Optimization with constraints: economic and financial examples
The economic and financial examples below provide ample motivation for using the optimization theory with constraints. For geometry, check out the post on level sets, isoquants and indifference curves.
Examples in Economics
Consumption theory. Let be a utility function of consuming two goods Denote their prices and the amount available for consumption. Then the budget constraint is The utility maximization problem is to
(1) maximize subject to
The dual problem to the utility maximization problem is the expenditure minimization problem:
(2) minimize expenditure subject to the desired level of consumption
where the is chosen by the consumer.
Production theory. Let be a production function, that is, are two inputs (for example, capital and labor) and the value gives the output produced using these inputs. in this context means the budget available for employing the inputs. The output maximization problem is to
(3) maximize subject to the cost constraint
This is absolutely similar to (1). Moreover, often both and are modeled as the Cobb-Douglas function
It's easy to formulate the dual problem of cost minimization:
(4) minimize the cost of production subject to the desired level of output
where the is chosen by the producer.
Examples in Finance
Now we consider investor optimization problems. There are stocks and the stock returns are denoted They are random because they are not predictable. The investor has initially dollars to invest in these stocks. Suppose he chooses to invest in stock 1,..., in stock so that the initial investment is split as Dividing this equation through by we get Here are called shares of the total investment. They obviously satisfy
Working with shares is better because they don't depend on and the results are equally applicable to small and large investors. Shares of investment are determined at the time investment is made and are deterministic. The total return on the portfolio is defined by
Riskiness of the portfolio is measured by standard deviation The basic fact is that a higher expected return is associated with a higher risk Now we can state two optimal portfolio choice problems:
(6) maximize subject to the chosen level of risk
(7) minimize risk subject to the chosen level of expected return
In both cases the investor varies the shares so as to obtain the best allocation of his money. The constants are chosen subjectively. A more objective problem obtains if we use what is called an expected return per unit of risk: Instead of the above two problems we can solve the maximization problem
In addition to explicitly written constraints in (6)-(7) we remember that the shares satisfy (5), which contains inequality constraints.