Raising the bar

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 $latex u(x,y)$ be a utility function of consuming two goods $latex x,y.$ Denote $latex
p_x,\ p_y$ their prices and $latex M$ the amount available for consumption. Then the budget constraint is $latex p_xx+p_yy=M.$ The utility maximization problem is to

(1) maximize $latex u(x,y)$ subject to $latex p_xx+p_yy=M.$

The dual problem to the utility maximization problem is the expenditure minimization problem:

(2) minimize expenditure $latex p_xx+p_yy$ subject to the desired level of consumption $latex u(x,y)=const$

where the $latex const$ is chosen by the consumer.

Production theory. Let $latex f(x,y)$ be a production function, that is, $latex x,y$ are two inputs (for example, capital and labor) and the value $latex f(x,y)$ gives the output produced using these inputs. $latex M$ in this context means the budget available for employing the inputs. The output maximization problem is to

(3) maximize $latex f(x,y)$ subject to the cost constraint $latex p_xx+p_yy=M.$

This is absolutely similar to (1). Moreover, often both $latex u(x,y)$ and $latex f(x,y)$ are modeled as the Cobb-Douglas function $latex Ax^\alpha y^\beta.$

It's easy to formulate the dual problem of cost minimization:

(4) minimize the cost of production $latex p_xx+p_yy$ subject to the desired level of output $latex f(x,y)=const$

where the $latex const$ is chosen by the producer.

Examples in Finance

Now we consider investor optimization problems. There are $latex m$ stocks and the stock returns are denoted $latex R_1,...,R_m.$ They are random because they are not predictable. The investor has initially $latex M$ dollars to invest in these stocks. Suppose he chooses to invest $latex M_1$ in stock 1,..., $latex M_m$ in stock $latex m,$ so that the initial investment is split as $latex M_1+...+M_m=M.$ Dividing this equation through by $latex M$ we get $latex M_1/M+...+M_m/M=1.$ Here $latex s_i=M_i/M$ are called shares of the total investment. They obviously satisfy

(5) $latex 0<s_i<1$ and $latex s_1+...+s_m=1.$

Working with shares is better because they don't depend on $latex M$ 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 $latex R=s_1R_1+...+s_mR_m.$

Riskiness of the portfolio is measured by standard deviation $latex \sigma (R).$ The basic fact is that a higher expected return $latex ER$ is associated with a higher risk $latex \sigma (R).$ Now we can state two optimal portfolio choice problems:

(6) maximize $latex ER$ subject to the chosen level of risk $latex \sigma (R)=const$

and

(7) minimize risk $latex \sigma (R)$ subject to the chosen level of expected return $latex ER=const.$

In both cases the investor varies the shares $latex s_{i}$ 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: $latex \frac{ER}{\sigma (R)}.$ Instead of the above two problems we can solve the maximization problem

(8) $latex \max \frac{ER}{\sigma (R)}$

In addition to explicitly written constraints in (6)-(7) we remember that the shares satisfy (5), which contains inequality constraints.