A Simple Model

Let $A$ be the total attention available in some interval — a morning, say — and suppose it is divided among $n$ tasks, with $a_i$ the share given to task $i$. The conservation claim from the previous chapter is just the constraint in Equation 1: the shares sum to the whole, no more and no less.

So far this is only bookkeeping. The content arrives when we ask what each share buys. A reasonable first guess is that the value returned by a task has diminishing returns in the attention spent on it — the tenth minute on a problem is worth less than the first. Write the return on task $i$ as in Equation 2, with $v_i$ a task-specific weight and the square root standing in for "diminishing":

Maximising total return $\sum_i r_i$ subject to the budget in Equation 1 gives a tidy result: attention should be spread so that the marginal return is equal across tasks. Pour everything into one task and you leave easy gains on the table elsewhere; spread it too thin and nothing clears the threshold of usefulness. The optimum is neither monomania nor scatter — a conclusion we will lean on in the final chapter.

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