Question

The utility you derive from exercises (X) and watching movies (M) is described by the function

U(X,M) = 100 – e^-2x – e^-M . Currently you have 4 hours each day you can devote to either watching movies or exercising. Set up the Lagrangian function for finding the optimal amount of time spent at each activity. Solve for the optimal amounts of time exercising and watching movies. Confirm that the bordered hessian is negative-semi-definite. Also, provide an interpretation of the Lagrange multiplier in this problem.

Answer 1

Solution:

Let us assume that the price of exercise per hour is denoted by P_x and the price of watching movie per hour is denoted by P_M.

The consumer has a total of 4 hours to spend on exercise or movies a day; so if he chooses to consume X hours of exercise and M hours of movies,

then his budget function could be written as follows—

P_x X + P_M M = 4.

The budget function in inclusive form can be written as, 4 – P_x X – P_M M = 0.

The utility function of the consumer is given as,

U (X, M) = 100 – e^-2x – e^-M

The objective of the consumer would be to maximize this utility with respect to the budget function computed above.

The Lagrange function of the consumer could then be written as follows—

Where, > 0, is the Lagrange multiplier.

Now in order to determine the optimum level of time spent on exercise and movie, we need to see whether the first order and second order condition(s) are satisfied or not.

In case of first order condition, we need to maximize the utility function with the time spent on exercise and movies separately and set the result to zero.

This can be shown as follows—

   ...(i)

   ...(ii)

   ...(iii)

Equation (iii) shows that the budget constraint is satisfied.

We will assume that the second order condition for maximization is satisfied for this case.

From equation (i) and (ii) we will get,

If we put this value into equation (iii) we’ll have,

This will be the optimum level of exercise for the consumer.

Now from the equation we will get,

This will be the optimum level of movie watching for the consumer.

conclusion:

We can observe that both the optimum level of exercise and movie watching depends on the price of them.

So, if we are provided with the price of the exercise and movie watching we can easily compute the optimum level of those by simply putting the values into the equations.