Intermediate Macro Problem Set - Problems from Blanchard:

1) Revisiting Balanced Budget Requirements

Suppose the economy is given by:

C = c₀ + c₁(Y - T)

T = t₀ + t₁Y

I = I^-

G = T

with 0 < c₁ < 1, 0 < t₁ <1

a. Derive an expression for equilibrium output. What is the multiplier?

b. Now suppose that G = G^- (i.e. that G is an exogenous constant). Derive an expression for equilibrium output and the multiplier. Is this multiplier larger, smaller, or equal to the multiplier derived in part a?

c. How is your answer to part b related to the question of whether the balance budget requirement (G = T) is destabilizing? Is the economy more stable when G = T, or when G = G? How do you know?

d. Consider the economy depicted by the IS-LM graph in Figure 5-5, p. 96 of the Blanchard book. Use this diagram to show the effects of increasing both G and T by one unit How does this change affect the IS and LM curves? How does it affect equilibrium output and the interest rate? Describe these effects in words.

2) Deriving the Slope of IS and LM Curves

2.1) The IS Curve

Suppose equilibrium in the goods market is given by:

Y(Y,  i,  G, T) = C(Y, T)+ I(Y, 0) + G

               +,-           +,-

a. Derive ∂Y/∂i, what is its sign? How does this relate to the slope of the IS curve we derived graphically in class? What does the steepness of the slope of the IS curve depend on?

b. 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.)

c. Consider the economy depicted by the IS-LM graph in Figure 5-5, p. 96 of the Blanchard book. Use this diagram to show the effects of the central bank increasing the interest rate by 1 percentage point. How does this change affect the IS and LM curves? How does it affect equilibrium output and the interest rate? Describe these effects in words.

Now 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.

d. Derive an expression for equilibrium output.

e. Derive ∂Y/∂i. What is its sign? What is the effect on output of a small increase in the interest rate?

f. 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? How do b₂, b₁, and c₁ relate to the expression you derived in part a?

g. Using the expression for equilibrium output you derived in part d, 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?

2.2) The LM Curve

Suppose equilibrium in financial markets is given by: M/P =YL(i)

a. Derive an expression for Y in terms of i (i.e. Y = ....).

b. Suppose the central bank holds the money supply constant. Using your answer to part a, derive dY/di. What is its sign? How does this relate to the slope of the LM curve we derived graphically in class? (Hint: if y = 1/f(x), then y' = -f'/f(x)²)

c. If the central bank now wants to keep the interest rate constant, instead of the money supply, how does it need to adjust the real money supply as income increases?

Now suppose that equilibrium in financial markets is given by: M/P = Y(.25 - i).

d. Derive an expression for Y in terms of i (i.e. Y .....).

e. Suppose the central bank holds the real money supply constant. Using your answer to part c, derive dY/di. What is its sign? If there is a small increase in i, by how much does output need to increase to maintain equilibrium in the financial market?

3) IS and LM Curves

Suppose that the goods market 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.

And that the financial market is given by: M/P = Y(.25 - i)

a. Derive an equation for equilibrium i in the goods market. Describe in words how the interest rate is determined by loanable funds in the goods market.

b. Derive an equation for equilibrium i in the financial market. Describe in words how the interest rate is determined by liquidity preference in the financial market.

c. Describe how it is possible for the interest rate to be determined by the two different relations you derived above: loanable funds in the goods market, and liquidity preference in the financial market.