Freetime, the supply of labor and saving

The F.I.R.E. movement

F.I.R.E. stands for Financial Independence Retire Early, and the movements consists of people that aim to save so much in their early working life, that they achieve financial independence and even the opportunity to retire early before they even hit there mid age. In the movement there are even 30-somethings that declared themselves retirees!

But in many ways, FIRE is a concept that works best in good economic times, where it can be relatively easy to secure a well-paying job and stock markets are buoyant. The basis for many "retirement" plans is to save enough so that a 4% real return on their investments is enough to pay for their yearly costs. In this way, a nest-egg can last in perpetuity assuming that your investments, over the long term, earn a real return of 4%.

When Corona came, many in the FIRE movement had to change their plans somewhat. Perhaps they lost their jobs and couldn't save as much as previously. When the stock market fell, many also had to reevaluate how much saving they needed.

In the end, how much someone changes their saving or decisions about how much they will work over their lifetimes in response to a change in their income and budgets will vary by individual. These types of decisions about how much to work and save over time is what we will be studying in this lesson.

Supply of labor

When you get a higher wage, will you work more or less?

In the figure above we have time on the x-axis -- we all have 24 hours in a day. We define leisure broadly as time we are not at work, and we look at leisure as a good that we value. But isn't leisure free?
Economists would say that leisure has a cost - an alternative cost. You could be working more, earning more money. So the cost of leisure is the wage you could be earning working instead. And with that extra money, we could have extra consumption of market goods.
If w represents our wage, then we can say that our leisure costs w
On the y-axis is our consumption of market goods - things we can buy with our wage.
This figure then shows the trade off between buying things and time spent on leisure.
We can then continue with the analysis as we would with any other two goods, like apples and bananas.
We again have an indifference curve that is bowed inwards to represent a preference for a good mix of leisure and consumption.
But there is one big difference compared to consumption of other goods. With other goods we could increase our budget. But in this case we are limited to a "budget" of 24 hours in a day.
We have also modelled in a certain minimum income. The line with the label "MIN" is meant to represent that even if we have 0 wages, we still have some basic consumption. We can perhaps think about this as the consumption we receive without necessarily having to pay for it: Parks, schools, roads, etc.
We can find an optimal combination of consumption and leisure in a similar way as previously. Press the "Increase wages" button. We see now that we can buy more without having to work more, and thus the budget line grows steeper.
We could get the same result if the price of all consumptions goods was reduced.
If we press the "Optimal leisure" button then we see that with a higher wage we decide to work more? This seems intuitive enough, but will it always be true according to our theory?
We again need to think about price and income effects.
Press the "Compensating income" button. In this scenario, the compensating income idea becomes even more abstract. The idea is when wages increase, how much time do we need to take back så that we land on the same indifference curve..
From A to (B) we have our price effect. Consumption has become relatively less expensive (we need to give up less leisure), while leisure has become relatively more expensive. Compensating for the income effect (reducing our total time), we see that the price effect will lead to less leisure and more consumption.
But then we also have to consier the income effect - with the same amount of time we can have more of both leisure time and consumption of goods. The income effect is represented by shifting from (B) to B.
So here comes the big question: Is leisure a normal or inferior good?
In this figure we show leisure as a normal good. When we get a higher wage, leading to the ability to have more of both consumption and leisure, then we assume that the income effect on leisure will be positive.
In the figure we also show the price effect as being dominant: The effect of working more when leisure becomes more expensive outweighs the income effect of being able to have both more consumption and leisure. But it doesn't necessarily need to be that way. Someone with other preferences (another indifference curve) might decide that with a higher wage, they will cut back on the time spent at work, and instead enjoy life.

Supply of labor: Using algebra

Definitions:

x: Consumption of market goods
p: average price of goods (price index)
w: wage
m: Minimum income ("MIN")
L: Hours of labor
F: leisure ("Freetime"): 24-L

We can write total income as:

$$Income=w*L+m$$

We write our budget constraint letting our income be equal to our consumption:

$$px=wL+m$$

Inserting our definition of leisure, $F=24-L$

$$px=w*(24-F)+m$$

In our figure, we had (x) alone on the vertical axis, so let's divide by p.

$$x =\frac{-w}{p}*F + \frac{w}{p}*24 + \frac{m}{p}$$

The slope of the budget line will be, $\frac{-w}{p}$. In other words our real wage is how much consumption one hour of work can buy.

