After the shipwreck…

Imagine Robinson, the lone survivor of a shipwreck. He is stranded on an island, where he has very few alternatives. He can stay on the beach ("leisure"), or he can use his time to produce some goods. For instance, he could try catching fish in the sea or climbing a coconut tree to grab fresh coconuts. He could also construct a shelter. (We assume for now that there is a lot of driftwood, so we do not to count driftwood as capital in the production process.)

To sum up: Robinson enjoys two things: leisure, and consumption. He does not like work per se, and so would be very happy to spend all his days doing nothing but leisurely activities, like learning to surf. However, to get the goods that he enjoys, he needs to work. How is he going to choose how much time to work and how much time to spend on the beach?

Before proceding, we need to make assumptions about Robinson's preferences and his ability to produce. How much extra consumption or additional goods does he need to motivate him to work an extra hour? If he has close to no consumption, then he is more than willing to give up leisure time to work and produce more goods. Hence, he does not need to be compensated by many units of goods to work one hour more. When he has a lot of goods at disposition, such as when he had been fed and had shelter, then he doesn't feel the same need for extra goods: he'd rather keep some of his time for leisure. Hence, to remain as happy, working one extra hour, he will require a big increase in consumption.

What about his ability to produce? If he is working very few hours, then working one hour allows him to produce a lot more goods than before. If he has already been working for a long time, then one extra hour won't allow him to produce much more. Hence there is diminishing productivity in labor!

So, how much will Robinson work? Assume he is working 5 hours a day. Does he want to work one more hour? Taking into account diminishing productivity of labor, he can know how many goods he can produce by working the sixth hour, and compare this to the amount of goods he would like to be compensated for that hour of work. If he produces more goods in the last hour (the 6th) than he needs to be compensated from that extra labor, he will work more. If he gets less, then he won't. Since he can do this for every starting work effort, it seems clear that, as long as working a bit more yield more than enough product, he will increase his work load, and vice versa. So the point where he doesn't want to increase or decrease his work effort is where the last hour worked yields just enough extra goods to compensate him for his work: no more, no less. From that point, working one hour more produces less than what he needs to be compensated, hence he would be less happy by working more. And working one hour less does increase his leisure time, but the number of goods that he produces decreases, and that decrease is more important in terms of loss of well being than the gain due to the extra time off!