Econ 32 - IS/L odel Notes Handout: IS/L odel IS Cuve Deivation Figue 4-4 in the textbook explains one deivation of the IS cuve. This deivation uses the Induced Savings Function fom Chapte 3. Hee, I descibe an altenative deivation of the IS cuve using the 45 -line/expenditue function model fom Chapte 3. The esults is the same but the gaphs diffe. IS Cuve: All combinations of inteest ates and GD fo which the spending balance model is in equilibium. Deivation: The deivation begins with the Keynes Coss model developed in Chapte 3. Bea in mind that I p and C a fom this model both depend on the inteest ate in the economy,. Gaphically, the income detemination model goes on the top set of axes and below it a set of axes with GD gaphed on the hoizontal axis and the inteest ate gaphed on the vetical axis. E po A po E E E 0 45 o E p = A p + c. Begin at the equilibium point E o in the spending balance model. E o identifies the equilibium level of GD (o ) and planned expenditue (E po ) in the model. In tun, the equilibium level of planned expenditue depends on two factos, I p and C a that depend on the inteest ate. To locate A po, thee must also be some coesponding inteest ate, o, in the economy. o E 0 o 2. ap the equilibium level of GD, o, and the coesponding inteest ate, o, into the bottom panel ( ). ap means locate the equilibium point E o in the lowe gaph, by locating o and o in this space. Note that, by definition, E o is on the IS Cuve.
Econ 32 - IS/L odel Notes 2 E p A p E E E p = A p + c E E 0 p E 45 E E 0 3. Change the inteest ate. Suppose that inteest ates ise fom o to. An incease in inteest ates educes both I p and C a, which educes A p fom A p0 to A p. This shifts the E p line down and educes equilibium GD fom o to and leads to a new equilibium point, E. 4. ap the new equilibium point in the spending balance model into the bottom panel ( ). The coodinate can be found by extending the vetical dashed line staight down. The coodinate must be located somewhee above the pevious inteest ate o because the inteest ate was assumed to incease. Note that the new equilibium point E is also, by definition, on the IS Cuve. o E E 0 IS o oe points on the IS Cuve could be found by epeating this pocedue and finding additional equilibium points. This would esult in a linea elationship between and. oving along the IS Cuve inteest ates move in the opposite diection as GD.
Econ 32 - IS/L odel Notes 3 Stong and Weak olicy onetay olicy Fiscal olicy E 0 E 7 L E 7 E0 E5 IS Stong onetay olicy 7 Stong Fiscal olicy 2 E 0 L E 2 IS 2 Weak onetay olicy Oiginal E0 E5 5 Weak Fiscal olicy Hoizontal L L 4 E0 E 4 IS 6 E 6 E0 E5 4
Econ 32 - IS/L odel Notes 4 oe on the Algeba of the IS/L odel The gaphical pesentation to the IS/L model has a coesponding analytical epesentation. In many ways, the model in equations povides moe insight than the gaphical vesion of the model. The appendix to Chapte 5 contains a discussion of the algebaic solution to this model. Hee, I pesent a slightly diffeent vesion of this mateial. Recall fom Chapte 3, the effect of changes in the exogenous vaiables of the model on GD depended on the multiplie multiplie = maginal leakage ate And also ecall that the equilibium condition fom the Chapte 3 model of income detemination was () = ka p (2) In a nutshell, the IS cuve deivation in Chapte 4 simply makes Autonomous lanned Expenditue depend on the inteest ate. A p = A p b (3) Hee, the paamete b simply eflects how sensitive A p is to changes in the inteest ate. Recall fom the IS Cuve deivation, when changed, I p and C a changed, and the intecept of the E p line shifted aound; when, the intecept fell and. This equation simply shows this algebaically. b is elated to the effect of changes in the inteest ate on I p and C a. This also changes the equilibium condition fom the model. Simply substitute the expanded equation fo A p into the equilibium condition = k(a p b) (4) In fact, this new equilibium condition is also an expession