The optimal condition is then when the slope of the indifference curve (Marginal Rate of Substitution) is the same as the slope of the budget constraint, $\frac{-w}{p}$. This is the point where our preferences between consumption and leisure is equal to the market price of work: our wage.

How much should we save?

Saving or consumption?

Sometimes we think of saving as the opposite of consumption, but that's not quite right.
A microeconomist would rather look at saving as shifting consumption from one period (periode 1) to consumption at a later period (periode 2).
In the figure below, s represent saving in period 1. Here we can think of saving as both positive (moving consumption from this period to the next) as well as negative (moving consumption from the next period to this one, that is, taking out a loan).
The dot in the figure is the place where you have zero in saving.
When we have 0 in saving, then your consumption in period one is equal to your income in period one (m1)
Likewise, with 0 in saving, your consumption in the second period is equal to your income in that period (m2).
There is no period 3 - you die after period 2 (dark, right?).
Notice that we set prices in the periods to be the same (p1=p2)
If you want to increase your current consumption (periode 1) then you have to borrow money and in turn have less consumption in period 2
You have to pay interest, r, on the loan.
If you want higher consumption in period 2, you will need to save in period 1.
Then you get interest, r on the money you save.
We assume that the interest rate on saving and loaning money are the same.
What happens when we increase the interest rate, r? Why?

Saving: With Algebra

What we saw in the previous section was that the slope of the budget curve (representing the relative "price" of consumption between the two periods) was the interest rate. Or to be precise: $-(1+r)$. That is to say that if we want one more unit of consumption today, we will need to give up (1+r) units of consumption in the next period. (Jeg har også brukt antakelsen at prisene er like i de to periodene.)

The ends of the budget line represent the extremes. If you will have all your consumption now, and nothing in the next period, then you will have $m_1+\frac{m_2}{(1+r)}$

We can interpret $\frac{m_2}{(1+r)}$ as the present value of the income you earn in period 2.

If you want all your consumption in the next period, then your consumption will be $(m_1*(1+r) + m_2)$.

So far we have assumed that the price levels are the same in the two periods. If we have two different price levels: $p_1$ and $p_2$, then the slope of the budget curve becomes: $-(1+r)\frac{p_1}{p_2}$

If s represents saving, then we can write our consumption in the first period as:

$$p_1*x_1 = m_1-s$$

Moving s over to the left side:

$$s=m_1-p_1*x_1$$

And the same for period 2:

$$p_2*x_2 = m_2 + (1+r)*s$$

And then putting s on the left side:

$$(1+r)*s = p_2*x_2 - m_2$$

This just shows that the difference between our income and consumption in period 2 is equal to our saving in the first period with added interest.

Dividing by (1+r) we get.

$$s=\frac{p_2*x_2}{1+r}-\frac{m_2}{1+r}$$

Putting together our two equations where s is on the left side:

$$\frac{p_2*x_2}{1+r}-\frac{m_2}{1+r} = m_1-p_1*x_1$$

Then moving the income terms to the right side and consumption terms to the left we get:

$$\frac{p_2*x_2}{1+r} + p_1*x_1 = \frac{m_2}{1+r} + m_1$$

Here we see a simple, intuitive result that we nonetheless can sometimes forget. The sum of our consumption over our two periods must equal the sum of our income, taking into account of interest.

So when a nation, like the US, takes out a loan in order to increase public consumption, what does that mean? Simply put, that borrowed money means that the US will need to accept less public consumption at a later point.

Finally, we can rearange so that $x_2$ is alone on the left-hand side:

$$x_2 = \frac{(1+r)*m_1 + m_2}{p_2} - (1+r)\frac{p_1}{p_2}x_1$$

The slope of the budget line is: $- (1+r)\frac{p_1}{p_2}$ and if the price levels over the two periods are the same, the equation simplifies to $- (1+r)$. So here we get an interpretation of what an interest rate represents: It measures the relative price of consumption between now and later.

