 25 to 30) and reduces the level
of output in the economy (since employment has fallen from 1625 to
1550).