Week 4 Intertemporal decisions Flashcards
What are Intermtemporal decisions?
These are decisions that happen across periods, months, days etc.
What is the difference between the representative consumer in the intratemporal model and this model?
In this model consumers are not identical, they are different
What is the difference between the intratemporal model and this intertemporal model in terms of income?
Income falls from the sky in the intertemporal model
Whereas income was produced in the intratemporal model.
For simplicity of our two period model, what are we going to assume about income?
Income is exogneous ( no work/leisure).This allows us to focus on the consumption-savings decision.
Is there money in this economy?
No not yet
Lets say in the current period ( we are looking at Tyrion, whos a lannister ( GAME OF THRONES), he doesn’t work, he just gets income from family, which are bottles of whine, but we also have to pay taxes to the government, what expression can we show in the current period?
Y - T
In the current period with Tyrion’s dispoable income, what can he do with his dispoable income?
He can consume( how many bottles of whine he is going to drink or save ( how many bottles of whine hes going to save)
What can Tyrion do if he wants to consume a little less of his dispoable income Y-T so savings = s>0?
Well he can go to the credit market and go and buy a bond ( this is a promise that in the futrue he will receive 1+r bottles of whine, in exchange for 1 bottle of whine today. ( Tyrion is a lender, as he consumes less than his dispoable income today )
But what about if Tyrion loves to drink and consume more than Y-T, then what can Tyrion do, so S< 0?
Tyrion might issue a bond, on the credit market, he will pay 1+r bottles of whine, in the next period, if you give me now 1 bottle of whine. Tyrion is a borrower ( as he consumes more than his dispoable income)
What is the current period budget constraint?
Y - T = C + S
What bond assumptions will we have?
We assume everyone is a Lannister: they always pay their debts
- Bonds are indistinguishable: no risk associated with them
- They are bought and sold by individual consumers: no financial intermediation
- Buying and selling bonds carries the same real interest rate
In the future period, what will Tyrion have and what is his future period budget constraint?
In the next period , he has some bottles of whine sent deom his family and still has to pay taxes.
Depending on the situation in the current period, Tyrion is either going to have positive savings or debt to be repaid
The future budget constraint is :
Y’ - t’ + ( 1+r)S
Where ( 1+r) is the bottle of whine he pays on top like interest
and S is his debt.
So bonds and credit markets are a way of what?
They are a way in which periods become interconnected.
LETS make it clear, what is whine in the first period and 2nd period?
They are 2 different goods, whine in the first period comes from the sky, whereas whine in the second period comes from the sky, but im still in the first period
If there are 2 goods in the economy, what does it mean?
They both have their own relative price, so we will express all the prices in terms of consumtpion of whine the first period.
If i buy 1 bond and i have to give 1 bottle of whine, what is the price of the bond?
1
What is the expression of if i buy 1 bottle of whine tommorow, i have to give away what today?
So what is the expression 1 / 1+r
This is the relative price of one bottle of whine tommorow in terms how many consumption goods( bottles of whine i give away today) = Present value
What are we going to do with the current period budget constraint and future period budget constraint?
We are going to put them together
So first of all
Period 1 = C + S = Y - T
Period 2 = Y’ - T’ + (1+r)S
( If we have negative savings we subtract)
If we rearrange Period 1, we get S = y - t - c
then we can plug in the second equation
C’ = y’-t’+(1+r)(y-t-c)
We can expand to get
C’ = y’-t’+(1+r)(y-t) - (1+r)c
Next move -(1+r)c to the left hand side
C’ + ( 1+r)c = y’-t’ + (1+r)(y-t)
Last step divide by (1+r)
What is this called
It is called the intertemporal budget constraint
So all the consumption goods given today and tommorow = given the resources he receives in the 2 periods
How is the intertemporal budget constraint similar to micro economics budget constraint?
Lets say we have 2 goods x and y and each have their own relative price, = income to spend. this is the budget constraint
If i divide everything by the price of the first good, i will get the quanitiy of the first good + the relative price of 2 goods times quanitiy of second good = income calculated in terms of goods of x i can buy with my income
The intertemporal budget constraint is written in this format
So i have 2 goods C and C’, the relative price of the 2 goods is c’/1+r and the income in terms of how many consumption goods of x i can buy = the right handside of the intertemporal budget constraint.
What is the right handside of the intertemporal budget constraint called?
The lifetime wealth which is the quanitiy of resources avaliable for the consumer ( in current consumption goods) to spend on consumption overe 2 periods)
What letter are we going to indicate with for the lifetime wealth?
a
How can we draw the intertemporal budget constraint on a diagram, where we have c on the vertical axis and c’ on the horizontal axes?
If we write the intertemporal budget constraint in a another way C’ = -(1+r)C + (1+r)a
The slope = -(1+r)c amd the vertical intercept is ( 1 +r)a and the horizontal intercept is just a.
There is a special point on the diagram called what and either point of this special point, the consumer is what.
The endownment point ( the bundle the consumer consumes is equivalent to his own dispoable income), so no saving at all, consumer doesnt use the credit market
Either point of this point the consumer is either a borrower or a lender.
