1. A Model with a Small and a Large Country (100 points) Let us consider a one-period model with two countries. Each of them is denoted by a subscript i {1,2}. Country 1 is populated by 1 perfectly competitive household and by 1 perfectly competitive firm who owns K unit(s) of physical capital. Country 2 is populated by 10 perfectly competitive households and by 1 perfectly competitive firm who owns 10K unit(s) of physical capital. Each household in country i {1,2} is endowed with T unit(s) of time that can be allocated between leisure: x; and work: [;. Each unit of time allocated to work gets paid a real wage w. The preferences of household i are represented by the following logarithmic utility function: U; = Blnc; + Inx; where ; stands for the consumption of the tangible produced commodity and 8 (0,1) denotes a preference parameter. Each firm in country i produces ; unit(s) of a tangible commodity using L; unit(s) of labour and Kj; unit(s) of physical capital according to the following Cobb-Douglas production function: Y; = AL{K]" where a(0,1), A>0 are production parameters. The profit of the firm in country i denoted by /7 is equally shared among the household(s) of country i who each receives a dividend income ;. 1) Write down the labour market equilibrium condition and the good market equilibrium condition under autarky in both country 1 and country 2. (10 points) 2) If each country simply uses all its physical capital and time endowments to produce, how much larger the economy/output of country 2 would be relative to country 1? (5 points) 3) Derive expressions for the competitive labour supply of the household in country 1: [, the competitive labour demand: L%, and the maximized profit: l'l{"\"x of the firm in country 1 given the wage rate w. (15 points) Derive expressions for the competitive labour supply of the household in country 2: I3, the competitive labour demand: L, and the maximized profit: [15/** of the firm in country 2 given the wage rate w. (15 points) Under autarky, derive the competitive equilibrium expression for the real wage in country 1: wy and in country 2: w;. Which country has the higher real wage? (25 points) Write down both the labour market equilibrium condition and the good market equilibrium condition under free trade. (10 points) If the world good market is in equilibrium, then show that the world labour market must also be in equilibrium according to Walras' law. (5 points) Under free trade, derive the competitive equilibrium expression for the real wage: w\". How would free trade affect each country? (15 points)