Assignment #5 Econ 0: Intermediate Macroeconomics December, 009 Keynesian Cross Consider a closed economy. Consumption function: C = C + M C(Y T ) () In addition, suppose that planned investment expenditure is I, the level of taxes is T, and government spending is G. Use your knowledge of the Keynesian cross (goods market / aggregate expenditures model) as a starting point to answer the following questions.. Write equilibrium output level Y as a function of T, M C, C, I, and G. (hint: write down the equilibrium condition for the goods market, i.e. AE = Y ; solve for Y ) AE = C + I + G = C + M C(Y T ) + I + G = Y Y = M C [ C M C(T ) + I + G]. Dene tax multiplier M T Y ; government expenditure multiplier M T G Y. Given the equilibrium output expression from the previous part, write G M T and M G, the scal policy multipliers, in terms of constants. Which multiplier is larger in magnitude? M T = M C M C M G = M C 0 < M C < M T < M G () Kelly ; UW-Madison. TAs Lihan Liu and Scott Swisher.
. From this part forward, assume the following: C = 00, M C = G = T = 00. If Y = 500, what is planned expenditure? Given your answer, would you expect equilibrium Y to be higher or lower than Y = 500? Explain briey. planned expenditure = AE = C + I + G = 00 + (500 00) + 00 + 00 = 600 Given Y = 500, AE > Y implies that inventories are drawn down, so rms will scale up output in order to keep inventories constant. Therefore, we'd expect that equilibrium Y is greater than Y = 500..4 What is the equilibrium level of Y? Is it consistent with your guess in the previous part? Y = M C [ C M C(T ) + I + G] = [00 (00) + 00 + 00] = 800 Yes, our guess was in the correct direction; Y = 800 > 500 in equilibrium..5 Now, suppose that G is reduced to 00 units, but T remains unchanged. What is the new equilibrium output level? Y = M C [ C M C(T ) + I + G] = [00 (00) + 00 + 00] = 500.6 Given your answer to the previous part, calculate Y. Is it consistent with G the government expenditure multiplier M G you computed in part (.)? Y 500 800 = G 00 00 = 00 00 = M G = M C = The answers are consistent; Y G = M G..7 Now, suppose that T is reduced to 80 units with G = 00. What is the new equilibrium output level? Y = = M C [ C M C(T ) + I + G] = [00 (80) + 00 + 00] = 040.8 Given your answer to the previous part, calculate Y. Is it consistent with T the tax multiplier M T you computed in part (.)? Y T Again, the answers are consistent; Y T = M T. = = 040 800 80 00 = 40 0 = M T = M C M C = = =
Dynamics of the Goods Market Let's add an explicit dimension of time to our model of the goods market (Keynesian cross). Here, subscript t stands for time, so all variables are now indexed by t. C and M C are still treated as constants; they don't vary over time. The economy is open to international trade and international capital ows; this is reected in the equations. Implement the following economic model in Excel. Aggregate expenditure: Y t+ = C t + I t + G t + (X t M t ) () Consumption: Investment: Government expenditure: Taxes: Exports: Imports: Monetary policy: Exchange rates: C t = C + M C(Y t T t ) (4) I t = 0.Y t 50r t (5) G t = 50 + 0.Y t 00r t (6) T t = 50 + 0.5Y t + 50r t (7) X t = 0.Y t 0ɛ t 0.(X t M t ) (8) M t = 0.Y t + 00r t + 0ɛ t + 0.(X t M t ) (9) r t+ = 0.0005Y t + 0.000ɛ t + 0.000G t (0) ɛ t+ = ɛ t + 0.000(X t M t ) + 0.5(r t+ r t ) ()