Optimal consumption: Between now and later

Optimal consumption, now and later

In this figure we are shown an example of optimal consumption between two time periods.
We end up in a place where we save in the current period, in order to have higher consumption in a later period.
The slope of the budget line is equal to the interest rate
At the optimal point, the slope of the indifference curve (our relative preference for consumption now versus consumption later) is equal to the interest rate.
Is this intuitive?
Let's say that this were not the case. The market interest rate is higher than your relative preference for consumption now versus consumption later. Then you could reduce your consumption now, save more and get a return, r, on those savings and have a higher consumption later. With this combination you would also be moving to a higher utility level. Thus the previous point was not optimal (you could do better!)

Quiz

Answer True or False for the following statements

Problems

You make a wage of 20 euro and you have a job where you can freely choose the number of hours you work (like a taxi driver). You also get a certain fixed income that is independent of hours worked. This is equivalent to 40 euro per day.

Du tjener 20 euro i lønn and du har en jobb der du står fritt til å øke eller redusere dine antall timer (kanskje som en taxisjåfør som kan velge sine egne timer.) Du har også noe fast income som du får uansett om hvor mange timer du jobber (kanskje leieinntekt fra en utleiedel av dit hus.) Dette tilsvarer 40 euro per dag.

a.) How much does an hour of leisure cost?

b.) Given that you are limited to 24 hours in a day, what is the maximum consumption you can achieve?

c.) Draw a budget curve with consumption on the y-axis and x-axis?

d.) In the following year, we have inflation of 20%, that is all prices for consumption goods have increased on average 20%. Your wages and fixed income are unchanged. How does your budget curve change?

e.) Say you have an indifference curve as above where the price effect is bigger than the income effect of a price change. Will inflation as in d.) lead you to choose more or less leisure time?

a.) One hour of leisure costs what you could have made in wages - 20 euro

b.) 24*20 euro + 40 euro = 520 euro.

c.) The slope is -20 - how much consumption you have to give up in order get an extra hour of leisure.

d.) When inflation is 20% (a factor of 1,2) then our real consumption is reduced to (5200/1,2).

The slope of our budget curve has become less steep, our real wage has been reduced, thus we have to give up less in order to have an extra hour of leisure.

e. Your real wage has decreased, thus the price effect will lead you to want more leisure. Prisvridningseffekten vil derfor føre til at du ønsker å ha mer fritid. But your real income has also been reduced, and assuming that leisure is a normal good, we will want less of it when income is reduced. Without more specific information on the shape of our indifference curve, the sign of the total effect is unknown.
Saving and consumption in two periods

Let us assume you earn 10,000 euro in period 1 and 15,000 euro in period 2. The interest rate is 10%.

a.) What is the maximum amount of consumption that it is possible to have in both periods (assuming you have enough to pay back a loan)?

b.) Draw a budget curve that represents possible combinations of consumptions in periods 1 and 2. What is the slope of the curve?

c.) If the interest triples to 30%, what happens to the budget curve? There is a point that does not move, what is this point represent?

a.) You can choose to have maximum $10,000 + \frac{15,000}{1.1} \approx 23,600$ euro of consumption in the first period.

Or maximum $10,000*1.1 + 15,000 = 26,000$ consumption in the second period.

b and c.) The blue curve reprents the budget curve with a 10% interest rate. The slope is -1.10: The factor that indicates how much extra you get in consumption the second period by giving up consumption in the first period. When the interest rate increases, the slope of the line increases (the dotted line). The point that does not change is where there is no borrowing or saving - you consume exactly your income in each period.
Optimal saving and consumption

The figure below shows an individuals optimal consumption between two periods given an interest rate of 20%.

a.) Does this individual save or borrow money in the first period?

b.) If the interest rate decreases to 5%, how will this affect consumption?

c.) When the interest rate is reduced, what will be the price and income effects?

d.) Is it possible to say what sign the total effect will have?

a.) Initially, they have more consumption than income, thus they must borrow money at 20% interest rate.

b.) The new budget curve is shown in the figure. Since the interest rate has decreased, then can now borrow more money and have higher consumption in the first period.

c.) Consumption in the first period has become relatively cheaper, thus the price effect shifts consumption from period 2 to 1.

Since the individual had initially borrowed money, a lower interest rate means a higher real income. This will lead to higher consumption in both periods.

d.) For period 1, both the price and income effects are positive, and thus the total effect will also be positive. For period 2, the price and income effects are of opposite sign, so the total effect is unknown.