Recall the advertising launch campaign from the previous assessment. Now suppose you are given a budget of $V million and are tasked with maximising the number of pre-orders.
mathematics
Description
1. Recall the advertising
launch campaign from the previous assessment. Now suppose you are given a
budget of $V million and are tasked with maximising the number of pre-orders.
It costs $1000 to rent a billboard and $C million to sign a celebrity
spokesperson.
Finally, suppose
where b is the number of billboards rented
and s is the number of celebrity spokespeople signed.
a) (1 point) Write the constrained
maximisation problem.
b) (4 points) Can the inequality constraint
hold with strict inequality? Justify your answer using a complementary
slackness argument.
c) (3 points) Relate your answer in part
(b) to the interpretation of the Lagrange multiplier’s value.
d) (4 points) Find the equation relating
the optimal number of celebrity spokespeople to hire with V and C.
e) (3 points) Holding everything else
fixed, how does the optimal number of spokespeople change in C, i.e. what is s′(C)? Interpret its sign.
2. Evaluate, showing your work.
3. (12 points) Consider a Leontief economy
consisting of two sectors: wine and cheese. Workers can either be assigned as
cheesemakers, to produce one pound of cheese each, or winemakers, to produce
one litre of wine each. Every worker must be fed half a pound of cheese and
half litre of wine to stay productive.
a)
(6 points) Suppose the economy aims to export 50 pounds of cheese and 100
litres of wine. Let c be the number of cheesemakers and w the
number of winemakers. Summarise this economy in matrix form.
b)
(3 points) Does this economy have a unique solution? Justify your answer without
solving the system.
c)
(3 points) Now suppose each worker must be provided a third of a pound of
cheese (instead of a half). How many cheesemakers and winemakers are needed in
this economy?