X is a lender and Y is a borrower
X is a lender on the assumption in the current period he consumes less than dispoable income, meaning positive savings, this means he can consume more than his dispoable income in the future.
Y is a borrower as in the current period we consumes more than his dispoable income, meaning negative savings, this means he can consume less than his dispoable income in the future.
How do we know what propotion of consumption today and tommorow to have?
We draw indifference curves, where there slope = - MRSc,c’
Which tells you how happy are you to exchange 1 unit of consumption today, for one unit of consumption tommorow, and be happy as before.
We know Tyrion is very impatient and bit of a alocholic and Ned stark is patient, which colour of indifference curve is Tyrion and Ned?
As tyrion is very impatient he has a large negative MRS, meaning he is very happy to give away a lot of consumption tommorow for a bit today ( the steeper the indifference curve the more impatient)
Draw the optimum choice for a borrower and a lender?
What was happens when when we increase current dispoable income and what happens to lifetime wealth?
If there is an increase in current dispoable income, in the current period then the next period will not change,( if current income goes up, maybe because taxes are lower etc), then the endownment point will increase from y0-t0 to y1-t1, and shift to the left
The slope didnt change and so the slope is the same as before, your lifetime wealth increases
a0 = y0 - t0 + (y0’-t0’/1+r )
whats in brackets stays the same but the y0 -t0 increases to y1-t1
a1 = y1-t1 + (y0’-t0’/1+r)
What happens to optimal choice with an increase in current dispoable income for a lender?
The consumer consumes more in both periods, savings increase, consumers act to smooth consumption over time, this means from his preferences he doesnt have extreme amounts of consumption in one period than the other, he wil save some consumption and use credit market for this?
( still a lender after an increase in current dispoable income)
What happens to the optimal choice of a lender if there is an increase in future dispoable income, what happens to consumption?
Before the increase in future dispoable income he has a lender, ( consuming less than endownment point)
But after the increase in future dispoable income he has become a borrower( consumers above endownmnet)
Why?
I will have more lifetiime wealth to spread across the 2 periods, this means that i can consume a little bit more today using credit markets, because i know in the future i will be able to pay it off.
consumption goes up in both periods, although the consumption increase i have in the second period is not as big as the increase in future dispoable income ( because you are trying to spread over 2 periods)
What happens when there is an increase in real interest rate on our diagram of lender and borrower
1) the slope of the budget constraint changes ( it becomes steeper
2) endownment doesnt change
3) the horizontal intercept a0 to a1 went down , as interest rate went up, your lifetime wealth goes down as (1+r) increased ( refer to the intertemporal budget constraint)
4) the budget constraint become steeper rotating at the endownment point
5) the vertical intercept increases as r1 increases.
What is the optimal choice for the consumer, given there is an increase in the interest rate?
It depends whether the consumer is a borrower or lender.
What is the optimal choice for a lender with an increase in real interest rate?
At X0, i am a lender consuming less than endownment point in the current period and more than endownment in future
with an increase in real interest rate for a lender
we move to X1 in which i am still a lender as endownment didnt change
As you can see consumption increases in the future and current consumption falls showing an income and subsitutuion effect.
Show the income and subsitution effect, of an increase in real interest rate, showing the optimal choice of lender?
1) take a fake buget line of the new budget line, and make it parralel, so it is tangent to the indiiference curve of the original budget line
2) the movement from X0 to D isolates the subisituion effect, reducing current consumption and increase future consumption ( this is called intertemporal subsituiton, the price of future consumption is lower, so you want to subisutute current consumption for future consumption ) remember the price of future consumption is 1/1+r if interest rate goes up, the price of future consumption goes down.
3) the movement from point d to x1 is the income effect, as the 2 goods are normal goods, both increaase
What does the income and subistution effect, with an increase in real interest rate show, for the optimal choice of a lender and what does it tell us about savings?
what would it mean if subsitution effect dominated, what would it mean if income effect dominated, how do you calculate savings?
1) future consumption goes up for sure as the income and subsitution effect go in the same direction
2) for current consumption the income and subsitution effect go in different directions( subsitution effect means less consumption and income effect means more consumption)
so whatever dominates tells us what happens to current consumption and consequently savings
if subsitution effect dominated, than it would mean more savings, as you consume less today ( to calculate savings, its the difference between unchanged endownment and current consumption)
if income effect dominated, then it would mean you consume more today, meaning less saving
What happens to the optimal choice of a borrower from an increase in the real interest rate?
For this case we have drawn endownment point higher up for space, for the increase in interesst rate, he is a borrower consuming more of his dispoable income today,
after the increase in interest rate, you see i consume more in the future period and less in the current period
there is an income and subsitution effect
( borrower to a borrower)
Show the intertemporal subsitutution effect from an increase in real interest rate if i am a borrower, what is different about the income effect, compared to a lender?
1) The intertemporal subsitiution effect is highlighted from x0 to D. showing a decrease in current consumption and an increase in future consumption, as if the real interest rate goes up 1/1+r, the price of future consumption goes down, so i want to consume more future consumption and less current consumption.
