Monday, May 26, 2008

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 Vf(x)+Vw(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.)

4 comments:

  1. I believe you are correct about the math. The derivative of the net value function should be zero at equilibrium. This indicates a local maximum, where any small change will only decrease the net value.

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  2. Actually, I think I'm over-complicating the math.

    I think the key function is the marginal value function, not the value function; the total value of n units of something is the integral of the marginal value function.

    So the value (not the derivative) of the net marginal value function is zero at equilibrium.

    The marginal value functions for both food and work are monotonic and mostly strictly decreasing, so the they can intersect (if we invert the sign of the marginal value of work), and the net marginal value function can be zero.

    They would be guaranteed to intersect, but they are bounded; the food marginal value function has a discontinuity or asymptote at 9 units, and the work marginal value function has a discontinuity, asymptote or boundary at however many hours there are in a day.

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  3. This, of course, leaves out other variables... some of which I only vaguely understand.

    For instance, I HATE yard work and have always paid someone to mow my lawn, namely, the neighbor's kid (kids now, since one has gone off to college and his sister has taken over). I don't even own a lawn mower. Yet I know people who like to mow the lawn, not so much because they like to mow, but because for them, it is a spouse-sanctioned hour escape from the house and anything they might be asked to do while in it, or more generously, simply to get some alone time while everyone else is home in the evening. First, I don't understand this because I do a lot of the housework and if I was out of the house it simply wouldn't get done. And second, I don't understand it because I hate mowing so much. And third, I just have a study I retreat to to be alone, though my toddler often follows me there.

    So I wonder if in the village there might not be some people who go out and do an 11th hour on the food simply to get out of the house in the evening when everyone else is underfoot (as they are all done with the fields for the day).

    And then someone else might decide that, like me, they HATE doing work in the field, and so to minimize it, they work very hard, on late hours, to come up with a faster growing crop or whatever so that they can only work eight hours a day in the field and get enough food.

    I know, I'm just making it overly complicated... and you'll probably get to some of that later.

    ReplyDelete
  4. At this point, I'm mostly averaging out extra variables. A family on average works 10 hours per day for 10 units of food, but each family has some definite point of equilibrium.

    Remember, I'm talking about subjective value and marginal value, which are not derived from a priori truths but rather hypothesized to explain the presence of non-extremal behavior.

    We can use the exact same concepts to analyze a particular family's or individual's specific equilibrium. We observe the same things about the individual we observe statistically about the village: that he neither eats nor works at an extremal value. There must be an equilibrium, so we then choose marginal value function that result in the observed equilibrium.

    That looking at the equilibrium of marginal value functions is invariant over scale is an additional empirical confirmation of the concept.

    What keeps this sort of reasoning from being tautological or theological is the criterion of simplicity: If we were to find that we had to have a large number of unobservable hypothetical subjective entities to account for behavior, then we would erode confidence in our theory.

    But the simplicity stays. Individuals who spend a lot of time caring for their lawns, for instance, report that they enjoy the effort and/or enjoy the result.

    The observation is universal: no matter what the economic or economic-like activity, when we see equilibrium-like behavior we also find two competing directly reported or one-step inferred values. (An inferred value is value directly inferred, for example, from an objectively described ideology.)

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