Question

You need to find optimal exercise hours for your healthy life. Suppose you have the following utility function

(0.1) U =2ln(H)−(E)²
where H refers to your level of health that gives increases your utility by preventing potential ill- ness and E refers to the time you spend on exercise. Here, the negativity in front of (E)² represents disutility from exercise.
You choose the amount of time you want to spend on exercise. You also "choose" the health level that you would like (implicitly). Your health levels of course depend on how much time you spend exercising. Mathematically:
(0.2) H =9∗E.
Here 9 can be interpreted as a productivity term: you increase your health by 9 by exercising one hour more.
Using the above information, determine how many hours you should spend on exercise and what should be your optimal health level in terms of numbers (you should provide numerical values).
Assume that the productivity increases to 16 after using a commercial application for health management. Find new optimal exercise hours and health levels. Explain the change using the income or substitution effects.

Answer 1

The original utility maximization problem yields optimal exercise hours of 1/3 and an optimal health level of 3. When the productivity term increases to 16, the new optimal exercise hours are 1/4, and the optimal health level is 4. Changes are due to the income and substitution effects adjusting to the higher 'real wage' for health resulting from increased productivity.

Step-by-step explanation:

The initial utility function given is U = 2ln(H) - E², and the health function is H = 9E.

After substituting H in the utility function, we get U = 2ln(9E) - E². To find the optimal exercise hours (E), we take the derivative of the utility function with respect to E and set it equal to zero: dU/dE = 2/(9E) - 2E = 0. Solving this equation gives us E = 1/3 and hence H = 9(1/3) = 3, which is the original optimal health level.

When the productivity term increases to 16 after using a commercial application for health management, the new health function is H = 16E. Repeating the above process with the new health function, the optimal exercise hours are found to be E = 1/4, leading to an optimal health level of H = 16(1/4) = 4.

The change in optimal exercise hours and health levels can be explained using the concepts of income effect and substitution effect. The increase in productivity can be seen as an increase in 'real wage' or 'purchasing power' for health, leading to a reevaluation of how much time is allocated to exercise versus other activities.

This relation highlights how utility maximization adjusts to changes in constraints, similar to how leisure and work decisions are balanced within the labor-leisure budget constraint.