FIFTH TAKE-HOME EXERCISE SET [12 Points Total] L. Tesfatsion DUE DATE: Thursday, April 11, 9:30 A.M. Econ 302/Spring 96 **PLEASE NOTE: As always, no late assignments will be accepted. Each exercise below concerns a Hall and Taylor Chapter 8 economy satisfying the following six properties UNLESS OTHERWISE SPECIFIED: (1) The economy exists over time periods T=0,1,...; (2) Potential GDP is given by a **constant** value Y* in each period; (3) Money demand is given by M^D/P = kY - hR, implying that the expected inflation rate does **not** affect money demand. (4) At the beginning of period 0 the economy is in internal balance; (5) the actual inflation rate INF(0,1) from period 0 to 1 is zero; (6) The expected inflation rate satisfies (*) INF^e(T,T+1) = 0.6 x INF(T-1,T) , T = 1,2,... . ---------------------------------------------------------- ---------------------------------------------------------- EXERCISE 5.1 [4 Points Total; 2 points for each part]. [HINT: For some parts of this exercise, use the HT6 formula [1 + q^2 + q^3 + ... ] = 1/[1-q], which holds for any number q with absolute value less than 1.] Part (a): Suppose that GDP Y(1) for period 1 rises above the potential GDP level Y*, but that Y(T) = Y* in all subsequent periods T = 2, 3,... . Using the Hall and Taylor expectations-augmented Phillips curve, determine what happens to the **actual** inflation rate over periods T = 2, 3, ... How would your answer change if, in relation (*), the constant 0.6 were changed to 1.5? #Be sure to justify your assertions#. Part (b): Suppose, instead, that GDP Y(1) for period 1 rises to a level Y' that is higher than Y*, and that Y(T) **remains** at the higher level Y' in all subsequent periods T = 2, 3,... Using the Hall and Taylor expectations- augmented Phillips curve, explain what happens to the **actual** inflation rate over periods T = 2, 3, ... How would your answer change if, in relation (*), the constant 0.6 were changed to 1.5? #Be sure to justify your assertions#. ------------------------------------ NOTATIONAL NOTE: Throughout my discussion here and elsewhere in class lecture notes I use INF(T,T+1) instead of the Hall and Taylor notation "pi"(T,T+1) for the inflation rate from T to T+1 because my editor-generated special pi symbol turns into garble when uploaded to the Web site. Under the assumptions of this exercise, the expectations-augmented Phillips curve takes the form Y(T) - Y* (1) INF(T,T+1) = .6INF(T-1,T) + f ----------- , T = 1, 2, ... Y* where f is some positive constant and the initial inflation rate from 0 to 1, INF(0,1), satisfies (2) INF(0,1) = 0. For expositional simplicity in answering this question, let A(T) denote f multiplied by the GDP gap in any period T: Y(T) - Y* (3) A(T) = f ----------- , T = 1, 2, ... Y*. --------------------------- Answer Outline for Exercise 5.1, Part (a): For this part, Y(1) increases to some level Y' greater than Y* in period 1 and thereafter returns to Y*. Consequently, A(1) is greater than zero ; A(T) = 0 for T = 2, 3, ... . Given the initial condition (2), relation (1) can thus be used to generate the following values for the actual inflation rate INF(T,T+1) in periods T = 1, 2, ... : INF(1,2) = .6INF(0,1) + A(1) = .6 x 0 + A(1) = A(1) INF(2,3) = .6INF(1,2) + A(2) = .6 x A(1) = .6A(1) INF(3,4) = .6INF(2,3) + A(3) = .6 x [.6A(1)] = .36A(1) INF(4,5) = .6INF(3,4) + A(4) = .6 x [.36A(1)] = .22A(1) and so forth. In general, for arbitrary T, one has INF(T+1,T+2) = (.6)^T x A(1) . The term (.6)^T gets closer and closer to 0 as T gets very large---that is, (.6)^T converges to 0. Consequently, the actual inflation rate converges to 0 as T gets arbitrarily large. Note that *any* positive number strictly less than 1.0, when successively multiplied by itself, converges to 0; so this same answer would be obtained if INF^e(T,T+1) = rINF(T-1,T) for **any** positive number r strictly less than 1.0. Now suppose, instead, that INF^e(T,T+1) = 1.5INF(T-1,T). Using the same approach as above, one finds for arbitrary T that INF(T+1,T+2) = (1.5)^T x A(1) . But any number **greater** than 1.0, when successively multiplied by itself, gets arbitrarily large---that is, it diverges to infinity. Consequently, in this case the actual inflation rate becomes arbitrarily large over time, so that the initial increase in Y destabilizes the price