Question

Collaborative Exercise Jack is looking to replace the recently sunk Black Pearl In an uncharacteristically well-behaved moment, he considers building a new ship with his pirate's hoard of $20,000. But what design is best? The effectiveness of a ship depends on the number of sails, s, the number of guns it carries, g, and the number of barrels that will fit mits hold, h, according to the equation Effectiveness SSg h Ja sail costs $700. a gun costs $800, and the hold costs $500 per barrel how many sails, guns and barrels should Jack's ship cary?

Answer 1

Effectiveness/Utility = s^0.5g^0.3h^0.7

Total Income = $20000

Budget Constraint is given by 700s + 800g + 500h = 20000

Marginal Rate of Substitution of sails for guns, MRS_sg = MU_s/MU_g

Similarly, MRS_sh = MU_s/MUh,

The MRS between two goods is equal to their price ratio at optimal solution,

MRS_sg = 0.5s^(-0.5)g^(0.3)h^(0.7)/s^(0.5)0.3g^(-0.7)h^(0.7) = 700/800 => g/s = 21/40
or g = 21s/40

Similarly, MRS_sh = 0.5s^(-0.5)g^(0.3)h^(0.7)/s^(.5)g^(0.3)0.7h^(-0.3) = 700/500 => h/s = 21/40
or h = 49s/25

Plugging the values of g and h in the budget constraint, we get

700s + 800(21s/40) + 500(49s/25) = 20000

2100s = 20000

s = 9.5 (approximately)
g = 21s/40 = 5
h = 49s/25 = 18.6

Hence, Jack's ship should carry 10 sails, 5 guns and 19 barrels.