Initial conditions (at time t = 0): Y 0 = 500 () X 0 = 0 () M 0 = 00 (4) r 0 = 0.05 (5) ɛ 0 = (6) Constants: C = 50 (7) M C = 0.8 (8) You should have the following columns in your Excel le: t, Y t, C t, I t, G t, X t, M t, T t, ɛ t, and r t. You do not have to print out your tables, only your graphs (when asked to do so).. Translate the model formulas above into Excel formulas, and then run the model for t = 0,,..., 5. Write out the rst two periods, t = 0 and t =, and then copy the t = row down until t = 5 to run the model. Don't get discouraged! This is really just an exercise in writing Excel formulas. See Excel le, tab.-.5.. Graph Y t, C t, I t, and NX t = X t M t vs. t (time). What trends do you observe? lease print this out. See Excel le.. Graph G t and T t vs. t (time). What trends do you observe? See Excel le..4 Graph ɛ t and r t vs. t (time). What trends do you observe? See Excel le..5 Graph S private,t and S public,t vs. t (time). What trends do you observe? lease print this out. See Excel le. 4
.6 Copy your result from (.) - (.5) into a new spreadsheet (save the old one!). Let's say that there is a preference shock at t = 75, so now M C = 0.95 from t = 75 onwards; people want to consume more. To translate this into the Excel le, make a column for the marginal propensity to consume, and ll in M C = 0.8 from t = 0 to t = 75; M C = 0.95 from t = 75 to t = 5. Your formula for C t should now refer to M C t, this new column, not M C = 0.8 for all t. Reprint your graphs from parts (.) and (.5). Interpret. See Excel le, tab.6..7 Copy your result from (.6) into a new spreadsheet (save the old one!). Let's change the consumption function to: C t = C + M C(Y t T t ) 5r t ; only the 5r t term is new. Since consumers have to borrow money at rate r t to buy expensive items like automobiles and housing, they are less likely to increase consumption by taking out a loan with high real interest rates. Reprint your graphs from parts (.) and (.5). Interpret. See Excel le, tab.7. 5
IS / LM Model Assume a closed economy. Consumption function: Investment function: Fiscal policy: Money demand: C = 00 + 0.75(Y T ) (9) I = 00 500r (0) G = T = 00 () ( M )d = Y 0000r () Money supply: ( M )s = 000 (). Derive the IS curve. (hint: write interest rate r IS as a function of Y ) AE = C + I + G = Y 00 + 0.75(Y T ) + 00 500r + G = Y 00 0.75(00) + 00 500r + 00 = 45 500r = 0.5Y 500r = 45 0.5Y r IS = 0.7 0.000Y. Let =. Derive the LM curve. (hint: write interest rate r LM as a function of Y ) ( M )s = ( M )d 000 = 000 = 500 = Y 0000r 0000r = Y 500 r LM = 0.000Y 0.05. Solve for IS / LM equilibrium (r, Y ). r IS = r LM 0.7 0.000Y e = 0.000Y e 0.05 0.000Y e = 0. Y e = 00 r e = 0.7 0.000(00) = 0.7 0. = 0.06 = 6% (r e, Y e ) = (0.06, 00) 6
.4 Suppose that government purchases are raised from G = 00 to G = 50. All else equal, describe the IS curve shift. What is the new IS / LM equilibrium (r, Y )? The IS curve shifts to the right; no direct eect on the LM curve. New IS curve: 00 + 0.75(Y 00) + 00 500r + 50 = Y r IS = 0.9 0.000Y LM curve: r LM = 0.000Y 0.05 Equilibrium: r IS = r LM 0.9 0.000Y e = 0.000Y e 0.05 0.000Y e = 0.4 Y e = 00 r e = 0.9 0.000(00) = 0.9 0. = 0.07 = 7% (r e, Y e ) = (0.07, 00).5 With G = 00, the money supply is increased from ( M )s = 000 to ( M )s = 00. All else equal, describe the LM curve shift. What is the new IS / LM equilibrium (r, Y )? The LM curve shifts to the right; no direct eect on the IS curve. IS curve: r IS = 0.7 0.000Y New LM curve: ( M )s = ( M )d 00 = 00 = 600 = Y 0000r 0000r = Y 600 r LM = 0.000Y 0.06 Equilibrium: r IS = r LM 0.7 0.000Y e = 0.000Y e 0.06 7
0.000Y e = 0. Y e = 50 r e = 0.7 0.000(50) = 0.7 0.5 = 0.055 = 5.5% (r e, Y e ) = (0.055, 50).6 With G = 00 and ( M )s = 000, suppose that the price level rises from = to = 4 due to an oil price shock (cost-push ination). All else equal, describe the LM curve shift. What is the new IS / LM equilibrium (r, Y )? The LM curve shifts to the left; no direct eect on the IS curve. IS curve: r IS = 0.7 0.000Y New LM curve: ( M )s = ( M )d 000 = 000 = 50 = Y 0000r 4 0000r = Y 50 r LM = 0.000Y 0.05 Equilibrium: r IS = r LM 0.7 0.000Y e = 0.000Y e 0.05 0.000Y e = 0.95 Y e = 975 r e = 0.7 0.000(975) = 0.7 0.0975 = 0.075 = 7.5% (r e, Y e ) = (0.075, 975) 8
