4. (15 marks) Consider a two-country life-cycle model. The two countries, country 1 and 2, differ only in the individuals propensity to consume. The lifetime utility of a generation-t individual is given by ulicy. Co+1) = (1) Co+1) 9.4 in country 1 and Ces C +1) = G)" (Cat.) 2 in country 2 where and are consumption when young in country 1 and 2, cod+1 and C+1 are consumption when old in country 1 and 2. Assume that capital is completely immobile between the two countries and fully depreciates after production in both countries. However, assume that labour is completely mobile. That is, workers can choose the country in which they choose to live and supply labour and that firms can hire labour domestically or from abroad. Assume young individuals supply one unit of labour and choose to live and work (while young) in the country with the highest real wage. Assume once they have made that choice, they reside in that country for the rest of their lifetime. The two countries begin with the same population size: with N young individuals and N old individuals in each country. Assume that in total, across both countries, 2N individuals are born each period (i.e. there is no growth in the world population). The remaining notation of the model is as follows: Q41 and ar, the savings (capital holdings) for old age of a generation-t individual in country 1 and 2, respectively - W: the worldwide real wage rate per worker in period t, . and 14: the real interest rate in period t in country 1 and 2 respectively, The production function of the representative firm in country 1: Y, = AKL?-?, and that of the representative firm in country 2: Y = A*(K)(L) - As- sume initially that A* = A (i.e. total factor productivity in the two countries is identical). (a) (2 marks) Write down the conditions that characterise profit maximisation by firms in each country. (b) (2 marks) Write down the market clearing condition for the world labour market and solve for the market clearing global real wage as a function of each country's capital stock and the global supply of labour. (e) (2 marks) Formulate the utility maximisation problem for a generation-t individ- ual in country 1, and find the optimal savings, 2:41. Similarly, find the optimal savings of a generation-t individual in country 2, 4-1. (If you know what the solutions are for Q+1 and 4:41, you can simply write them down without using the first-order conditions to derive them). 4 (d) (2 marks) Are the capital-labour ratios and real interest rates the same across countries? Explain why, or why not. (e) (2 marks) Now assume that total factor productivity in country 2 increases so that A* > A. Show how the global wage is affected by the increase in total factor productivity in country 2. (f) (2 marks) Explain what happens to the labour flows (i.e. migration) across the two countries as a result of the increase in total factor productivity in country 2. (E) (3 marks) Finally, explain what happens to the capital-labour ratios and real interest rates as a result of the migration that occurs when A> A. 4. (15 marks) Consider a two-country life-cycle model. The two countries, country 1 and 2, differ only in the individuals propensity to consume. The lifetime utility of a generation-t individual is given by ulicy. Co+1) = (1) Co+1) 9.4 in country 1 and Ces C +1) = G)" (Cat.) 2 in country 2 where and are consumption when young in country 1 and 2, cod+1 and C+1 are consumption when old in country 1 and 2. Assume that capital is completely immobile between the two countries and fully depreciates after production in both countries. However, assume that labour is completely mobile. That is, workers can choose the country in which they choose to live and supply labour and that firms can hire labour domestically or from abroad. Assume young individuals supply one unit of labour and choose to live and work (while young) in the country with the highest real wage. Assume once they have made that choice, they reside in that country for the rest of their lifetime. The two countries begin with the same population size: with N young individuals and N old individuals in each country. Assume that in total, across both countries, 2N individuals are born each period (i.e. there is no growth in the world population). The remaining notation of the model is as follows: Q41 and ar, the savings (capital holdings) for old age of a generation-t individual in country 1 and 2, respectively - W: the worldwide real wage rate per worker in period t, . and 14: the real interest rate in period t in country 1 and 2 respectively, The production function of the representative firm in country 1: Y, = AKL?-?, and that of the representative firm in country 2: Y = A*(K)(L) - As- sume initially that A* = A (i.e. total factor productivity in the two countries is identical). (a) (2 marks) Write down the conditions that characterise profit maximisation by firms in each country. (b) (2 marks) Write down the market clearing condition for the world labour market and solve for the market clearing global real wage as a function of each country's capital stock and the global supply of labour. (e) (2 marks) Formulate the utility maximisation problem for a generation-t individ- ual in country 1, and find the optimal savings, 2:41. Similarly, find the optimal savings of a generation-t individual in country 2, 4-1. (If you know what the solutions are for Q+1 and 4:41, you can simply write them down without using the first-order conditions to derive them). 4 (d) (2 marks) Are the capital-labour ratios and real interest rates the same across countries? Explain why, or why not. (e) (2 marks) Now assume that total factor productivity in country 2 increases so that A* > A. Show how the global wage is affected by the increase in total factor productivity in country 2. (f) (2 marks) Explain what happens to the labour flows (i.e. migration) across the two countries as a result of the increase in total factor productivity in country 2. (E) (3 marks) Finally, explain what happens to the capital-labour ratios and real interest rates as a result of the migration that occurs when A> A
