Macroeconomics III:
Week 8
Chapter 9
Introduction to the Two-Period Model:
- Shows us how consumers make intertemporal decisions (economic trade off)
regarding consumption & saving
A consumer’s consumption-savings decision involves a trade-off between current and future
consumption. By saving, you give up consumption in the present to consume more in the
future. By dissaving (through borrowing) the opposite.
Initial Assumptions
1. N consumers that live for 2 periods: the current & the future
2. They receive exogenous income (don’t need to consider work-leisure decisions)
Notation:
lowercase letters= variables at the individual level
uppercase letters= aggregate variables.
prime (‘)= future period.
Budget Constraints:
c= consumption
s= saving
y= real income
t= tax
(y-t)= disposable income
The consumer’s current-period budget constraint:
If s>0, consumer is a lender
If s<0, consumer is a borrower
Assumptions:
• the financial asset= bond
• all bonds are default free
(No risk of default – not like this in real world. E.g. Zambia is defaulting on their bonds
due to Covid)
• all bonds can be traded directly in the credit market
(In reality- it is not true- normally consumers who want to borrow must go through an
intermediary, like commercial bank)
• both consumers and the government can issue bonds
1
, (In reality- consumers can’t issue bonds. It is mostly the government & some firms)
• the lending rate=borrowing rate.
(In reality- this does not happen. The borrowing rate tends to be higher than the lending
rate to account for various risks)
bond= financial instrument that makes or pays a fixed income stream of payment. Usually
capital market/ long-term financing
One bond issued in the current period promises to pay 1+r units of consumption in the
future period. So, one unit of current C can be exchanged for 1+r units of future C
r= real interest rate
1
So, the relative price of future C in terms of current C is:
1+ r
The consumer’s future-period budget constraint=
There is only s because it is the final period. Consumer decides to finish the period with no
assets. Consumer decides to consume ALL disposable income + interest rate on the principal
amount on savings
Now, we take the current & future budget constraints and write them as a single lifetime
budget constraint=
Solve the future-period budget constraint for s:
Substitute s in the current period budget constraint:
(current period budget constraint= c+s=y-t)
Consumer’s Lifetime Budget Constraint
- Rearrange to obtain the lifetime budget constraint:
2
, Or better:
Present value of life time consumption= present value of life time income – present value of
life time tax
Present value of life time disposable income= life time wealth=
we=
Simplified Lifetime Budget Constraint and slope intercept:
C’=
*This is the equation used to graph the consumers lifetime budget constraint
Consumers Lifetime Budget Constraint:
Consumers lifetime budget constraint= blue line
slope= -1+r determined by the real interest rate (r)
3
, we(1+r)= what could be consumed in the future if the consumer saved all their current
disposable income & consumed their lifetime wealth in the future. (Point B) B: If c=0,
c’=we(1+r)
we= what could be consumed if the consumer borrowed the max amount possible against
future disposable income and consumed all the wealth in the current period. (Point A) A: If
c’ =0, c= we (life time wealth) look at equation & you will see
E= endowment point (the consumption bundle if I consume all my Yd in the current & future
period) Allows us to distinguish between a lender & borrower At point E tells us that you
only consume your disposable income (current & future). At this point s = 0
A consumer’s IC:
1. More is preferred (more consumption (current or future) makes the consumer better
off)
2. Diversity in the consumption bundle (C smoothing dislike for having large
differences in current and future period)
E.g. A consumer has large current consumption & small future consumption
B consumer has large future consumption & small current consumption
(consumer must be given large Q of small future consumption to give up a small
amount of current consumption)
3. Current & future C are normal goods (if disposable income increases, consumption
(current and future) increases graphically increase in disposable income = shift to
the right of the consumer budget constraint, hence current & future consumption
increases)
Slope of IC=MRS
4
Week 8
Chapter 9
Introduction to the Two-Period Model:
- Shows us how consumers make intertemporal decisions (economic trade off)
regarding consumption & saving
A consumer’s consumption-savings decision involves a trade-off between current and future
consumption. By saving, you give up consumption in the present to consume more in the
future. By dissaving (through borrowing) the opposite.
Initial Assumptions
1. N consumers that live for 2 periods: the current & the future
2. They receive exogenous income (don’t need to consider work-leisure decisions)
Notation:
lowercase letters= variables at the individual level
uppercase letters= aggregate variables.
prime (‘)= future period.
Budget Constraints:
c= consumption
s= saving
y= real income
t= tax
(y-t)= disposable income
The consumer’s current-period budget constraint:
If s>0, consumer is a lender
If s<0, consumer is a borrower
Assumptions:
• the financial asset= bond
• all bonds are default free
(No risk of default – not like this in real world. E.g. Zambia is defaulting on their bonds
due to Covid)
• all bonds can be traded directly in the credit market
(In reality- it is not true- normally consumers who want to borrow must go through an
intermediary, like commercial bank)
• both consumers and the government can issue bonds
1
, (In reality- consumers can’t issue bonds. It is mostly the government & some firms)
• the lending rate=borrowing rate.
(In reality- this does not happen. The borrowing rate tends to be higher than the lending
rate to account for various risks)
bond= financial instrument that makes or pays a fixed income stream of payment. Usually
capital market/ long-term financing
One bond issued in the current period promises to pay 1+r units of consumption in the
future period. So, one unit of current C can be exchanged for 1+r units of future C
r= real interest rate
1
So, the relative price of future C in terms of current C is:
1+ r
The consumer’s future-period budget constraint=
There is only s because it is the final period. Consumer decides to finish the period with no
assets. Consumer decides to consume ALL disposable income + interest rate on the principal
amount on savings
Now, we take the current & future budget constraints and write them as a single lifetime
budget constraint=
Solve the future-period budget constraint for s:
Substitute s in the current period budget constraint:
(current period budget constraint= c+s=y-t)
Consumer’s Lifetime Budget Constraint
- Rearrange to obtain the lifetime budget constraint:
2
, Or better:
Present value of life time consumption= present value of life time income – present value of
life time tax
Present value of life time disposable income= life time wealth=
we=
Simplified Lifetime Budget Constraint and slope intercept:
C’=
*This is the equation used to graph the consumers lifetime budget constraint
Consumers Lifetime Budget Constraint:
Consumers lifetime budget constraint= blue line
slope= -1+r determined by the real interest rate (r)
3
, we(1+r)= what could be consumed in the future if the consumer saved all their current
disposable income & consumed their lifetime wealth in the future. (Point B) B: If c=0,
c’=we(1+r)
we= what could be consumed if the consumer borrowed the max amount possible against
future disposable income and consumed all the wealth in the current period. (Point A) A: If
c’ =0, c= we (life time wealth) look at equation & you will see
E= endowment point (the consumption bundle if I consume all my Yd in the current & future
period) Allows us to distinguish between a lender & borrower At point E tells us that you
only consume your disposable income (current & future). At this point s = 0
A consumer’s IC:
1. More is preferred (more consumption (current or future) makes the consumer better
off)
2. Diversity in the consumption bundle (C smoothing dislike for having large
differences in current and future period)
E.g. A consumer has large current consumption & small future consumption
B consumer has large future consumption & small current consumption
(consumer must be given large Q of small future consumption to give up a small
amount of current consumption)
3. Current & future C are normal goods (if disposable income increases, consumption
(current and future) increases graphically increase in disposable income = shift to
the right of the consumer budget constraint, hence current & future consumption
increases)
Slope of IC=MRS
4
