Consumer adaptation

The FIRE 4% Rule

Mr. Money Mustache blog post on the 4% rule.

A major goal for many in the FIRE (Financial Independence Retire Early) movement is, of course to retire early. They want to build up enough savings and investments so that they can eventually live off the returns of these investments. But how do you know you will have enough money to retire?

This question, in general, requires knowing both about a persons preferences and budget. How much money do you need to live a comfortable life? And then how much do you need accumulated investments to maintain that lifestyle?

Mr. Money Mustache has a simple rule, which he calls the 4% rule. This says that should be able to live off 4% of your total investment wealth. So if you have 1 million euros, you can use 40,000 Euro per year. If you have a preference for more spending, you will need to acquire more savings before you retire.

The logic of the 4% rule an assumption based on historical experience that in the long run you can make 6% returns by investing in a broad index of the stock market (by buying index funds, for example). Then you assume that on average 2% of this will disappear in inflation per year, and then you have 4% real return. If these assumptions hold, then the original principle you have saved up will never be used, just the return on the principle.

As it happens, this 4% rule is also used in slightly different context. The Norwegian state stashes away most of the income it receives from petroleum extraction in an investment fund. The idea was that this fund should benefit not just current citizens of Norway, but also future citizens as well. In order to maintain the principle of the fund, the Government has had a stable policy of only extracting 4% of the funds wealth into the government budgets. This was eventually lowered to 3%, which was seen as a more realistic expectation for future real returns on the fund.

What does this have to do with microeconomics? Deciding when we have accumulated enough wealth to retire requires that we have some idea of both our preferences and our budget. In the previous lessons we have discussed indifference curves, but how do we decide which indifference curve we should be on and where on that indifference curve we should be. That is, what combination of goods do we value? We will try to build a model to help us think about exactly this question in this lesson.

Decisions and consumption

In the figure we have a budget curve and an indifference curve. The prices and the budget are fixed. For the moment we will assume that they will not change.

First, notice that on the indifference curve there are combinations of apples and bananas that you can not afford. These are all at points that lie above and to the right of the budget curve.
But we don't care about those point. We can simply move down the indifference curve until we move below the budget curve. By definition of the budget curve, we can afford these combinations. By definition of the indifference curve, we are equally happy--we are left no worse off.
We also see that we can move even further towards the origin. Again, as long as we are on the indifference curve, we are exactly as happy as before. But now we are beneath the budget curve. We have more money left!
Since we have money left over in our budget, we can increase our own well-being by buying more of both apples and bananas.
You can simulate this by clicking "increase utility". Continue to press this button until you no longer can increase both apples and bananas.
Which point have you landed on? Hopefully, you are at the optimal point: The point where it is no longer possible to increase your utility and wellbeing.
You can check this by clicking on the "optimal consumption" button.

Decisions and consumption 2

We can do a similar exercise as the one above, but now starting with a indifference curve that initially lies above the budget line.

When the entire indifference curve lies above the budget line, then none of the combinations of apples and bananas on the curve are achievable
We need to adjust our expectations and accept a level utility level.
We can simulate this by pressing the button "Reduce utility", until we just reach the budget curve. This will be our optimal utility.

Conditions for optimal consumption

The green line represents the slope of the indifference curve at a certain point.

We recall that the slope of the indifference curve represents the marginal rate of substitution (MRS)--how many bananas are we willing to give up in order to get a apple. We can write this as:
$$MRS = \frac{U_{apples}}{U_{bananas}}$$
Initially, we have a quite steep slope. That is, we are willing to give up quite a lot of bananas in order to get more apples.
Press the button "Increase # bananas."
What happens with the slope (MRS)? Why?
Now press "Optimal consumption" to find the optimal amount.
Press the "Increase # apples" or "Reduce # apples" until you get to the optimal point.
What is the relationship between the slope of the indifference curve ($MSR = U_{apples}/U_{bananas}$) and the slope of the budget curve (relative prices) $-P_{apples}/P_{bananas}$?

Conditions for optimal consumption

The budget condition: An optimal combination of goods must lie on the budget curve. If a combination lies above the budget line then you can't afford it. If a combination lies below the budget line, then you can increase your utility by buying more of both.
The tangeant condition: The slope of the indifference curve (MRS) is equal to the slope of the budget curve. We can interpret this as the relative prices being in line with your own relative preferences. Say for example that it cost twice as much to buy an apple as a banana, but you prefer them equally (MRS is one). Then you could buy one less apple and instead two more bananas and end up better off.

When prices change...

By changing the price of apples, we can generate the individuals demand curve.

If we reduce the price of apples, then the budget line shifts outward: We can buy more apples.
Then we can also shift out our indifference curve and we move to a new optimal combination.
We buy more apples at the new optimal amount, but...
But we also buy fewer bananas when apples have become cheaper. We call this the price effect when a good becomes cheaper.
But it doesn't always need to be the case that when apples become cheaper that we buy fewer bananas. When apples become cheaper, then the total amount we can buy increases (we say that our real income has increased). This will tend to increase the amount of bananas we buy. We call this the income effect. If the income effect is larger than the price effect of cheaper apples, then we would in total increase the number of bananas we buy.
In the lower figure, we plot the relationship between price and optimal consumption of apples. This turns out to be our individual demand function for apples!

