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 <b< 1. The parameter b is the marginal propensity to consume, and the parameter a is a constant sometimes called autonomous consumption. Suppose also that investment is a linear function of the interest rate: 

I(r) = c-dr, 

where c> 0 and d > 0. The parameter d measures the sensitivity of investment to the interest rate, and the parameter c is a constant sometimes called autonomous investment. 

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. 

b. 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. 

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. 

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. d. Solve for r as a function of Y, M, and Pand the parameters e and f.

e. Using your answer to part (d), determine whether the LM curve is steeper for large or small values off and explain the intuition. 

f. How does the size of the shift in the LM curve resulting from a $100 increase in M depend on 

i. the value of the parameter e, the income sensitivity of money demand? 

ii. the value of the parameter f, the interest sensitivity of money demand? 

g. 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, exogenous policy variables M, G, and T, and the model's parameters. This expression should not contain r. 

h. Use your answer to part (g) to prove that the aggregate demand curve has a negative slope.

 i. 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. 

Answer 1

a) First let us consider the relation between consumption and disposable income

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

This equation allows us to calculate consumer spending as a function of disposable income and taxes (T).

Now, Total Income (Y) = Consumer spending + Investment (I(r)) + Government spending (G)

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

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

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

Y = (a - bT + c - dr + G) / (1 - b) ...... equation 1

b) The IS curve is usually plotted with r (rate of interest) on Y axis and Y(Total Income or GDP) on X axis

equation 1 could be rearranged as:

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

r = Y(1 - b)/d + (a - bT + c + G)/d

This equation is of the form y=mx +c . The slope is then given my m

Comparing above equation with y= mx + c form, m = slope is IS curve = (1-b)/d

It is now obvious that the slope of IS curve is inversely proportional to interest sensetivity of investment, d.

c) Consider equation 1 :

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

If the tax is cut by $100, Y= (a - b ( -100) + c - dr + G)/(1-b) = (a + 100b + c - dr +G)/(1-b)

Thus, the income will increase by $(b/(1 - b)) X 100

Similarly, if G is increased by $100, the income will increase by $(1 / (1 - b)) X 100

Now, since b is less than 1, (1 / (1 - b)) X 100 (b / (1 - b)) X 100

Therefore, increase in income caused by $100 increase in G is greater than that caused by $100 cut in taxes. Therefore, an increase in Government spending by $100 will cause a bigger horizontal shift.

d) Here,  demand for money = real amount of money supply for equilibrium in markets

The real amount of money supply is given by, real amount of money supply = M/P , where M is the money supply and P is the price level

M/P = eY - fr

fr = eY - M/P

r = (eY - M/P)/ f ...... equation 2

e) From equation 1, we can write r= (e/f)Y - (M/Pf)

This is again the form y =mx +c , where m=e/f and c= -M/ Pf

Thus, slope of the LM curve is e/f. Therefore, smaller values of f will lead to larger values of m and the LM curve will be steeper for smaller values of f.

Note : I could answer only 4 subquestions as per rules. Though I could have answered more forgive me for lack of time.