Appendix Algebra of the ISLM Model

Transcription

1 Chapter 23: Monetary an Fscal Polcy n the ISLM Moel Appenx Algebra o the ISLM Moel The use o algebra to analyze the ISLM moel allows us to exten the multpler analyss n Chapter 22 an to obtan many o the results o Chapters 22 an 23 very quckly. Basc Close-Economy ISLM Moel The goos market can be escrbe by the ollowng equatons: Consumpton uncton: C = C + mpc (Y T) () Investment uncton: I = I (2) Taxes: T = T (3) Government spenng: G = G (4) Goos market equlbrum conton: Y = Y a = C + I + G (5) The money market s escrbe by these equatons: Money eman uncton: M = M + ey (6) Money supply: M s = M (7) Money market equlbrum conton: M = M s (8) The uppercase terms are the varables o the moel; G, T, an M are the values o the polcy varables that are set exogenously (outse the moel); an C, I, an M are autonomous components o consumer expenture, nvestment spenng, an money eman that are also etermne exogenously (outse the moel). Except or the nterest rate, the lowercase terms are the parameters, the gvens o the moel, an all are assume to be postve. The entons o these varables an parameters are as ollows: C = consumer spenng I = nvestment spenng G = G = government spenng Y = output T = T = taxes

2 M M s = money eman = M = money supply = nterest rate C = autonomous consumer spenng = nterest senstvty o nvestment spenng I = autonomous nvestment spenng relate to busness conence M = autonomous money eman e = ncome senstvty o money eman = nterest senstvty o money eman mpc = margnal propensty to consume IS an LM Curves Substtutng or C, I, an G n the goos market equlbrum conton an then solvng or Y, we obtan the IS curve: Y = ( C + I mpct + G ) (9) mpc Solvng or rom Equatons 6, 7, an 8, we obtan the LM curve: M M + ey = (0) Soluton o the Moel The soluton to the moel occurs at the ntersecton o the IS an LM curves, whch nvolves solvng or Y an smultaneously, usng Equatons 9 an 0, as ollows: M Y = C + I mpct + G mpc + e/ + M () [ e( C + I mpct + G) + M ( mpc) M( mpc) ] = ( mpc) + (2) Implcatons The conclusons reache wth these algebrac solutons are the same as those reache n Chapters 22 an 23; or example:

3 . Because all the coecents are postve, Equaton ncates that a rse n C, I, G, an M leas to a rse n Y an that a rse n T or M leas to a all n Y. 2. Equaton 2 ncates that a rse n C, I, G, an that a rse n M or T leas to a all n. M leas to a rse n an 3. As, the nterest senstvty o money eman, ncreases, the multpler term mpc + e/ ncreases, an so scal polcy (G, T ) has more eect on output; conversely, the term multplyng M, = mpc + e/ ( mpc + e) eclnes, so monetary polcy has less eect on output. 4. By smlar reasonng, as, the nterest senstvty o nvestment spenng, ncreases, monetary polcy has more eect on output an scal polcy has less eect on output. Open-Economy ISLM Moel To make the basc ISLM moel nto an open-economy moel, we nee to nclue net exports n the goos market equlbrum conton so that Equaton 5 becomes Equaton 5': Y = Y a = C + I + G + NX (5') As the scusson n Chapter 23 suggests, the net exports an exchange rate relatons can be wrtten NX = NX he (3) E = E + j (4)

4 where NX = net exports NX = autonomous net exports h = exchange rate senstvty o net exports E = exchange rate (value o omestc currency) E = autonomous exchange rate j = nterest senstvty o exchange rate Substtutng or net exports n the goos market equlbrum conton (Equaton 5') usng the net exports an exchange rate relatons an then solvng or Y as n the basc moel, we obtan the open-economy IS curve: [ C + I mpct + G + NX he ( hj ) ] Y = + mpc (5) The LM curve s the same as n the basc moel, an so the solutons or Y an are as ollows: Y = + hj + hj C + I mpct + G M + M + NX h mpc + ( + hj ) e/ E (6) ( mpc) + ( + hj ) [ e( C + I mpct + G + NX he) + M ( mpc) M( mpc) ] = e (7) Implcatons. As the IS curve n Equaton 5 ncates, nclung net exports n aggregate eman proves an atonal reason or the negatve relatonshp between Y an (the ownwar slope o the IS curve). Ths atonal reason or the negatve relatonshp o Y an s represente by hj n the term ( + hj). 2. Equatons 6 an 7 ncate that all the results we oun or the basc moel stll hol. 3. Equaton 6 ncates that a rse n NX leas to a rse n Y an that an autonomous rse n the value o the omestc currency E leas to a eclne n Y.

5 4. Equaton 7 ncates that a rse n NX leas to a rse n an that a rse n E leas to a eclne n by Ason Wesley Longman, a vson o Pearson Eucaton