2) The movement from D to X1, highlights the income effect, this goes in the opposite direction as to a lender, ( now i feel poorer, with the new interest rate, as i have to pay higher interest rates, so you will consume less of current and future income.
What can we say about current and future consumption and savings from an increase in real interest rate for a borrower?
Current consumption definitely falls, as income and subsitution effect go in the same direction.
This implies sayings definetly increasse( you wont to save more as the debt in the future will be higher, if you keep consuming)
For future consumption, consumption depends on whether which effect is dominant ( if subsitution effect is dominant future consumption will increase, but if income effect is dominant than future consumption will fall.
What is the governement budget constraint in the current and future period?
So governement can either issue bonds in which case T = 0 in the current period
or raise taxes which in this case B = 0,
or could do both
So in the second period the government needs to finance G’ plus the repayments of the bonds, which market interest rate, this is done via taxes.
Derive the government intertemporal budget constraint ( same as the way we derived consumer)
B = t -g
plug into second
G’ + (1+r)(T-G) = T
to get
This tells us on the LHS, is the value of governement spending calculated at current consumption value
Price of future government spending is G’/1+r
On the RHS shows how its financed via taxes
So we know that the government needs to finance government spending lets say by bonds, and we know that the subsitution effect is stronger than the income effect meaning that, if interest rates go up, i save more, so equibrium what must equal?
net savings by consumers and borrowers after taxes = bonds issued by gov
How does S^p(r) look on the diagram?
Well we know that the intertemporal subistution effect is stronger than income effect, so as interest rates increases, net savings increase, means bonds issued increase. so there is a linear relationship.
this determines real interrest rate
How can i manipulate the government budget constraint, to find out the total amount of taxes an individual pays?
Replace T which what its expanded to
then divide by n on both sides to get the total amount of taxes that an indvidiual has to pay over the next 2 periods calculated at current consumption prices
How can we manipulate the consumer intertemporal budget constraint into the governement budget constraint then mave the 2 sides equal?
We can sub taxes into the consumer intertemporal budget constraint, and we know taxes, equal the right hand side, so sub that back into the intertemporal budget constraint.
What this says, is that its doesnt matter when he pays the taxes, what happens iis the government spending plans, if government spending plans don;’t change the tax burden of consumer is the same
What does this imply about the intertemporal budget constraint
The intertemporal budget constraint will not change if there is a change in the timing of taxes.
If he pays more taxes today than tommorow, this doesnt matter for the consumer, what matters is the total amount of the tax burden
Lets look at an example of ricardian equivalence, so if the government want to do a tax cut but still preserve the amount of government spending, how can we show ricaridan equivalence
So if -x is the reduction of taxes in the future i need to pay taxes by x(1+r), so by subbing into the into the government budget constraint, simplify everything
for consumers you sub what t is equal to and you see that the 2 objects are equal
so the endowment point changes, but not the intertemporal budget constraint, thus optimal choice of consumer is not changed
Lets look at the previous example on a diagram, lets say i am a lender, show ricardian equivalence?
1) The inital endownment point is e0, and the consumer is optimally choosing the point A., when there is a tax cut, repayed by an increase in taxes in the future, the new endownment point is going to be e1 ( so tax cut means dispoable income is higher, and tax increase means dispoable income is lower. but there is no change in the intertemporal budget constraint, the endownment point changes, so the consumer is saving the entire tax cut ( not changing optimal choice because in the future he will know he has to pay it back.) ( so consumption today and tommorow are not going to change)
How does ricardian equivalence influence the credit market?
A tax cut will mean there is an increase in bonds issued, to finance this tax cut, the savings curve move to the right, by the same amount bonds issued ( so saves tax cut), bottom line interest rate doesnt change.
The people of Winterfell live for two periods. They have some income 𝑦 = 𝑦 ′ = 𝑦‾ in both periods. They can consume (𝑐) or save/borrow (𝑠) in the first period. They can consume (𝑐 ′ ) in the second period. Assume for the moment that 𝑡 = 𝑡 ′ = 0. The real interest rate is 𝑟.
The people of Winterfell have standard preferences for 𝑐 and 𝑐 ′ . Some of them are more patient than others, therefore when choosing their optimal intertemporal allocations some are borrowers and some are lenders.
a. Write the intertemporal budget constraint, and show that it can be written as: (Slow and Dumb)
The people of Winterfell live for two periods. They have some income 𝑦 = 𝑦 ′ = 𝑦‾ in both periods. They can consume (𝑐) or save/borrow (𝑠) in the first period. They can consume (𝑐 ′ ) in the second period. Assume for the moment that 𝑡 = 𝑡 ′ = 0. The real interest rate is 𝑟.
The people of Winterfell have standard preferences for 𝑐 and 𝑐 ′ . Some of them are more patient than others, therefore when choosing their optimal intertemporal allocations some are borrowers and some are lenders.
Draw the budget constraint faced by the people of Winterfell. Show what the optimal choice of a lender will look like, by drawing its indifference curve and marking optimal choice. What is the optimality condition for this consumer? (Slow and Dumb)
