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Economics 318: Macroeconomics
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In depth lecture notes made throughout the semester of ONLY the macroeconomics component, presented by Dr van Lill.
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Chapter 11: A Real Intertemporal Model with InvestmentThe reason this chapter is so important is because it forms the foundation for chapters 11
through 18. It is a real model. The economists in the slides are: Robert Lucas, Ben Bernanke
and Paul Romer.
Learning Outcomes
• Explain the decisions made by the representative consumer in the real intertemporal
model
• Explain the decisions made by the representative firm in the real intertemporal
model
• Show how the firm’s investment decision is structured, and determine how changes
in the environment faced by the firm affect investment
• Construct the output supply and demand curves
• Show how a competitive equilibrium is determined in the real intertemporal model
• Use the real intertemporal model to explain the effects of particular shocks to the
economy
The Real Intertemporal Model
It includes a lot of different components that have been discussed already.
• Includes current and future periods
• Representative Consumer – consumption/savings decision – microeconomic
component that we are introducing – we take all consumers and aggregate their
preferences. Here, the consumer makes 2 different consumption/savings decisions
and a work/leisure decision.
• Representative Firm – hires labour and invests in current period, hires labour in
future (involved in the labour market and investment) – we take all firms and
aggregate their costs
• Government – spends and taxes in present and future, and borrows on the credit
market
Representative Consumer’s Budget Constraint
The consumer’s current-period budget constraint: C + SP = w(h – l) + - T.
The consumer’s future-period budget constraint: C’ = w’(h’ – l’) + ’ – T’ + (1 + r)SP
The (h – l) is the hours worked and leisure trade off. This is slightly different to what we had
before. The main difference is the introduction of labour on the consumer side – how hard
people want to work. The labour supply is with the consumer and labour demand is with the
firm. How much you decide to consume and save is all found on the consumer side. The
consumer makes labour supply decisions in the current and future periods, and a
consumption/savings decision in the current period. So, consumption and saving decisions
aren’t completely endogenous of labour market decisions. We associate the idea of
working, and with working comes a wage – this is where the ‘w’ comes in.
The p suggests we are getting profit from firms, so the implicit assumption is that we are the
owners of the firms.
,Minus T means that we are getting taxed. So we can consume on the left side and the right
side is our income to a certain extent.
We can combine the above equations to form a life time budget constraint:
𝐶′ 𝑤′(ℎ − 𝑙′) + ′ − T′
C + 1 + 𝑟 = w(h – l) + - T + 1+𝑟
Marginal Conditions for the Consumer
The consumer makes 3 choices: current labour supply, future labour supply, current savings.
Each can be summarized by a standard marginal condition. We need to make the decision in
the current period of if we want leisure or if we want to consume, and this is reflected in the
wage. When the wage increases, it makes it more attractive for you to work. The same thing
happens in the future period. So the marginal rate of substitution between leisure and
consumption and the marginal rate of substitution between future leisure and future
consumption is a function of the wage. The wage is going to determine by how much you
are willing to trade off one for the other. The optimality conditions:
• Current period: MRSl,c = w
• Future period: MRSl’,c’ = w’
• Intertemporal Choice: MRSc,c’ = 1 + r
Consumer’s Current Labour Supply Behaviour
Labour supply is something the consumer has to decide on. There are 3 basic factors:
1. Current labour supply increases the real wage (substitution effects are assumed to
dominate income effects) – if the real wage goes up, we supply more labour
2. Current labour supply increases with an increase in the real interest rate through an
intertemporal substitution effect – if the real interest rate goes up, you want to save
more, and in order to save more, you need to work more in the current period, and
thus leisure decreases.
3. An increase in lifetime wealth (e.g. taxes fall) reduces labour supply – this reduces
labour supply because we want to consume more leisure as leisure is a normal good
The way we think about how labour supply is going to be constructed is based on these 3
factors.
The decisions that consumers make determines the
labour supply curve. This graph:
Figure 11.1: The Representative Consumer’s
Current Labour Supply Curve. The marginal
condition here: MRSl,c = w
We assume that the substitution effect is stronger
than the income effect. This graph says that with
increases in the real wage rate, current labour
supply increases.
, Figure 11.2: An Increase in the Real Interest
Rate Shifts the Current Labour Supply Curve
to the Right. This is an important point and is
something we will see often. We care about
the (1 + r) component, and will need it when
defining equilibrium conditions. The marginal
condition: MRSc,c’ = 1 + r
An increase in the real rate, given w and w’,
results in an increase in the price of current
leisure relative to the price of future leisure.
So, if we experience an increase in r, we will
see an outward shift in the labour supplied.
This is because if interest rates increase, there
is an opportunity for us to save more, so we will work more in order to capitalise this.
Example: Paul is self-employed ad the market rate rises. Paul faces higher returns on
savings, so if he works more now in the current period, and saves proceeds, he can both
consumer more and work less in the future.
Figure 11.3: Effects of an Increase in Lifetime
Wealth. This means disposable income is going to
increase, which also means an increase in the
quantity of leisure. This is because leisure is a
normal good. An increase in lifetime wealth
increases quantities of current and future leisure.
So the curve, for a given wage rate, shifts to the
left.
The Current demand for Consumption Goods
Marginal propensity to consume (MPC) is the increase in demand for consumption goods
induced by a one-unit increase in current real income. Intertemporal substitution: the
demand for consumption goods decreases with an increase in the real interest rate. An
increase in lifetime wealth increases the demand for consumption goods. This means we
know what the shape of the demand curve is going to look like.
Figure 11.4: The Consumer’s Current Demand for
Consumption Goods Increases with Income. The
slope is going to be equal to MPC – the amount of
total income you consume. This decreases with
higher values of Y. The slope flattens out with an
increase in Y. Proportionally, we would expect low
income earners to spend more of their income. This is
reflected on the graph – MPC is higher for low income
, levels, and as we move towards high income levels we see MPC flatten out. This is why the
curve has to be concave.
Figure 11.6: An Increase in Lifetime
Figure 11.5: An Increase in the Real Interest
Wealth Shifts the Demand Curve for
Rate Shifts the Demand for Consumption
Consumption Goods Up.
Goods Down.
Representative Firm
We know that the firms current and future production technology are given by: Y = zF(K , N)
and Y’ = z’F(K’ , N’) respectively. The z component is going to be out current technological
augmentation of a production function, and z’ is going to represent future technology. An
important assumption is that z’ is known in the current period. So, when we make decisions
on how productive our firm is going to be in the future, we know this information already.
The evolution of the firm’s capital stock: K’ = (1 – d)K + 1
The firms current and future profits are determined as follows:
= Y – wN – 1 and ’ = Y’ – w’N’ + (1 + d)K’
This means: the total output that they produce, minus whatever they pay labourers minus
investments that they make. Future profits are the future outputs produced, minus what
the pay labourers in the future + what they bring from the previous period in terms of
capital.
The firm is going to try and maximise their profits by that the profits from today, and the
profits of tomorrow, but discount the profits of tomorrow by 1 + r to get it in present value
′
terms: V = + 1 + 𝑟 . It can make future decisions today because it knows everything about
the future economy. So it makes decisions on N, N’ and I.
Figure 11.7: Demand Curve for Current Labour Is
Representative Firm’s MPN Schedule
The demand curve for current labour demanded is
going to be a downward sloping curve, and the slope
of this is determined by the marginal productivity of
labour: MPN = w – this is because MPN declines as
labour input increases (see chapter 4).
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