Homework #4 - Answers. The IS-LM Model Due Mar 18

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1 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 1 of 12 Homewok #4 - Answes The - Model Due Ma Fun with the Keynesian Coss: a. Use the geomety of the Keynesian Coss diagam shown at the ight to deive that the govenment puchases multiplie is 1/(1 MPC), whee MPC is the slope of the planned expenditue line,. In the figue, planned expenditue has inceased (fo any given income) by the amount of an incease in govenment puchases, G. Use the labels in the figue (a, b, c, ) to denote the distances involved (e.g., ab is the distance fom 1 to 2, o ). Fist, identify the following: 45 f e c d a b 1 2 ' ab G ef MPC Two othe distances that =ab de/cd cd, df Then use these to deive the govenment puchases multiplie: G = ab ef cd = ( df de) = df cd 1 de cd 1 = 1 MPC

2 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 2 of 12 b. In the diagam at the ight, suppose that the economy stats, in time peiod t=0, out of equilibium at income 0. Suppose also the following dynamic pocess fo adjusting when out of equilibium: in each peiod, t, the economy poduces an output equal to the level of planned expenditue fom the pevious peiod. Find and label in the figue the levels of income fo time peiods 1, 2, and e c. In the diagam at the ight, daw the new planned expenditue cuve that will exist if thee is a cut in taxes of T=½ inch. Label T so as to show how you detemined whee the new cuve should be. (Hint: If taxes ae cut by some amount, say $1000, then at any total income consumes will buy the same amount that they would have bought befoe if thei income had been ) Identify also the old and new equilibium levels of income, 1 and 2 and the change in income. 45 Hee is quite a bit smalle than T. Note that the multiplie fo a tax cut is MPC/(1-MPC), which is smalle (in absolute value) than 1 if MPC<1/2. =½ in. T 1 2

3 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 3 of 12 d. (Optional) Retun to the diagam of pat (a) and identify the planned expenditue cuve that would eflect both the incease in govenment puchases and an accompanying tax incease sufficient to finance it. What is the multiplie ( / G = / T) in that case? Can you show that in the figue? As seen in pat (c), a tax change shifts the cuve hoizontally by the amount of the change in taxes. So hee, with an incease in G accompanied by an equal incease in T, the cuve will shift both to the ight by T and up by G, as shown below: f ' e c T G d 45 a 1 2 b Consideing point c, it moves fist to the ight and then up by the same amount, taking it back to the 45 line whee it is theefoe the new equilibium. Theefoe, in this case = G = T, o / G = / T = 1. It is sometimes said that the balanced budget multiplie is one.

4 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 4 of The following ae the equations of the - model, hee including a featue that taxes ae not simply given but depend on income though a tax function, T(). Cuve: = C( T ( )) + I( ) + G Cuve: M / P = L(, ) a. Diffeentiate the model totally and solve fo the govenment spending multiplie, d/dg, in tems of the vaious slopes of the functions: C' = maginal popensity to consume out of disposable income, T' = maginal tax ate, I' = effect (deivative) of the inteest ate on investment, and L, L = the effects (patial deivatives) of the inteest ate and income on liquidity pefeence. Totally diffeentiate the two equations: d = C ( d T d ) + I d + dg 2 dm / P ( M / P ) dp = L d + L d Fo the cuent poblem, dm=dp=0. Solve the second equation fo d: d = ( L / L ) d Substitute this fo d in the fist equation: d = C ( 1 T ) d ( I L / L ) d + dg Subtact the fist two tems on the ight fom both sides and then divide by dg and by the coefficient of d: d 1 = dg 1 C (1 T ) + ( I L / L ) b. Fom this, what would the multiplie be if taxes did not depend on income? Does the income tax (T'>0) make the multiplie lage o smalle? If T'=0, the multiplie is d 1 = dg 1 C + ( I L / L ) Having T'>0 makes the C' tem smalle and, since that tem appeas with a minus sign in the denominato, makes the denominato lage and the multiplie smalle. An income tax makes the govenment spending multiplie smalle. c. Can you tell whethe this multiplie fom pat (a) is lage o smalle than one? What featues of behavio tend to make the multiplie smalle, and what make it lage? No, we can t tell. We know that I'<0, L <0, and L >0, so the tem I L / L is positive. If it is lage than C'(1-T'), then the denominato will be lage than one and the multiplie smalle than one. In geneal, the multiplie will be lage the lage in absolute value ae C and L and the smalle ae T', I', and L.

5 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 5 of We nomally daw the cuve as downwad sloping and the cuve as upwad sloping, as appopiate fo ou usual assumptions about behavio. How would eithe o both of these cuves look diffeent if the following unusual assumptions wee made? ithe descibe o daw you answes, being sue in eithe case to make clea what you mean. a. Investment does not depend on the inteest ate. cuve is vetical (because now, as the inteest ate falls, thee is no component of aggegate demand that inceases). b. The MPC is zeo. This makes the cuve steepe than it would othewise be, but it does not cause it to be vetical. It makes the multiplie equal to 1, athe than lage than one, but this is enough fo a fall in the inteest ate that inceases investment to still cause an incease in income. c. The MPC is one. This eithe causes thee to be no equilibium in the goods maket at all, o it causes all values of to be equilibium fo a paticula level of investment and thus the inteest ate. The eason is that it makes the expenditue function have the same slope as the 45 line, and theefoe eithe be paallel to it o coincide with it fo all. Thus, if the cuve exists at all, it is hoizontal.

