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This problem asks you to analyze the IS-LM model algebraically
This problem asks you to analyze the IS-LM model algebraically. Suppose consump- tion is a linear function of disposable income: C(Y −T)=a+b(Y −T), where a > 0 and 0 < b < 1. Suppose also that investment is linear function of the interest rate: where c > 0 and d > 0. I(r) = c − dr,
(a) Solve for Y as a function of r, the exogenous variables G and T, and the model’s parameters a, b, c, and d. [3]
(b) How does the slope of the IS curve depend on the parameter d, the interest rate sensitivity of investment? Refer to your answer to part (a), and explain the intuition. [5]
(c) Which will cause a bigger horizontal shift in the IS curve, a $100 tax cut or a $100 increase in government spending? Refer to your answer to part (a), and explain the intuition. [5]
(d) Suppose that demand for real money balances is a linear function of income and the interest rate: L(r,Y)=eY −fr. Solve for r as a function of Y , M, and P and the parameters e and f. [3]
Expert Solution
answer;
(a) In goods market equilibrium, Y = C + I + G
Y = a + b(Y - T) + c - dr + G
Y = a + bY - bT + c - dr + G
(1 - b)Y = a + c - bT + G - dr
Y = (a + c - bT + G - dr) / (1 - b)
(b) From above relationship,
Y = a + bY - bT + c - dr + G
dr = (a - bT + c + G) + Y(b - 1)
r = [(a - bT + c + G) + Y(b - 1)] / d = [(a - bT + c + G) / d] + [Y(b - 1) / d]
Slope of IS curve = dr/dY = (b - 1) / d
The higher (lower) the value of d, the higher (lower) the interest sensitivity of investment to interest rate, the lower (higher) the slope of IS curve and the flatter (steeper) the IS curve is.
(c) MPC = b
Spending multiplier = 1 / (1 - MPC) = 1 / (1 - b)
Tax multiplier = - MPC / (1 - MPC) = - b / (1 - b)
Since [1 / (1 - b)] > [- b / (1 - b)], Spending multiplier > Tax multiplier and therefore, the $100 increase in government spending will lead to a bigger horizontal shift in IS curve.
(d) In money market equilibrium, (M/P) = L(r, Y)
M/P = eY - fr
fr = eY - (M/P)
r = [eY - (M/P)] / f
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