Numerical Problems

1. Suppose an economy is characterized by the following equations. All figures are in billions of dollars.

   C = 400 + ⅔(Y_d)

   T = 300 + ¼(Y)

   G = 400

   I = 200

   X_n = 100

   1. Solve for the equilibrium level of income.
   2. Now let G rise to 500. What happens to the solution?
   3. What is the multiplier?

2. Consider the following economy. All figures are in billions of dollars.

   C = 180 + 0.8(Y_d)

   T = 100 + 0.25Y

   I = 300

   G = 400

   X_n = 200

   1. Solve for the equilibrium level of real GDP.
   2. Now suppose investment falls to $200 billion. What happens to the equilibrium real GDP?
   3. What is the multiplier?

3. Suppose an economy has a consumption function C = $100 + ⅔ Y_d. Autonomous taxes, T_a, equal 0, the income tax rate is 10%, and Y_d = 0.9Y. Government purchases, investment, and net exports each equal $100. Solve the following problems.

   1. Draw the aggregate expenditures curve, and find the equilibrium income for this economy in the aggregate expenditures model.
   2. Now suppose the tax rate rises to 25%, so Y_d = 0.75Y. Assume that government purchases, investments, and net exports are not affected by the change. Show the new aggregate expenditures curve and the new level of income in the aggregate expenditures model. Relate your answer to the multiplier effect of the tax change.
   3. Compare your result in the aggregate expenditures model to what the aggregate demand–aggregate supply model would show.
4. Suppose a program of federally funded public-works spending was introduced that was tied to the unemployment rate. Suppose the program was structured so that public-works spending would be $200 billion per year if the economy had an unemployment rate of 5% at the beginning of the fiscal year. Public-works spending would be increased by $20 billion for each percentage point by which the unemployment rate exceeded 5%. It would be reduced by $20 billion for every percentage point by which unemployment fell below 5%. If the unemployment rate were 8%, for example, public-works spending would be $260 billion. How would this program affect the slope of the aggregate expenditures curve?