Equilibrium in a Very Simple Dynamic Model With No Production.

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 credit (or capital) market and thus a real interest rate, r. So this is an economy made of individuals like those found in BBM#2 (recall from class). The following table shows each type’s endowment and (utility-maximizing) consumption in each period, as functions of r. All parameters (α, β, γ) are positive. Person Type Period-1 Consumption Period-2 Consumption Period-1 Endowment Period-2 Endowment Type A a/(2(1+r)) a/2 0 a>0 Type B b/(1+r) b(1+r) b>0 b>0 Type C g/2 (g(1+r))/2 g>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 ¾ what is it?) b) Was the condition that you applied in part (a) a necessary condition for these to be utility-maximizing choices or a sufficient one? Explain. c) Assuming there are N A , N B and N C people of types A, B, and C, respectively, write the condition that determines the equilibrium interest rate in this economy. d) Solve for the equilibrium interest rate, r*. Show the equilibrium in a precisely drawn diagram. e) Show precisely (using math!) how an increase in N A affects r*. Show also how an increase in N C affects r*.  Explain any differences that you detect in the two answers. f) “An increase in a country’s population almost certainly raises the real interest rate.” Using the intuition that you (hopefully) gained from this specific model, comment on this statement

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