Value and cost

Part II: Value and cost
Part I: The definition of the village

Why does each family labor 10 hours per day, and why do they consume 10 pounds of food per day? They could work only 9 hours a day and still survive, or they could work 11 hours per day and have 11 pounds of food. Why 10?

The answer requires an understanding of marginal value and marginal equilibrium. These concepts will appear frequently in the study of economics, so it's worthwhile to take some time to understand them.

The more food a family produces, the less subjective value the additional food has. Obviously a family must produce 9 pounds of food to survive, but the 10th pound of food has value if for no other reason than making a small reduction in food supply non-life-threatening. Even though one can survive on the edge of starvation, people still value some additional food. The additional value, though, tends to decrease. We want to be well fed, but (for most of us) there's little additional value to eating when we're not hungry. And there are finite physical limits: a person can only eat so much, use so many rooms, wear so many clothes. There's a point at which additional life support doesn't add any additional value and production of that life support is pure waste.

Likewise, the more a person works, the more tired they become, each hour worked adds more fatigue than the previous hour worked. Thus the subjective cost of producing each pound of food increases over time.

Since the value of food is always falling, and the cost of work is always rising, they will intersect at some point. At this point, the subjective value of the food produced is equal to the subjective cost of the work required to produce the food. This is the point at which a family will stop working and start relaxing.

Most importantly, we can come to this conclusion without knowing much at all about subjective value and subjective cost besides that the value always falls and the the cost always rises. We don't need to be able to actually calculate or determine the actual subjective value of food or subjective cost of labor. We can instead infer the characteristics of these phenomena from the observation that the families neither work as little as necessary for survival nor as much as they possibly could work without immediately dropping dead.

The marginal value is the difference between the value of some and the value of more. This concept allows us to discuss the value of food in excess of that necessary to survive without complicating our analysis with the extremal value of food necessary to survive.

We can observe that the marginal value of food always decreases as the total amount of food increases. The value of having 10 units of food vs. 9 units of food is still rather large, the difference between being on the edge of starvation and eating well. The value of having 11 units of food vs. 10 units of food is smaller, the difference between eating well and overeating. At some point, one cannot usefully consume even more food — it will simply go to waste — so the marginal value drops to zero at some finite point.

We can define labor as any activity that has negative intrinsic subjective value (i.e. the activity as an end in itself has negative value) and positive consequential or instrumental value. In other words, you would avoid plowing the fields if it didn't result in food you could eat. (A positive consequential value of an activity that has positive intrinsic value is just a bonus, and an activity that has negative intrinsic value and zero or negative consequential value is just a waste of time and effort.)

It's important to understand that this analysis is empirical: We infer that the marginal value of food is always decreasing and the marginal cost of labor is always increasing by observing that the villagers work 10 hours per day when it is physically possible that they might work more or less. That this analysis fits with our intuitive notions of value and cost is additional support.

There are other observations that support this empirical analysis.

First, we can observe how people behave when the direct physical relationship between work and food temporarily changes, such as during a drought or especially favorable weather.

Second, we can use marginal equilibrium to explain complicated value/cost trade-offs. We can for instance look at life-support in detail. We assume that the time spent specifically growing food, building and maintaining shelter, making clothing, etc. is Pareto efficient. We then find that the functions we have to supply to make the Pareto efficiency work out closely match our intuitive understanding of the relatively costs and values, and none of the presumed functions are strongly counterintuitive.

Third, we can examine exceptions, in the sense that exceptions do indeed prove the rule. If we find some situation with extremal behavior, e.g. the family either works the minimum or maximum possible amount, there should be some independently determinable feature of that situation that doesn't apply to equilibrium situations.

(I need help with the math. The marginal value feels like a derivative: the marginal value of one unit of something over and above n units is V(n+1)-V(n); if we take the limit of V(n+x)-V(n)/x as x goes to zero, we should get an instantaneous marginal value that looks a lot like the first derivative of the value function. It also feels like the first derivatives of the food and work value functions should have some special relationship at the point of equilibrium, or the derivative of the net value function V_f(x)+V_w(x) should have some special character at equilibrium, perhaps being zero. My calculus, sadly, just isn't good enough to to define that relationship more rigorously.)