level. Answer Outline for Exercise 5.1, Part (b): For this part, Y(1) rises above Y* and **stays** at this higher level in all future periods. Thus A(1) is greater than zero ; A(T) = A(1) for T = 1, 2, 3, ..... Consequently, given the initial condition (2), relation (1) can be used to generate the following values for the actual inflation rate INF(T,T+1) in periods T = 1, 2, ... : INF(1,2) = .6INF(0,1) + A(1) = .6 x 0 + A(1) = A(1); INF(2,3) = .6INF(1,2) + A(1) = .6[A(1)] + A(1) = [1 + .6] x A(1); INF(3,4) = .6INF(2,3) + A(1) = .6[1 + .6A(1)] + A(1) = [1 + .6 + (.6)^2] x A(1); INF(4,5) = .6INF(3,4) + A(1) = .6[1 + .6 + (.6)^2] x A(1) + A(1) = [1 + .6 + (.6)^2 + (.6)^3] x A(1); and so forth. In general, for arbitrary T, one has INF(T+1,T+2) = [1 + .6 + (.6)^2 + ... + (.6)^T] x A(1) . Using the HINT stated at the beginning of this exercise, it follows that the actual inflation rate INF(T,T+1) converges to 1 -------- x A(1) = 2.5 x A(1) 1 - .6 as T gets arbitrarily large. Consequently, the inflation rate stabilizes to a constant rate; but, unlike Part (a), this constant rate is **not** zero. If, instead, INF^e(T,T+1) = 1.5INF(T-1,T), then by the same type of argument one obtains for arbitrary T that (4) INF(T+1,T+2) = [1 + 1.5 + (1.5)^2 + ... + (1.5)^T] x A(1) . In this case, however, you cannot use the formula in the HINT because 1.5 is greater than 1. Indeed, the bracketed term in relation (4) diverges to infinity as T gets arbitrarily large. ------------------------------------------------------ ------------------------------------------------------ EXERCISE 5.2 [4 Points Total; 2 points for each part]. Hall and Taylor, Chapter 8, ANALYTICAL exercise 4, page 226-227. #Be sure to justify your assertions and label your graphs carefully#. NOTE: For each part of this exercise, assume that the government increase in defense spending occurs suddenly at the beginning of period 1, disrupting the internal balance of the economy. In the first part, assume [instead of relation (*)] that expectations of inflation remain at zero. In the second part, assume that expectations of inflation satisfy relation (*). ADDITIONAL NOTE FOR ANSWER KEY: Hall and Taylor do not clearly state whether the increase in defense spending in period 1 is maintained in future periods. In the answer below it is assumed that the increase is maintained in all future periods. However, answers that assume that defense spending returns to its original level after period 1 will also be accepted. -------------------------------------------------------- Answer Outline for Exercise 5.2: The sudden increase in defense spending at the beginning of period 1 results in a sudden increase in government expenditures from G to some higher level G'. This increase in G results in an upward shift of the IS curve (for each Y, a higher R because the R-intercept of the IS curve is larger) but no change in the position of the LM curve. Also, the AD curve moves out to the right (for each P, a higher Y). Since it is assumed here that the increase from G to G' is maintained in all future periods, and no further changes in exogenous variables take place, the IS curve and the AD curve remain at their new positions in all subsequent periods. Since the economy was originally assumed to be in internal balance, implying Y(1)=Y*, the GDP level Y(1)' that results at the new IS-LM equilibrium point resulting from the change in G to the higher level G' must be greater than potential GDP Y*. Thus, a **positive** GDP gap opens up in period 1. -------------------- Assume, first, that the **expected** inflation rate is 0 in each period T. Then the actual inflation rate from period 1 to period 2 that results after the increase in G to G' is determined by Y(1)' - Y* (5) INF(1,2)' = 0 + f ----------- , Y* so INF(1,2)' is positive. By definition, the inflation rate is the percentage rate of change in the price level from period 1 to period 2. Thus, P(2)' - P(1) (6) INF(1,2)' = --------------- , P(1). where P(2)' denotes the price level that results in period 2 after the increase in G to G' in period 1. Combining (5) and (6), it follows that P(2)' must be **higher** than P(1). This shifts **up** the LM curve in period 2 (for each Y, a higher R because the R-intercept of the LM curve is less negative). If the GDP gap at the new IS-LM equilibrium point, say (Y(2)',R(2)'), is still positive---that is, if Y(2)' is still greater than Y*---then the price level P(3)' that results for period 3 will be greater than P(2)'. However, since Y(2)'-Y* is smaller than Y(1)'-Y*, the inflation rate from period 2 to period 3 will be **smaller** than the inflation rate from period 1 to period 2. Alternatively, if Y(2)'-Y* is zero, P(3)' will be equal to P(2)'. And if Y(2)'-Y* is negative, P(3)' will actually be less than P(2)'. In general, in any future period T, one has P(T+1)' - P(T)' Y(T)' - Y* INF(T,T+1)' = ----------------- = f --------------- P(T)' Y*. Consequently, the inflation rate is directly proportional to the GDP gap. Thus there cannot be any episodes of stagflation in which the price level is increasing at the same time there is unemployment (a negative GDP gap). It follows from the above observations that internal balance can only be reestablished if the inflation rate returns to **zero** (all change in the price level **ceases**) but the price level has increased to a high enough level P' so that the LM and IS curves again intersect at the potential GDP level Y*. In this case, as depicted in the following graph, the long-run outcomes for GDP Y, the price level P, the (real) interest rate R, consumption C, investment I, and net exports NE relative to what their values would have been with **no** increase in defense spending in period 1 are as follows: Y = Y* (THE SAME) ; R INCREASES to a higher level R' ; P INCREASES to a higher level P' ; C is the SAME (because Y is the same) ; I DECREASES to a lower level I' (because R is higher) ; NE DECREASES to a lower level NE' (because Y is the same and R is higher) . This situation is graphically depicted below, where: is, lm, and ad denote the original IS, LM, and AD curves for the original level of G; is' and ad' denote the IS and AD curves that result after G is increased to G'; and lm' denotes the LM curve that results after P has increased to P' and internal balance is reestablished at Y=Y*. .(k is' lm' R is' lm' | is' lm' | is is' lm' | is is' lm' lm | is is' lm' lm R' | is is' lm | is lm' is' lm | is lm | lm' is lm is' R | lm' lm is' | lm' lm is is' | lm' lm is ------------------------------------------------ Y 0 lm' lm Y* Y(1)' is lm ad' P ad | ad' | ad | ad ad' . | . | ad' . | ad . P' | ad' | . ad' | ad . ad' | . P | ad | . ad ------------------------------------------------ Y 0 Y* .)k ------------------- If, instead, the expected inflation rate depends on last period's actual inflation rate in accordance with relation (*), then the pattern of adjustments will be the same until the GDP level returns to potential. At that point the expected inflation rate will still be positive, implying that the actual inflation rate will be positive and the price level will continue to rise. Thus, the actual GDP level will fall below potential. Given relation (*), note that the only actual inflation rate consistent with internal balance Y=Y* is **zero**. Consequently, the only way in which internal balance can be reestablished is to have the economy's actual GDP level cycle around the potential GDP level Y* until inflation finally dies **out**. In this case the **ultimate** effects on Y, R, P, C, I, and NE are exactly the same as determined and graphically depicted for the first part of the exercise, above, where the expected inflation rate was simply assumed to be zero in every period. However, the short run **dynamic adjustment path** of the economy is different when relation (*) replaces the assumption that the expected inflation rate is always zero. The AD graphs depicting the dynamic path of the economy towards the reestablishment of internal balance are similar to the HT figures 8-7 and 8-8 depicting the dynamic path of the economy towards the reestablishment of internal balance after an initial increase in the money supply. TECHNICAL NOTE FOR ANALYTICALLY MINDED STUDENTS: The answers above explain what must be the case in order for internal balance to be reestablished. However, will internal balance **necessarily** be reestablished following a period 1 increase in government defense expenditures G? To answer this question, one has to examine the following three equations that describe the HT economy in any period T = 1, 2, ...: (AD Curve): Y(T) = V(G) + B/P(T), where dV(G)/dG is