6 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 6 of 12 d. Demand fo money does not depend on the inteest ate. The cuve is vetical (because now demand will equal supply of money only at the paticula level of income,, fo which that is tue fo all ). e. Demand fo money does not depend on income. The cuve is hoizontal (because now demand will equal supply of money only at the paticula level of the inteest ate,, fo which that is tue fo all ). f. Supply of money ises endogenously as a esult of inceases in the inteest ate. The cuve is less steep than it would othewise be, but it is still upwad sloping. The eason is that now, as the inteest ate inceases, it inceases supply of money in addition to deceasing demand fo it, and it theefoe equies a lage incease in income to offset these changes by inceasing demand.

7 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 7 of In each case below, you ae given the - diagam with an initial equilibium. Show how one o both of the cuves change fo the following exogenous changes in the model. Then, fom that, detemine the qualitative changes (+,, 0,?) in the indicated vaiables (whee? means the vaiable could go eithe way). Give easons fo you esults, whee asked. a. A eduction in govenment puchases 1 ' 1 C Why? Because income falls. M 0 Why? Because M is exogenous and did not change.

8 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 8 of 12 b. An incease in the money supply. ' I + Why? Because the inteest ate falls. L + Why? Because L=M and M went up. O because fell and ose, both of which aise L.

9 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 9 of 12 c. A downwad shift in the consumption function (less consumed at each T). 1 ' 1 L 0 Why? Because L=M and M has not changed. (Note that you can t tell fom and, since the fall in lowes L while the fall in aises L.) C Why? Because fell and the consumption function shifted down.

10 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 10 of 12 d. An upwad shift in the investment function (moe investment at each inteest ate). 1 ' S (=national savings) + Why? Since pivate savings depends positively on income, and income has gone up. I + Why? Because I=S and S has isen. Note that you can not get it diectly fom the I function, since the shift of the function and the ise in wok in opposite diections.

11 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 11 of In pat (c) of question 4 above you found (I hope) that the downwad shift in the consumption function caused a fall in output. Suppose now that policy makes ty to pevent this by expansionay monetay o fiscal policy. Fo each of the policies indicated below, assume that the policy is used in exactly the ight amount to offset the effect on, so that the combined effect of the shift of the consumption function and of the policy is fo equilibium not to change at all. Detemine in each case how pivate savings, govenment savings, and national savings in the new equilibium compae to what they wee befoe all this happened. a. A change in the money supply An incease in the money supply shifts the cuve out to '' as shown and the inteest ate falls futhe to a new A equilibium at point C. Thus investment 1 is lage and pivate savings must also be B lage than it was initially, since govenment savings is unchanged and S C must equal I. (Pivate savings inceased with the shift of the consumption function ' and is not futhe alteed since is unchanged.) 1 '' b. A change in govenment puchases An incease in govenment puchases estoes the old level of by shifting the cuve back to its initial position, so equilibium etuns to point A. Thus thee is no change in, and theefoe no change in I. Since S=I, national savings must be unchanged. But now the govenment is saving less, so pivate savings must be inceased, exactly as in pat (a). c. A change in taxes. How does the effect on the govenment budget hee compae to pat (b)? A tax cut also shifts the cuve back to its initial position and estoes equilibium at A, whee investment and theefoe national savings ae at thei initial levels. As in pat (b), govenment savings is educed, and theefoe pivate savings must be highe. In fact, both of these changes must be lage than in pat (b), because the tax cut aises pivate savings even moe than the oiginal shift of the consumption function. Indeed, the tax cut itself must be lage than the incease in govenment puchases in pat (b) because the tax multiplie is smalle than the govenment spending multiplie.

12 Winte Tem 2004 Alan Deadoff Homewok #4 - Answes Page 12 of Suppose that at a moment in time neithe the goods maket no the money maket is in equilibium, so that the economy is neithe on the cuve no on the cuve. a. Based on what you know about how these two makets adjust towad equilibium, which maket would you expect to move moe quickly towads equilibium? Why? The goods maket adjusts though a athe cumbesome pocess: if expenditue doesn t match output, then fims espond by poducing moe o less. This changes incomes, which in tun cause a change in expenditue that also must be esponded to by fims changing output. This pocess continues, and theefoe can be expected to take some consideable time. The money maket, on the othe hand equies only that the inteest ate adjust to whateve the levels of supply and demand fo money happen to be. The inteest ate is detemined in the bond maket, whee bond pices (and thus inteest ates) ae detemined moment to moment by bond tades. So the inteest ate, and thus the money maket can adjust vey quickly. Theefoe it makes most sense that an economy will move moe apidly to the cuve than to the cuve. b. Based on you answe to (a), show in the - diagam at the ight, and then on the time axes below, how you would expect income and the inteest ate to espond to a sudden lage incease in the money supply that occus at time t 0. ' 1 1 t 0 time t 0 time