Two ways of finding the best choice: comparing totals and comparing changes

What optimization means. To optimize is to pick, from the options that are actually open to a decision maker, the one that serves her objective best. Economics assumes this kind of behaviour throughout: a rational consumer picks, among the bundles she can afford, the one that gives her the greatest satisfaction, and a firm looks for the output at which its profit, total revenue minus total cost, is higher than at any other output.

• Every option has a cost as well as a benefit. The cost of a choice includes what is given up to make it: the benefit of the next best alternative, its opportunity cost.
• There are two ways of carrying out the search for the best option. One compares the total net benefit of every option (optimization in levels); the other compares the change in net benefit from one option to the next (optimization in differences).

(i) Optimization in levels. Meaning. The decision maker works out the full net benefit of each available option and chooses the option whose net benefit is the largest. A firm that computes its profit, total revenue minus total cost, at every possible output and picks the output with the highest profit is optimizing in levels. The same logic lies behind cost and benefit analysis, where the total benefit of a project or an activity is set against its total cost to judge whether it is worth taking up. Steps.

1. List the options and express every benefit and every cost in one common unit, usually rupees, counting opportunity costs such as the value of the time used.
2. Work out the total net benefit of each option: total benefit minus total cost.
3. Choose the option with the highest total net benefit. By construction, every other option then gives less.

(ii) Optimization in differences. Meaning. Instead of totals, the decision maker looks at what changes when she moves from one option to the next, that is, at the margin. Rational decision makers weigh incremental changes: they compare the marginal benefit of one more unit with its marginal cost. For a firm, profit keeps rising as long as marginal revenue is above marginal cost and falls once marginal revenue is below it, so profit is greatest where the two are equal. Steps.

1. Express benefits and costs in one common unit, as before.
2. Work out the marginal benefit and marginal cost of each move from one option to the next. Costs that stay the same whichever option is chosen, such as a fixed cost already committed, do not change at the margin and drop out of the comparison.
3. Keep moving while marginal benefit exceeds marginal cost, and stop at the option beyond which marginal cost would exceed marginal benefit. At that option, moving to it made the decision maker better off and moving on would make her worse off.

Example: tuition batches in Patna. Our own figures. Meera, a postgraduate student, earns ₹1,500 a week in fees for each evening tuition batch she takes. Her total cost, mainly the value of study time she gives up, rises faster as she takes more batches.

• In levels: the total net benefit column is largest at 4 batches (₹2,400), so she takes 4.
• In differences: each of the first four batches adds more to benefit than to cost (+1,000, +800, +500, +100), but the fifth adds ₹1,800 of cost against ₹1,500 of benefit (-300). She stops at 4 batches.

Why the two methods agree. A total is built up from its marginal parts: total utility from n units, for example, is the sum of the marginal utilities of the first n units. In the same way Meera's total net benefit at 4 batches, ₹2,400, is the sum of the four changes, 1,000 + 800 + 500 + 100. So the option with the highest total is exactly the option at which the changes stop being positive, and both methods pick it. Optimization in differences is often easier, because it needs only the changes caused by the decision and can ignore costs that do not change.

Conclusion. Optimization in levels compares the total net benefit of every option and picks the largest; optimization in differences compares marginal benefit with marginal cost for each step and stops where a further step would cost more than it brings. Done correctly, both lead to the same best choice.