This problem asks you to analyze the IS- LM model algebraically. Suppose consump

Question

This problem asks you to analyze the IS- LM model algebraically. Suppose consumption is a linear function of disposable income: C(Y -T ) = a + b(Y-T), where a > 0 and 0 0. The parameter d measures the sensitivity of investment to the interest rate, and the parameter c is a constant sometimes called autonomous investment. Solve for Y as a function of r, the exogenous variables G and T, and the model's parameters a, b, c, and d. How does the slope of the IS curve depend on the parameter d, the interest sensitivity of investment? Refer to your answer to part ( a), and explain the intuition. 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. Now suppose demand for real money balances is a linear function of income and the interest rate: L(r, Y) = eY-fr, where e > 0 and f > 0. The parameter e measures the sensitivity of money demand to income, while the parameter f measures the sensitivity of money demand to the interest rate. Solve for r as a function of Y, M, and P and the parameters e and f. Using your answer to part ( d), determine whether the LM curve is steeper for large or small values of f, and explain the intuition. How does the size of the shift in the LM curve resulting from a $100 increase in M depend on the value of the parameter e, the income sensitivity of money demand? the value of the parameter f, the interest sensitivity of money demand? Use your answers to parts (a) and ( d)to derive an expression for the aggregate demand curve. Your expression should show Y as a function of P; of exogenous policy variables M, G, and T; and of the model's parameters. This expression should not contain r. Use your answer to part (g) to prove that the aggregate demand curve has a negative slope. L Use your answer to part (g) to prove that increases in G and M, and decreases in T, shift the aggregate demand curve to the right. How does this result change if the parameter f, the interest sensitivity of money demand, equals zero? Explain the intuition for your result.

Explanation / Answer

a.) Y = C + I + G

C(Y-T) = a +b(Y-T)

I(r) = c - dr

Y = a + b(Y-T) + c - dr + G

Y = bY - dr + a + c + G - bT

Y = [(a + c + G - bT) / (1-b)] - [dr / (1-b)]

b.) In the IS curve we plot Y and r. Higher the d, flatter the IS curve would be and vice versa.

The intution behind this is that if the investment is very sensitive to interest rate(d is large), then a small change in r would have a great impact on I and thus a great impact on Y.

c.) We see that the multiplier for G is 1 / (1-b)

whereas for T is -b / (1-b).

As b<1 we know that the multiplier for G is greater. This means that the $100 increase in govt. spending will have a larger impact than the $100 tax cut.

The intution behind this is that tax cut leads to increase in disposable income of people. But some of the increase in income goes to savings and the net increase in consumption is smaller.

d.) Real money demand L(r,Y) = eY - fr

Real Money Supply = M/P

where M is nominal money supply.

P is the price

At equilibrium in Money Market, money demand = money supply

eY - fr = M/P

r = [eY - (M/P)] / f

e.) The LM will be steeper for small f and vice versa.

The slope of the curve = e/f which is larger for small f.

The intution behind this is that if the money demand is highly sensitive to r, than a small decrease in r would have to be compensated by a very high increase in Y if the Real Money Supply remains constant.

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