3 Consider an economy populated by a representative household that lives forever. Time is discrete. Household's preferences are given by Z'ogtql] (4) l=l] Page 3 of 4 Copyright 2020 v01 (5} University of Southampton 4 ECONBOIDWI with )3 E [0,1). They also have a unit oftime in each period, which they entirely devote to work for any positive wage because they have no disutility from it. So they supply ft: = 1 if the salary in: > 0. The budget constraint is 6: + kt-i-l E well: + [l + n - (Bk: (5) where Q,kg+1,hf 13 l] for all t and 45 E [0,1). In each period t static firms produce final goods through the follow- ing production function 14 = AK?H'\"E1_\". where E is the total capital stock in the economy. This enters separately into the production function from K: which represents the rm demand of capital input. Instead E is not an individual choice. As usual' or 6 (0,1) and A > 0. Firms sell nal goods and hire labour H; and capital K: from the household in perfectly competitive markets to maximize profits given by 1: = AKng-\"EH th, mm. (a) (a) Show that the production function is consistent with there being a representative firm and a competitive equilibrium. {b} Write down the planning problem. {c} Show that the competitive equilibrium is inefficient. {d} Design a policy intervention such that the competitive equililr rium achieves efficiency. END OF PAPER [10] [10] [10] [10]