Energy and Entropy 2021 (2 years)
In economics, the term utility is roughly related to overall
happiness. Many things affect your happiness, including the amount of
money you have and the amount of coffee you drink. We cannot directly
measure your happiness, but we can measure how much money you
are willing to give up in order to obtain coffee or bagels. If we
assume you choose wisely, we can thus determine that your happiness
increases when you decrease your amount of money by that amount in
exchange for increasing your coffee consumption. Thus money is a
(poor) measure of happiness or utility.
Money is also a nice quantity because it is conserved---just like
energy! You may gain or lose money, but you always do so by a
transaction. (There are some exceptions to the conservation of money,
but they involve either the Fed, counterfeiters, or destruction of
cash money, and we will ignore those issues.)
In this problem, we will assume that you have bought all the coffee
and bagels you want (and no more), so that your happiness has been
maximized. Thus you are in equilibrium with the coffee shop. We will
assume further that you remain in equilibrium with the coffee shop at
all times, and that you can sell coffee and bagels back to the coffee
shop at cost.*
Thus your savings \(S\) can be considered to be a function of your
bagels \(B\) and coffee \(C\). In this problem we will also discuss the
prices \(P_B\) and \(P_C\), which you may not assume are
independent of \(B\) and \(C\).
It may help to imagine that you could possibly buy out the local supply of coffee,
and have to import it at higher costs.
The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative
relationships between your savings and the quantity of coffee and
bagels that you have. What are the units of these prices? What is
the mathematical definition of \(P_C\) and \(P_B\)?
Write down the total differential of your savings, in terms of
\(B\), \(C\), \(P_B\) and \(P_C\).
- Solve for the total differential of your net worth. Your net worth
\(W\) is the sum of your total savings plus the value of the coffee and
bagels that you own. From the total differential, relate your amount of
coffee and bagels to partial derivatives of your net worth.