## 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.

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?