This model examines what happens in an economy with production and exchange. In order to do this, we model a society with two producers, a woman and a man. Each produces one good and trades a portion of his/her goods for the other's goods. One key idea in the model is a simple "law of diminishing returns" from consumption. Another is the balance between the satisfaction of consumption and the dissatisfaction of work. The third is the relative pleasure one gets from one's own goods.
VARIABLES
30.1
Why don't producers simply produce and consume their own goods? By looking at the economy we live in today that should be somewhat obvious.
Many of us would be very hungry and cold if we were forced to grow or catch our own food (even cook our own food from scratch), make our own clothes, and build our own house.
The question is: How much to trade? and How does this decision affect the economy?
Although it is not necessary to assume that one unit of x is traded for one unit of y, we will assume that producers trade equal portions of work.
This is a fairly reasonable assumption.
Say they each spend 8 hours a day producing their goods.
Would X be likely to give Y 4 units of x that took 4 hours each to produce in exchange for 4 units of y that took Y only 1 hour to produce? Probably not.
They would be most likely to trade equal proportions of work.
Therefore, we will assume that they each trade a fraction q of their work, keeping p=1-q.
A "utility function" gives a mathematical way to explain why people would exchange goods.
Utility can be thought of as satisfaction.
To the woman, it is more satisfying to have 75 units of her product and 25 units of the man's product than it is to just have 100 units of her own product.
The 25 units of y have greater utility for her than the additional units of x, W[75,25]>W[100,0].
In modeling production and exchange, it will be necessary to assume that utility can be quantified.
This is not a very realistic assumption.
Who is to say how many "units of utility" a chocolate bar is worth? Or for that matter, what a "unit of utility" is? However, we do have some sort of feel for how much utility a certain item or action gives us.
For example, say someone were to offer you a new car.
Wouldn't it be fair to say that you would receive more utility from a 1994 Mercedes convertible that you would receive from a 1981 Yugo? How much more? Maybe double the utility, maybe 5 or 10 times the utility. (It would depend on each individual.) So let's say you'd receive 5 times the utility from the Mercedes.
If we arbitrarily assigned the Yugo 10 units of utility, then we would say the Mercedes is worth 50 units of utility.
The property we have just described says that some inputs to the utility function are worth more than others in terms of output.
We need our model utility function to have this property.
The next assumption involves diminishing utility.
This states that each successive unit adds less to the total utility than the previous unit.
For example, that Mercedes may have given you 50 units of utility the first time.
However, once you already have five Mercedes, is another one just like it going to give you an additional 50 units of utility? Probably not.
If this doesn't seem quite so obvious, think of someone offering you another ice cream cone after you have just eaten five ice cream cones.
Would that sixth cone give you as much utility as the first one?
If ownership is satisfying, but by a decreasing amount as goods increase, how can we quantify it? "Fechner's law" (that we saw in the Richter Scale Problem 21.1 of the text) can help us with that.
This says that the more you have, the less satisfied you are with new goods.
If a change in utility U is the differential dU and a change in goods g is dg, then this law says that the change in satisfaction, dU, for a fixed size change in goods, dg, gets smaller as g gets larger.
2) Integrate both sides of the above expression to find an expression for utility
3) We need to correct the mathematical problem with our first attempt at a law of diminishing satisfaction.
Suppose we can subsist on 1 unit of goods.
Positive satisfaction arises from increased goods.
Explain why the law
4) Integrate both sides of this law and use the fact that U=0 if g=0 to show
Another assumption, at least in the first part of the project, is that work is unsatisfying.
If this is the case, work contributes a negative satisfaction.
The simplest model of dissatisfaction is that it is proportional to the work done.
Let W[x,y] and M[x,y] denote the satisfaction or utility from production and exchange for the woman and man, respectively.
From the assumptions above, we can see that when we assume work is unsatisfying, work would contribute -rx and -ry to W and M, respectively, where r is a constant.
Our law of diminishing satisfaction will tell us the other component of utility.
The woman subsists, keeps px of her goods, and exchanges qy of the man's goods.
Her total goods then are 1+px+qy.
Now we have an model for satisfaction, or utility, but what does that tell us? Using the equations for satisfaction, along with the assumption that neither X nor Y will increase production unless his/her satisfaction is increased, we can formulate equations for
Each producer can only produce her/his own good, so each considers only the effect that a change in her/his good would have on her/his satisfaction.
How is this expressed mathematically in the dynamical system? Why do these independent decisions still effect both producers?
We want to explore the economic meaning of the relative sizes of the parameters p and r (recall that q=1-p.)
Now that you understand the way this economy adjusts production in order to improve itself, we want to investigate the dynamics.
We would hope that each producer strives to make things better and that in time the economy approaches an ideal state, maximizing pleasure and minimizing work.
There is some danger.
The independent decisions might make one producer worse and worse off....
Pleasing economic dynamics would be for the independent production adjustments to result in the whole economy moving toward a positive equilibrium.
Negative production or some sort of borrowing is not really built into our basic assumptions, so we need to give conditions that mean our model makes sense.
Once you have found an equilibrium point, you can use compass heading arrows from hand-sketched direction fields to determine the flow and stability of an equilibrium point.
The computer can help you confirm the stability or instability using numerical experiments, but it can also perform the symbolic computations needed to characterize stability as follows.
1) Explain in what sense the equation
means that, "the more you have, the less you are satisfied with new goods." How satisfied are you with the first tiny bit of goods when you don't own any?
How satisfied are you if you don't own any goods?
means that, "the more you have, the less you are satisfied with new goods, beyond subsistence."
for the man and the woman.
Do they receive equal satisfaction from a given level of production?
and
.
Explain the dynamics of the economic model
(We can choose a unit of time such that c = 1.)
Figure 30.1: Dynamic Production
The first step is to see how the model acts in general.
These equations are autonomous.
Why? This means that we can modify the computer program Flow2D to try some experiments on the economy.
Begin with p=7/8 and r=1/7. This means two things.
First, each producer likes his/her own good.
Why? Second, neither is very work averse.
Why? Try the case p=1/3 and r=1/7. What does this mean? What happens dynamically? Try the case p=7/8 and r=1. What does this mean? What happens dynamically?
Show that there is a positive equilibrium when p>r. How could you describe this inequality in terms of economic preferences? (Hint: Use the computer to show that the equilibrium is x=y=(p-r)/r.)
Our production and exchange economy has a positive equilibrium if people would rather work and use their goods, p>r, than goof off and have nothing.
This equilibrium is a stable attractor if each producer likes her/his own goods better than those of the other producer.
What is the associated mathematical condition and why does it say this? What happens when it fails?