Marginal willingnesss-to-pay

What does our individual demand function tell us about ourselves?

The demand curve for apples tells us how much we are willing to give up in order to get one more apple.

So far, we have interpreted this in terms of bananas, but we could substitute a currency like euros or dollars to make this more general.

If, at a certain point in the demand curve, we are willing to give up 2 bananas to get one more apple, and a banana costs 5 euros, then we are willing to pay 5*2=10 euros to get one more apple.

This is what we call the marginall willingness-to-pay (MWP), and we can write it as: $$MWP_{apples} = MRS*p_{banana} = \frac{U_{apples}}{U_{bananas}}*p_{bananas}$$

In other words, our marginal willingness-to-pay is like our marginal rate of substitution times the price of the alternate good!

Demand, marginal willingness-to-pay and consumer surplus

In the figure above we see an individual supply curve and a horisontal line that represents the price.
According to this figure, how many apples should we buy?
The answer is that we should continue to buy apples until our willingness-to-pay (our demand curve) meets the market price.
We can also define the space between our demand curve and the price as the consumer surplus. See Lesson 1for more info.

Quiz

Answer True or False for the following statements

Problems

Assume we have the following Cobb-Douglas utility function:

for consumption of apples and bananas

The price of bananas ($p_b$) is 4 euro per banana.

The price of apples ($p_a$) is 5 euro per apple.

We have an income of m=100

What is the marginal rate of substitution? (This requires some calculus.)

We start by deriving the utility function relative to $x_a$ and $x_b$ - these derivative represent how much our utility changes when we increase our consumption of apples or bananas (marginal utility):
$$U_a = \frac{\partial U}{\partial x_a} = x_a^{-0,5}x_b^{0,5} = (\frac{x_b}{x_a})^{0,5}$$ $$U_e = \frac{\partial U}{\partial x_b} = x_a^{0,5}x_b^{-0,5} = (\frac{x_a}{x_b})^{0,5}$$
We said that the MRS is how much we are willing to give up of the one good to get more of the other good. This we can get by dividing the marginal utility of an apple with the marginal utility of a banana:
$$MRS = \frac{U_a}{U_e} = \frac{(x_b/x_a)^{0,5}}{(x_a/x_b)^{0,5}} = (\frac{x_b}{x_a})^{0,5}(\frac{x_b}{x_a})^{0,5} = \frac{x_b}{x_a}$$
Write the two conditions needed for an optimu.?

First we need the tangeant condition (that the slopes of the indifference curve and budget line are the same.)
$$MRS = \frac{p_a}{p_b} => \frac{x_b}{x_a} = \frac{p_a}{p_b}$$
Then we need that the budget condition (that our optimal point is on the budget line.)
$$p_a x_a + p_b x_b = m$$
Find the optimal combination of apples and bananas.

With our two conditions, we have two equations and two unknowns: $x_a$ and $x_b$. We can then find our optimal value (which we denote with a *):
$$x_a^* = \frac{m}{2p_a} = \frac{100}{2*5} = 10$$ $$x_b^* = \frac{m}{2p_b} = \frac{100}{2*4} = 12,5$$
(We assume it is possible to buy a half-banana!)
Calculate out the total utility when you have purchased the optimal amount of bananas and apples. Can you provide some interpretation to this number?

We simply input the values we calculated for optimal bananas and apples into our
$$U=2*x_a^{0,5}*x_b^{0,5} = 2*\sqrt(10)*\sqrt(12,5) \approx 21,9$$
Interpreting a utility is in itself meaningless. A utility usually only has meaning when doing a comparison - different combinations of apples and oranges for example.
Let's say that from your current (optimal) consumption, you decide you want to buy 5 more bananas. How many apples do you need to give up in order to afford this? How has our utility changed?

Each apple costs 4 euros, so then we need 20 euros more. We get this by buying 4 fewer apples (5 euros more). That means we now have 17,5 bananas and 6 apples

Putting this in our utility function:
$$U=2*\sqrt(6)*\sqrt(17,5) \approx 20,5$$
Our utility goes down - a bad deal!
If the price of apples increased, how would this effect the optimal number of bananas to buy?

Looking at our formula for optimal number of bananas:
$$x_b^* = \frac{m}{2p_b}$$
In this particular function, the price of apples does not enter into our decision.
Can you write demand for apples and bananas as a function of prices, $p_b$, and $p_a$ and income, $m$.

We have already done this (see 3)

We obtained:
$$x_a^* = \frac{m}{2p_a} $$ $$x_b^* = \frac{m}{2p_b} $$
The formulas for optimal amounts of apples and bananas can be interpreted as the demand curves for those goods.