fo the IS cuve. To get this expession in slope-intecept fom, we need to solve it fo the vaiable gaphed on the vetical axis, = k(a p b) = ka p kb kb = ka p = k kb A p kb = b A p kb In this equation, kb is the slope of the IS Cuve. It is negative, so the IS cuve slopes down. The fist tem is the intecept of the IS cuve. The exogenous factos in A p shift this intecept - they shift the IS cuve left and ight. The L cuve emeges fom the money maket model. Inteest ates move to equilibate money demand and money supply, so the equilibium condition i the money maket is The L Cuve emeges fom this equilibium condition. condition fo, the vaiable gaphed on the vetical axis. S S d = = h f (5) Again, simply solve the equilibium
Econ 32 - IS/L odel Notes 5 = h S f Now we have a system of two equations (the IS Cuve and the L Cuve) in two unknowns. These two equations can be solved to get a educed fom equation fo GD. To ecap, we have (6a) IS Cuve = b A p kb Spending aket Equilibium = k(a p b) L Cuve = h f S f oney ) aket Equilibium = h f ( S Thee ae many ways to algebaically solve this system. Fo example, the IS Cuve could be set equal to the L Cuve, and the esulting equation solved fo. The appoach in the appendix is to plug the expession fo the L Cuve into the ight hand side of the expession fo the spending balance model equilibium, substituting fo [Note the typo in equation (7) in the text] = k(a p b) = k [ A p + bh f + b f ] S Add kbh/f to both sides and divide both sides by k to solve fo ( k + bh ) = A p + b S f f And divide both sides by the tem in paentheses on the left hand side. = A p + b S f This equation is a educed fom equation fo GD fo the IS/L odel. It shows the oveall impact on of changes in the exogenous vaiables in the model when both the money maket and the spending balance model ae in equilibium. Chapte 5 is pimaily concened with the elative stength of monetay policy and fiscal policy. How do S and G affect in the model? Note that G is pat of A. To make the analysis easie, the educed fom equation can be ewitten a bit = A p+ b f = S A p + b f S In this equation, the policy vaiables of inteest ae S and G which is pat of A p. The two messy factions in font of these tems can be simplified into two paametes Whee the paametes ae simply = k A S p + k 2 (7) (8) (9) k = (0)
Econ 32 - IS/L odel Notes 6 k 2 = b/f = ( b f k And again, the simplified Reduced Fom Equation Fo GD fom the IS/L model is ) = k A S p + k 2 () (2) Some Algebaic Insight The gaphical depictions of the conditions fo stong and weak policy discussed in Chapte 5 and shown above also have algebaic countepats. Insight into the conditions unde which monetay and fiscal policy ae elatively stong and weak can be seen fom the educed fom solution fo GD in the IS-L odel. Reduced Fom Solution: = k A p + k S 2 onetay olicy = k A p + k S 2 Fiscal olicy = k A p + k S 2 Effect of on depends on: k 2 = b k 2 ( b f ) k k = Size of the effect of G on depends on: b k f k 2 f k An Example: As the L Cuve gets steepe, fiscal policy gets weake. Recall: L Cuve = h f f S Slope of the L Cuve = h f The stength of fiscal policy depends on k. When the L cuve gets steepe, h f gets lage. Note that when h f gets lage, the denominato of k gets lage, and k gets smalle. That makes fiscal policy ( G) have a smalle effect on. This concept is illustated with a sample exam question on the following page. Answe this question befoe looking at the solution on the next page.
Econ 32 - IS/L odel Notes 7 Sample Exam Question: Suppose that money demand is not vey sensitive to changes in the inteest ate. In this case, monetay policy is stong. (Tue/False/Uncetain)
Econ 32 - IS/L odel Notes 8 Solution: Tue. If money demand is not sensitive to changes in the inteest ate, then the paamete f is small. As f gets smalle, h f (the slope of the L cuve) gets bigge - the L cuve gets steepe. The steepe the L cuve, the less effective is monetay policy. L with f small, h f lage with f lage, L IS h f small