Question

The utility you derive from exercise (X) and watching movies (M) is described by the function: ?(?, ?) = 100 − ?^−2? − ?^−?. Currently you have four hours each day that you can devote either to watching movies or exercising (? + ? = 4).

a) Set up the Lagrangian function to maximize utility.

b) Find the optimal amount of time spent at each activity.

c) Is the second-order sufficient condition for maximum satisfied?

Answer 1

Economics

The utility we derive from exercise (X) and watching movies (M) is described by the function:

Now, we have 4 hours each day to devote between exercise (X) and watching movies (M). Hence, our budget constraint is

X + M = 4..........BL

Now, we have to maximize U(X,M) subject to the budget constraint X+M = 4

Let us answer the following questions with the help of these informations.

(a) Let us set the Lagrangian function to maximize utility.

The problem is

Max U(X,M)=100 − ?^−2? − ?^−? subject to X+M=4

Hence, the Lagrange's function is

Where, k>0 is the Lagrange's Multiplier.

(b) Now let us check for the FOCs or First Order Conditions.

​​​​​​ Lx = dL/dX = 0

or,

or, ..........(1)

Lm = dL/dM = 0

or,

or, .........(2)

And, Lk = dL/dk = 0

or, 4 - X - M = 0

or, X + M = 4..........(3)

Now, deviding equation (1) with equation (2), we get

or,

Now, taking natural logerithm on both sides, we get

(-2X + M).ln(e) = ln(1)

We know, ln(e) = 1 and ln(1) = 0. Hence,

(M - 2X)×1 = 0

or, M = 2X..........(4)

Pitting this in equation (3) we get

X + M = 4

or, X + 2X = 4

or, X* = 4/3

And, from equation (4) we get

M* = 2X* = 2×4/3

or, M* = 8/3

Hence, we will spend (4/3) hours on exercise and (8/3) hours on watching movies.

(c) The second order sufficiency condition is that

​​​​​​Where,

or,

or,

or,

Now, let us put the values of Ux, Um, Uxx, Uxm, Umm in the equation below

  or,

or,

Now, from equation (4) in part (b) we put

M=2X.

or,

or,

Now, we also have 3M = 6X from equation (4)

or,

or,

Now, for any M>0, we get

Hence,

Hence, the second order sufficiency condition is satisfied.

Hope the solutions are clear to you my friend.