.7 Return to G = 00 and ( M )s = 000. Recall that money demand (liquidity preference) is given by the equation: ( M )d = Y 0000r. Leave price level as a variable and write an equation for the aggregate demand curve. (hint: (.) gave you an equation for r IS in terms of Y, and (.) gave you an equation for r LM in terms of and Y ; set r IS = r LM in IS / LM equilibrium and solve for as a function of Y ) IS curve: r IS = 0.7 0.000Y New LM curve: ( M )s = ( M )d 000 = Y 0000r 0000r = Y 000 r LM = 0.000Y 0 Equilibrium: r IS = r LM 0.7 0.000Y = 0.000Y 0 0 = 0.000Y 0.7 0 = 0.000Y 0.7 = 500 (0.Y 85) = 500 0.Y 85 = 500 Y 850 9
4 Retirement lanning This question goes through the mechanics of thinking about consumption smoothing over the course of your working and retirement years. In the problem you will go through the process three times: ) the general model; ) a model where you specify the various parameters; and ) a revised model where you act as a nancial advisor suggesting concrete steps you can take to result in a better consumption path for your lifetime. 4. First Time rocess You know the following information: Susy has a house that is worth $00,000 and a money market fund with $00,000. She also has a mortgage on her house and she currently owes $00,000 on that mortgage. Susy does not anticipate any change in her net worth or wealth position other than from saving a portion of her annual income. Susy is thirty years old, expects to live to be 80 years old, and would like to retire when she is age 60. Susy currently earns $50,000 and she expects her real income to remain constant over the course of her working life. Assume there is no ination over the course of Susy's life (this is a simplifying assumption to make the calculations far easier). 4.. What is the value of Susy's assets at this point in time? 4.. What is the value of Susy's liabilities at this point in time? 4.. What is Susy's net worth or wealth at this point in time? 4..4 What is the value of Susy's income over the course of her working life? 4..5 If Susy smooths out her consumption so that her annual level of consumption is constant over the rest of her life, what will be her annual level of consumption? Show your work in arriving at this level. Answer:. Assets = $400,000. Liabilities = $00,000. Net worth = Wealth = $00,000 4. Value of income over lifetime = (40)($50,000) = $,000,000 5. Consumption = ((Wealth + (years to retirement)(annual income))/(expected Years to Live) = ($00,000 + $,000,000)/ (50 years) = $44,000 0
4. Second Time rocess This time calculate your annual consumption level if you smooth your consumption over the course of the rest of your life. To do this calculation you will need to specify what your real income per year is projected to be (so, do research on a particular profession and the annual income associated with that profession-in your answer identify the profession you are using), how many years you plan to work, how many years you anticipate living, and what your current wealth position is. Be as specic as you can so that you can see what level of annual consumption you can aord given your assumptions. Answers will vary here according to the assumptions that students make. 4. Third Time rocess Having done the calculation in part (b), redo the calculation but with some change in one of the variables (e.g., you might decide to work more years, or you might consider a dierent occupation that has a dierent salary). What happens to your level of annual consumption with this change? Be very specic in your example and be thoughtful about how your choices today aect your future choices. Identify the choices that you feel are the most critical for you in determining your level of annual consumption over your lifetime. Answers will vary here according to the assumptions that students make.