Question

Suppose that the economy is given by:

C = c0 + c1(Y −T)

I = b0 + b1Y −b2i
with 0 < c1 < 1, 0 < b1 < 1, (c1 + b1) < 1, 0 < b2 < 1, and G, T are exogenous constants.

(a) Derive an expression for equilibrium output.

(b) Derive ∂Y/∂i, and invert this expression so that it is now ∂i/∂Y (hint: (3/2)−1 = (2/3). This is the IS curve we derived graphically in class. What is it’s sign? What is the effect on output of a small increase in the interest rate?

(c) What happens to the slope of the IS curve if b2 increases? What happens to the slope of the IS curve if (c1 + b1) increases? What is the economic intuition behind this?

(d) Using the expression for equilibrium output you derived in part b, derive ∂Y/∂G and ∂Y/∂T. What are the signs of these derivatives? How much does output change (and in what direction) if there is a
small increase in G? How much does output change (and in what direction) if there is a small increase in T? Now suppose equilibrium in the goods market is given by: Y (Y,i,G,T) (+,−,+,−) = C(Y,T +,− ) + I(Y,i +,− ) + G
(e) Derive ∂Y/∂i, and invert this expression so that it is now ∂i/∂Y (hint: (3/2)−1 = (2/3). This is the IS curve we derived graphically inclass. Whatisthesignofthisexpression? Whatdoesthesteepness of the slope of the IS curve depend on? How do the terms in this expression relate to the parameters b2,b1, and c1 in the IS curve you derived in part b?

(f) Through which channels does the interest rate affect output (i.e. what is the effects of changes in the interest rate on C, I, and G )? (Hint: To answer this question, you could take partial derivatives of C, I, and G with respect to i; or simply infer this from your answer to part a. )