2. Suppose we have a simple two-period macro model in which there are three types of individuals. There is no production in this model; people are endowed with income (goods) in each period. But we do have a financial market and thus a real interest rate, r. The following table shows each type's endowment and (utility-maximizing) consumption in each period, as functions of r. All parameters (a, b, Y) are positive. Person Type Period-1 Consumption a/(2(1+r)) Period-2 Consumption a/2 Period-1 Endowment 0 Period-2 Endowment a>0 Type A Type B B/(1+r) B(1+r) B>0 B>0 Type C 7/2 (/(1+r))/2 7>0 0 a) I have claimed that these consumption values maximize utilities for the respective individuals, even though I have not shown you the respective utility functions or explicitly done the optimization. Is there a quick check you can make (without doing any formal optimization) to determine whether these stated consumption demands are even reasonable? (Hint: for any type of individual, any "genuine utility-maximizing consumption levels must satisfy a specific condition what is it?) b) Assuming there are NA, NB and Nc people of types A, B, and C, respectively, write the condition that determines the equilibrium interest rate in this economy. c) Solve for the equilibrium interest rate, r*. d) Show mathematically how an increase in NA affects r*. Show also how an increase in Nc affects r*. Explain any differences that you detect in the answer. e) An increase in a country's population is likely to raise the real interest rate. Using the intuition that you (hopefully) gained from this specific model, comment on the situations in which this statement is likely to be true, and the situations in which this statement is likely to be untrue. 2. Suppose we have a simple two-period macro model in which there are three types of individuals. There is no production in this model; people are endowed with income (goods) in each period. But we do have a financial market and thus a real interest rate, r. The following table shows each type's endowment and (utility-maximizing) consumption in each period, as functions of r. All parameters (a, b, Y) are positive. Person Type Period-1 Consumption a/(2(1+r)) Period-2 Consumption a/2 Period-1 Endowment 0 Period-2 Endowment a>0 Type A Type B B/(1+r) B(1+r) B>0 B>0 Type C 7/2 (/(1+r))/2 7>0 0 a) I have claimed that these consumption values maximize utilities for the respective individuals, even though I have not shown you the respective utility functions or explicitly done the optimization. Is there a quick check you can make (without doing any formal optimization) to determine whether these stated consumption demands are even reasonable? (Hint: for any type of individual, any "genuine utility-maximizing consumption levels must satisfy a specific condition what is it?) b) Assuming there are NA, NB and Nc people of types A, B, and C, respectively, write the condition that determines the equilibrium interest rate in this economy. c) Solve for the equilibrium interest rate, r*. d) Show mathematically how an increase in NA affects r*. Show also how an increase in Nc affects r*. Explain any differences that you detect in the answer. e) An increase in a country's population is likely to raise the real interest rate. Using the intuition that you (hopefully) gained from this specific model, comment on the situations in which this statement is likely to be true, and the situations in which this statement is likely to be untrue