Two Models of Macroeconomic Equilibrium

Transcription

1 Two Models of Macroeconomic Equilibrium 1 The Static IS-LM Model The model equations are given as C η +γ(y T) (1) T τy (2) I α r (3) G T (4) L φy θr (5) M µ (6) Y C +I +G (7) L M (8) where η,α,,φ,θ,µ > 0 and γ,τ (0,1). (1) is a consumption function where consumption C is a linear function of disposable income Y T. Here Y and T respectively denote the national income and the total level of tax. (2) implies that the government collects a fraction τ of the national income as tax. (3) is the investment function that makes the total private investment expenditure I in the economy a function of the interest rate r. (4) states that he government runs a balanced budget and the government spending G is equal to the total tax revenue. (5) determines the total demand L for money as a function of Y and r. (6) implies that the total supply M of money is constant. Finally, (7) and (8) are respectively the market clearing conditions: The first clears the goods-and-services market so that the national income is fully expended. The second clears the money market so that the supply of and the demand for money is equal. The endogenous variables are Y,C,I,G,T,r,L and M. Since we have 8 equations, we might be able to solve this system. Let s first use (7) and (1)-(4) to write Y C +I +G η +γ(y T)+α r +T η +γ(y τy)+α r +τy η +α+γ(1 τ)y r +τy η +α+[γ(1 τ)+τy r Notice that there exists a relationship between r and Y such that Y η +α+[γ(1 τ)+τy r r η +α+[γ(1 τ)+τ 1Y r η +α [ γ(1 τ) τ +1Y r η +α [(1 γ)(1 τ)y r η +α (1 γ)(1 τ) Y (IS) 1

2 This is called the IS relation because it represents (r,y) pairs at which investment and saving are equalized. Now, let s return to (5), (6) and (8). We have L M φy θr M φy θr µ We thus find another relation between r and Y: φy +θr µ θr φy µ r µ θ + φ θ Y (LM) This is called the LM relation because it represents (r,y) pairs at which liquidity and the supply of money are equalized. Let me now rewrite IS and LM equations in matrix form: [ (1 γ)(1 τ) φ θ [ Y r [ η +α µ Since the system is not homogeneous, we need (1 γ)(1 τ) φ θ 0 to find a unique nontrivial solution (Y,r). We have indeed (1 γ)(1 τ) φ θ θ(1 γ)(1 τ) ( φ) θ(1 γ)(1 τ)+φ and since,φ,θ > 0 and γ,τ (0,1), we have θ(1 γ)(1 τ)+φ > 0. The inverse of the coefficient matrix satisfies [ 1 θ θ(1 γ)(1 τ)+φ φ (1 γ)(1 τ) The solution thus reads [ Y r [ 1 θ θ(1 γ)(1 τ)+φ φ (1 γ)(1 τ) [ η +α µ [ θ(η+α)+µ θ(1 γ)(1 τ)+φ φ(η+α) (1 γ)(1 τ)µ θ(1 γ)(1 τ)+φ Note that we have indeed solved the model because we can now express all endogenous variables of the model as functions of the exogenous variables and the parameters: θ(η +α)+µ Y(η,α,,φ,θ,µ,γ,τ) θ(1 γ)(1 τ)+φ φ(η +α) (1 γ)(1 τ)µ r(η,α,,φ,θ,µ,γ,τ) θ(1 γ)(1 τ)+φ C(η,α,,φ,θ,µ,γ,τ) η +γ(y T) η +γ(y τy) η +γ(1 τ)y(η,α,,φ,θ,µ,γ,τ) I(η,α,,φ,θ,µ,γ,τ) α r(η,α,,φ,θ,µ,γ,τ) 2

3 and so on... Notice that using the solution of the model, we can implement comparative static analyses. What happens to equilibrium income Y if the government increases the tax rate τ? What if the government increases the money supply µ? Or, consider a change in investors behavior such that they suddenly become very pessimistic and reduce α. What does happen to the equilibrium level of r? For all such questions, we are going to use the partial derivatives of the form 1.1 Fiscal Policy Y(η,α,,φ,θ,µ,γ,τ) τ Y(η,α,,φ,θ,µ,γ,τ) µ r(η,α,,φ,θ,µ,γ,τ) α The fiscal policy measure of the model we studied is the tax rate τ. This is a bit different from the usual description because the government in this version of the model always runs a balanced budget. Thus, an increase in government spending is initiated with an increase in τ given Y. To see how effective fiscal policy is in increasing real economic activity, we simply look at 1.2 Monetary Policy Y(η,α,,φ,θ,µ,γ,τ) τ ( ) θ(η +α)+µ τ θ(1 γ)(1 τ)+φ ( ) θ(η +α)+µ θ(1 γ) [θ(1 γ)(1 τ)+φ 2 θ(1 γ)[θ(η +α)+µ [θ(1 γ)(1 τ)+φ 2 > 0 The monetary policy measure here is simply the money supply µ. To see how effective monetary policy is in increasing real economic activity, we simply look at Y(η,α,,φ,θ,µ,γ,τ) ( ) θ(η +α)+µ µ µ θ(1 γ)(1 τ)+φ θ(1 γ)(1 τ)+φ > 0 Exercise: Under what condition(s) an expansionary monetary policy (dµ > 0) is most effective? Under what condition(s) an expansionary monetary policy (dµ > 0) is least effective? 1.3 The Policy Mix A bit more difficult is to find a policy mix to avoid the increase in r. When τ increases, IS gets flatter in (r,y)-plane. With LM not changing, the increase in Y occurs along with an increase in r: Y τ > 0 and r τ > 0 When r increases, I decreases in response. This is called crowding out. To avoid the crowding out, the government may wish to support the expansionary fiscal policy with an expansionary monetary policy. Since an increase in µ shifts the LM curve to the right, Y can increase where r remains constant. 3

