Optimization and Coffee Brewing

While at work last week, I began to spoon coffee grounds to make coffee.  How many coffee grounds should I scoop?  Too few and the coffee will be watery.  To many and the coffee will taste like mud.  Then it hit me, coffee making is a perfect example of a number of economic concepts.  So today I will use coffee brewing as a vehicle to discuss 3 key economic concepts .

OPTIMIZATION

What is the optimal number of scoops of coffee?  The answer to this question depends on the function you are trying to maximize.  Let us assume you want to maximize taste.  The relationship between taste and scoops is:

• Taste = 10-(Scoops-9)²

By taking the derivative of this function and setting it equal to 0, we can quickly see that the ideal number of scoops is 9.  But what about cost?  If you buy coffee, you know that the supermarket isn’t giving it away for free. Instead, your goal may be to minimize cost.  If each scoop costs $0.15, the the cot function is as follows.

• Cost = $0.15 * Scoops

To minimize cost, the the ideal number of scoops is 0 (i.e., just drink water).

Most people, however, are concerned with both taste and cost.  Foodies will care more about taste, but frugal individuals may care more about cost.  One way to balance these two concepts is through a utility function.  Assume that your utility function is as follows:

• Utility = Taste/80 – Cost
• Utility = [10-(Scoops-9)²]/80 – 0.15*Scoops

Because this function is also strictly convex, we can also take the derivative and set it equal to 0 to determine the optimal number of scoops.  When taking into account both taste and cost, the optimal number of scoops is 3.

CONSTRAINED OPTIMIZATION

In life, you can’t always get what you want.  In these cases, constrained optimization must be used.  For instance, let us assume that you want to maximize taste subject to a constraint that the cost of the coffee has to be less than $1.  The optimal taste level is 9 scoops.  However, you cannot afford 9 scoops since this will cost $1.35.  Instead, what you can afford is 6 scoops ($0.90).  If we were being precise, you could afford 6 ²/₃ scoops.  If you had a constraint of a $15, however, you wouldn’t want to use 100 scoops of grounds (your coffee would taste like mud).  In this case, the constraint is not binding and you would choose the unconstrained optimal number of scoops (i.e., 9).

PUBLIC GOODS

Let us return to the case where you care about both taste and cost in your utility function.  If you spend more money on coffee you’ll have less money to use for other goods or services.  At my job, however, the kind folks at Acumen supply us with free coffee.  Employees should view this free coffee supply as a public good.  By using more scoops, you do not take away coffee from someone else because there is always more.  Further, because the coffee is readily available, no one is prevented from drinking it.  Thus, free coffee at work is in essence a public good since it is non-rivalrous and non-excludable.

Because I personally do not pay for the coffee, I always choose the number of scoops of grounds to maximize taste.  [In my example, this was 9 scoops].  If I am brewing my own coffee I might hedge towards using slightly less than optimal number of scoops of grounds.

The following spreadsheet gives the numerical calculations behind all the optimizations stated in this blog post.