Problem 5 David is very bored and decides to create a mini economy in his bedroom. He creates a profit maximising company called Bowie Inc. which produces water balloons (W) and slingshots (S) using the labour that David himself provides to the company for an hourly wage which can be normalised to 1. Bowie Inc. produces water balloons and slingshots according to the production functions WB = √LW and SB = √LS, where LW and LB are the labour demanded for the production of each good. Bowie Inc. is of course owned by David so whatever profits the company makes David can use them to buy water balloons and slingshots at prices pW and pS. David’s preferences over these goods are represented by the utility function U(SD,WD) = √(WD·SD). David has plenty of time in his hands and he has a total of 20(N+1) hours he can sell to Bowie Inc.
a)Find David’s demand for water balloons and slingshots.
b)Find Bowie Inc.’s optimal demand of labour for the production of water balloons and for the production of slingshots. Find its optimal supply of these goods and its profits.
c)Find the prices pS and pW that clear all markets (for water balloons, slingshots and the labour market). Find the resulting equilibrium allocation.
d)With slingshots in the horizontal axis, find the mathematical expression for the PPF and show that at the equilibrium allocation you found in
(c) the MRPT equal (in absolute value) to the ratio of equilibrium prices and David’s MRS at the equilibrium allocation.
