Consider a nation that produces food (Qf) and clothing (Qc) according to the following production functions: Qc=sqrt (Lc) Qf=sqrt (Lf) where, Lc and Lf are the number of hours in production of clothing and food respectively. The economy is endowed with 200 hours of labour and the social welfare function of the nation is given as: W=sqrt(QcQf)
a) Derive the production possibilities frontier (Qc= f(Qf)) and calculate the first and second derivatives in order to determine its shape.
b) Determine the autarky equilibrium; calculate the equilibrium levels of food and cloth produced/consumed, and the level of social welfare.
c) Suppose that the nation is now able to trade at a given world price ratio of: [Pf/Pc]=2/1. With free trade,
(i) calculate the new production of food and cloth
(ii) calculate the new consumption of food and cloth [Assume they are divisible and keep the decimal points].
(iii) what is the pattern of trade (i.e., quantities exported and imported)? Show that free trade is beneficial for this nation.