2
Jan 18

Solution to exercise 6.1: how to use homogeneity

Solution to exercise 6.1: how to use homogeneity

Suppose a firm produces just one output y using three inputs x_1, x_2, and x_{3} according to the production function: y=x_1(x_2+x_3).  The prices of goods y,x_1,x_2,x_3 are p,w_1,w_2,w_3>0, respectively. We assume that the firm can neither produce negative quantities of y nor use negative quantities of the inputs, so that y,x_1,x_2,x_3\geq 0.

This is a good opportunity to learn using the homogeneity notion. Assuming, for simplicity, that a function f has positive arguments, we say that it is homogeneous of degree \alpha if f(tx_1,...,tx_n)=t^{\alpha}f(x_1,...,x_n) for all t>0. For example, our production function f(x_1,x_2,x_3)=x_1(x_2+x_3) is homogeneous of degree 2 and the cost function g(x_1,x_2,x_3)=w_1x_1+w_2x_2+w_3x_3 is homogeneous of degree 1. Let's see how this affects the properties of the profit function

\pi(x_1,x_2,x_3)=px_1(x_2+x_3)-(w_1x_1+w_2x_2+w_3x_3).

Suppose we scale the inputs by t>0, then

\pi(tx_1,tx_2,tx_3)=t^2px_1(x_2+x_3)-t(w_1x_1+w_2x_2+w_3x_3)

=t^2[px_1(x_2+x_3)-\frac{1}{t}(w_1x_1+w_2x_2+w_3x_3)].

This means the following: if we start with any bundle of positive inputs (x_1,x_2,x_3) and move along the ray (tx_1,tx_2,tx_3) to infinity, the value of the output px_1(x_2+x_3) stays fixed (positive), while the value of the inputs \frac{1}{t}(w_1x_1+w_2x_2+w_3x_3) tends to zero. For t sufficiently large, the value in the brackets [px_1(x_2+x_3)-\frac{1}{t}(w_1x_1+w_2x_2+w_3x_3)] becomes close to px_{1}(x_{2}+x_{3}). As there is the factor t^2 in front of the brackets, the profit function tends to infinity along such a ray. Since the initial bundle (x_1,x_2,x_3) is arbitrary, such rays cover the whole quadrant \{(x_1,x_2,x_3):x_1,x_2,x_3>0\}. So the profit tends to infinity along any ray and not only along the "diagonal" (x,x,x), as the guide says.

Homogeneity is a notion whose usefulness contrasts with its simplicity. See homogeneity of means, of varianceof standard deviation, of correlationof conditional variance, and application to the Gauss-Markov theorem. And the Cobb-Douglas function is also in this club.

20
Oct 17

Optimization with constraints: economic and financial examples

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

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

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

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

where the const is chosen by the consumer.

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

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

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

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

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

where the const is chosen by the producer.

Examples in Finance

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

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

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

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

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

and

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

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

(8) \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.

 

17
Sep 17

The Cobb-Douglas function and level sets

The Cobb-Douglas function and level sets

Here and here I started to discuss topics in optimization. This post is a preparatory step to the next topic: the Lagrange method.

The Cobb–Douglas production function made a big splash in economics. It is still used a lot. In case of just two inputs, capital and labor, its definition is

f(K,L)=AK^\alpha L^\beta.

For simplicity, in the videos below I assume A=1. We discuss the definition, homogeneity and the reason for choosing a multiplicative form.

Cobb-Douglas production function

Video 1. Cobb-Douglas production function

It is important to realize that the same function is used for modeling preferences. That is, the utility of consuming a pair of goods is measured by the function

f(G_1,G_2)=AG_1^\alpha G_2^\beta.

Thus, everything that has been said about production functions, applies to utility functions as well.

Level sets, isoquants and indifference curves

Let z=f(x,y) be a function of two arguments. For a fixed constant c, the level set \{(x,y):\ f(x,y)=c\} is the set of all pairs (x,y) at which the value of the function is the given constant. Geometrically, z=f(x,y) is a surface in the three-dimensional space. We cut it by a horizontal plane at height z=c. In the intersection of the plane and surface we get a curve. This curve, projected onto the (x,y) plane, gives the level set.

Level sets, isoquants and indifference curves define equivalence relations, see the definition and properties here.

The next video explains that isocosts, isoquants and indifference curves are all level sets for certain functions.

Level sets, isoquants and indifference curves

Video 2. Level sets, isoquants and indifference curves