positive. (Phillips Curve): INF(T,T+1) = .6INF(T-1,T) + f[Y(T)-Y*]/Y* . (Definition of Inflation): P(T+1) = P(T) + P(T) x [INF(T,T+1)] Period T Predetermined Variables: P(T), INF(T-1,T) Period T Endogenous Variables: Y(T), P(T+1), INF(T,T+1) Suppose the economy starts off in period 1 with the price level P(1) equal to some exogenously given value P for which Y(1)=Y* and with INF(0,1) = 0. Then the inflation rate INF(T,T+1) will equal zero and the price level P(T+1) will equal P(1) in every period T, and Y(T) will always equal Y*. In other words, if the economy starts off in internal balance with INF(0,1)=0, it will stay there. Now suppose that P(1) = P and INF(0,1) = 0 but that G suddenly increases to some higher level G', which results in an AD curve coefficient V(G') which is higher than the original coefficient value V(G). As depicted in the graphs above, the **modified** economy with G=G' has a **modified** internal balance point of the form Y=Y*, P=P', and INF = 0. And if the modified economy were to start from this modified internal balance point, it would stay there. But an important issue is whether the economy will indeed **adjust over time** to the internal balance point corresponding to G' if it **starts** in the internal balance point corresponding to G and then G suddenly increases. In this case the economy in the first period is no longer in internal balance because (as depicted above) the resulting Y(1)' will **not** be equal to Y*. In common mathematical language, the question is whether the internal balance point corresponding to G' is "stable" in the sense that the economy will move to this internal balance point regardless of the initial starting point for the economy. Letting X(T) = (INF(T,T+1),P(T)), it can be shown that the system of three equations above has the form X(T+1) = H(X(T)) , where H(X) is a highly nonlinear function of X. So the answer is that the internal balance point corresponding to G' is **not** obviously stable in the indicated sense. One can use well known results from difference equation theory to argue that, for appropriate settings of the exogenous coefficients a, b, etc. that enter into V and B, one can guarantee "local" stability (the economy will converge to the internal balance point corresponding to G' if it initially starts "close enough" to this internal balance point. But establishing that the economy will converge to this internal balance point starting from **any** initial starting point is another matter! And remember, for the dynamic adjustment path to make economic sense, Y and P would have to stay **positive* all along the adjustment path. It is extremely unlikely that this would be the case starting from an arbitrary initial starting point, even if convergence to the internal balance point takes place. These interesting issues could be examined using MacroSolve. HT would surely respond that what they have is a highly simplified model to be used to **illustrate** dynamic economic relations only, and that the above discussion takes the model too seriously. --------------------------------------------------------- --------------------------------------------------------- EXERCISE 5.3 [4 Points Total; 2 points for each part]. Hall and Taylor, Chapter 8, MACROSOLVE exercise 4, page 228. NOTE: For part a, explore at least **three different combinations** of expansionary fiscal policy and tight monetary policy to check the robustness of your policy conclusions. For part b, you might first want to review some of the U.S. data presented by Hall and Taylor in Chapters 1 through 3. ----------------------------- NOTE: See the closed reserve copy of this answer outline for the MacroSolve-generated figures and tables referred to in this answer key for exercise 5.3. ----------------------------- Answer Outline for Part (a): A table of values for key macro variables generated for the default MacroSolve AD/PA Closed Economy model is given in Table 1. For Part (a), students are asked to use this HT MacroSolve model as follows: Examine at least three different combinations of settings for G, t, and M relative to the HT default values that represent expansionary policy (increased G, decreased t) and tight monetary policy (decreased M) to see what the effects are on the following macro variables relative to their default values: Y, R, I, and