Question

Liquidity Trap in the IS-LM Model (50 points)

Consider a closed economy in which output is the sum of consumption, investment and government purchases

Y = C + I + G, and where C, I and G are respectively given by C = 5000 – 2000 r + 0.8(Y– T), I = 1500 – 3000 r, and G = 2500. Note also that lump-sum taxes T are given by 1250.

(a) (5 points) Recalling that national savings equals S = Y – C – G, find an equation relating savings S to Y and r.

(b) (5 points) Find an analytical formula for the IS curve. To do this, set savings (which depends on Y and r) equal to investment (which depends on r), and solve for Y as a function of r. Depict the IS curve graphically. In your graph, where the IS curve intersect the Y axis? And the r axis?

(c) (5 points) Suppose that money market equilibrium is given by M/P = 3,000 + 0.5 Y – 16,000i, where the nominal interest rate i is the real interest rate r plus expected inflation. Assume for the sake of this exercise that expected inflation is zero, so that the nominal and real interest rates are the same. Further, let M = 6,000 and P = 2. Solve for the LM curve and show it graphically on your diagram from part (b). Where does the LM curve intersect the Y axis?

(d) (5 points) At what point do your IS and LM curves intersect? What is the resulting level of income Y and real interest rate r?

Answer 1

(a) S = Y - C - G

S = Y - [5,000 - 2,000r + 0.8(Y - 1,250)] - 2,500

S = Y - 5,000 + 2,000r - 0.8Y + 1,000 - 2,500

S = 0.2Y + 2,000r - 6,500

(b) In equilibrium, S(Y, r) = I(r)

0.2Y + 2,000r - 6,500 = 1,500 - 3,000r

0.2Y = 8,000 - 5,000r

Y = 40,000 - 25,000r........[Equation of IS curve]

When Y = 0, r = 40,000/25,000 = 1.6 (Vertical intercept) and when r = 0, Y = 40,000 (Horizontal intercept). In following graph, IS0 is the IS curve.

LMo K. I-6 ISo Yo 40, 000 Y

(c) In money market equilibrium, Money demand equals Money supply.

M/P = 3,000 + 0.5Y - 16,000r [Since Inflation = 0, i = r]

6,000/2 = 3,000 + 0.5Y - 16,000r

3,000 = 3,000 + 0.5Y - 16,000r

0.5Y = 16,000r

Y = 32,000r.........[Equation of LM curve]

When Y = 0, r = 0 (Vertical intercept. LM curve is a straight line from origin)

In above graph, LM0 is the LM curve.

(d) IS0 & LM0 intersect at point A with interest rate r0 & output Y0 which are obtained by equating IS and LM.

40,000 - 25,000r = 32,000r

57,000r = 40,000

r = 0.7018

Y = 32,000 x 0.7018 = 22,457.6