Question

Akua is a film buff who loves spending all her free time watching movies. This year, she is determined to improve your knowledge by only watching short films (S) (which take an hour to watch including prep/travel time) and long epic films (E) (which take four hours to watch including prep/travel time). Besides, other schoolwork and jobs, she has 1200 hours this year to watch films. Akua seeks to maximize her utility from watching films which can be represented by the function: where S is the number of short films watched and E is the number of epic films watched, respectively. Note this is not the form of a utility function we have seen so far, but it satisfies the relevant conditions for being well-behaved. This is also not a monetary budget line, but assume income is not a binding constraint.

U(S, E) = 4V5 +4V2E

(a) First identify the objective function and the constraint for this problem.

(b) Now, write out the equation which represents the Lagrangian function.

(c) Identify Akua’s optimal number of movies of each type to watch.

(d) Estimate how much more Akua’s utility would increase if she spent 100 more hours watching films this year.

Answer 1

A) Objective function: Maximize U(S,E) = 4S^0.5 + 4(2E)^0.5

Subject to the constraint S+4E = 1200

B) Lagrange can be set as:

L = 4S^0.5 + 4(2E)^0.5 + d(1200-S-4E)

C) In order to find optimal values, derivate L with respect to S and E to find first order conditions and then equate them to 0

That is, dL/dS = 2S^-0.5 - d = 0

dL/dE = 2*2^0.5E^-0.5 - 4d = 0

From these equations find the value of d to equate them to each other

That is,

2S^-0.5 = 0.705E^​-0.5

2E^0.5 = 0.705 S^{​​​​​​0.5}

E = 0.124S

Now substitute this in budget equation constraint:

S + 4E = 1200

S + 4(0.124S) = 1200

S* = 800 approx

E* = 100 approx

Current utility = 4S^0.5 + 4(2E)^0.5 = 113.13 + 56.56 = 169.7 utils

D) When S+4E = 1300

Then,

S* = 869

E* = 107

New U = 176.4 utils

Thus, U increases by 176.4 - 169.7 = 6.71 utils