private saving S_p = [1-t]Y - C = [1-t]Y - (120 + 0.7754[1-t]Y) = [1-0.7754][1-t]Y - 120 . By trying three different combinations, you get some idea how robust your conclusions are. In other words, do your conclusions hold across **all** policy settings or do your conclusions depend on precisely how much G is increased, how much t is decreased, and how much M is decreased. Tabulated and graphical data for three different experiments are given in Tables 2-4 and Figures 1-3. From this data, it appears that actual GDP Y always converges back to potential GDP Y*=6000 (that is, the economy appears to be stable for the given exogenous parameter settings), so there is NO CHANGE in Y. Also, R always INCREASES and I DECREASES. Assuming the economy always returns to internal balance, this observed increase in R and decrease in I can be explained analytically. The increase in G and decrease in t rotate/shift the IS curve upwards (for each Y a higher R). But the intersection of the upwardly displaced IS curve and the FE line Y=Y*= 6000 then determine the new internal balance interest rate R', which must be higher than the original internal balance interest rate level R*=0.050. It then follows from the form of the investment function I = e-dR that investment must be LOWER at this new internal balance point. Finally, private saving S_p always INCREASES. [Note that you can use the MacroSolve option "Create Series" under "DATA" on the menu bar to construct and add the variable S_p = YPD-CONS to the list of MacroSolve variables available for plotting and graphing---you do not have to calculate S_p values by hand!] Again, assuming that Y always converges back to Y*, this increase in S_p can be explained analytically. Since t is being decreased in all experiments, and Y is returning to its original level Y*, disposable income [1-t]Y must ultimately INCREASE. By the calculation for private saving S_p, given above, this implies that S_p must also INCREASE. NOT REQUIRED: It is interesting to observe that the effects of expansionary fiscal policy and tight money policy on the **general price level** are ambiguous. The direction of movement in P depends on how far left the LM curve shifts when M is decreased. If the LM shift is "too much," then the price level will have to fall to bring the economy back to the internal balance level of GDP, Y*=6000. ------------------------------ Answer Outline for Part (b): For this part, students should investigate whether expansionary fiscal policy and tight monetary policy were in effect for the U.S. economy during the time interval 1981-1986. If so, what are the empirical outcomes for the U.S. economy with regard to Y, R, I, and S_p for this period? In particular, are these values consistent with the predictions of the HT MacroSolve model obtained in part A? If so, explain why. If not, consider (as best you can) what may be causing the deviation between empirical reality and the HT model predictions. From Figure 4 and Figure 5, it indeed appears that fiscal policy was expansionary in the U.S. during this period, in the sense that the real government budget deficit as a percent of GDP was trending upwards. However, the money growth rate over this period is rather spiky, and it does not drop sharply until about 1986. It appears that, if the U.S. Fed was indeed trying for a contractionary monetary policy, they only achieved their goal with a lag. As depicted in Figure 6, the real interest rate did spike sharply upwards over this period, and real investment as a percent of GDP is generally falling. As seen from the statistical data accompanying Figure 6, the real interest rate and real investment are negatively correlated over this period. As depicted in Figure 7, private saving S_p is rather spiky over this period, exhibiting a sharp dip during 1982. However, as seen in Figure 8, a brief but deep recession occurred in the U.S. in 1982-3. Thus, it is doubtful that GDP Y was at potential during this period. Recall that the prediction in Part (a) that S_p increases given an expansionary tax policy (a decrease in t) was generated assuming that Y remained at its potential level Y*. If Y is dropping, then the overall net effect on disposable income [1-t]Y of a decrease in t **and** a decrease in Y could be negative, resulting in a decrease in private saving.