4 Here is the question then: if the increase in τ is equal to dτ > 0, how much should µ increase to keep r constant? Since we want dr 0, we first take the following total differential using (IS): ( ) ( ) η +α (1 γ)(1 τ) dr d d Y Since we change only τ and µ, we can rewrite this as dr (1 γ) d[(1 τ)y 0 (1 γ) d[(1 τ)y 0 d[(1 τ)y 0 (1 τ)dy +Ydτ dy Ydτ 1 τ Now apply the same reasoning to LM to find dµ that shifts LM under the constraint dr 0: ( ) ( ) 1 φ dr dµ+ dy θ θ ( ) ( ) 1 φ 0 dµ+ dy θ θ dµ φdy Thus, the needed increase in µ is equal to as a function of dτ. dµ φdy ( ) φy dτ 1 τ Exercise: Is it possible to find a policy mix when θ 0? What if φ 0? 1.4 The Aggregate Demand The IS-LM model is usually an intermediate step to derive what we call the Aggregate Demand curve. The AD curve, on (P,Y)-plane, is the geometric location of all points where IS and LM relations hold simultaneously (P denotes the aggregate price level). The IS-LM model we studied so far does not (explicitly) include P. Now redefine the stock of money supply µ so that the LM curve looks like r µ P θ + φ θ Y (9) Here, µ denotes the nominal money supply and, thus, µ/p is the real money supply. Since we just replace µ with µ/p, the way by which we solve the model does not change. Hence, the model augmented with P implies θ(η +α)+ µ P Y θ(1 γ)(1 τ)+φ Notice that there exists an inverse function P P(Y) such that P P(Y) µ [θ(1 γ)(1 τ)+φy θ(η +α) This function defines the AD curve, and it has a negative slope. 4

5 Exercise: Find the slope of the AD curve defined by P P(Y) above. Exercise: A version of the Quantity Theory of Money argues that the equilibrium in the money market is characterized by µ κpy where µ denotes nominal money supply and κ > 0 is a parameter. Under what condition(s), the AD curve derived above from the IS-LM model reduces to the one implied by µ κpy? (Note: Your answer should be clarifying the significance of Keynes s contribution to the theory of money demand!) 5

6 2 Money, Inflation, Growth, and Unemployment Here is a toy version of a core macroeconomic model defined by four structural relationships among four endogenous variables. 1. Suppose that there exists a negative relationship between unemployment rate u and inflation rate π as in u u 0 (π π e ) u 0,,π e > 0 where u 0 is the natural rate of unemployment and π e is the expected inflation rate. The conjecture is that, when the expected inflation rate π e is lower than the actual inflation rate π, the expected increase in real wage is higher than its actual increase. People work more because of this illusion, and the unemployment rate u decreases below its natural level u When this occurs (u < u 0 ), the growth rate g of real GDP increases above its natural level g 0 because labor is an input of production (more labor means more output). Let the negative relationship between u and g be formalized as in g g 0 α(u u 0 ) α,g 0 > 0 3. As the quantity theory of money implies, the growth rate m of the money stock is equal to the sum of the inflation rate π and the growth rate g of real GDP: m π +g 4. In this economy, it is the central bank s responsibility to adjust m for price stability. Specifically, suppose that the central bank follows the monetary rule m µ η(π π ) µ,η,π > 0 where π denotes the target level of inflation rate π. According to this rule, the central bank decreases m below µ if the actual inflation rate π is higher than the target level π. Exercise: Solve the model using linear algebra. Specifically, express the model s endogenous variables (π,u,g,m) as functions of exogenous variables and parameters. u u 0 + (g 0 µ+π e +ηπ e ηπ ) η +α +1 π µ g 0 +ηπ +απ e η +α +1 g g 0 +ηg 0 +αµ απ e αηπ e +αηπ η +α +1 m µ+ηg 0 +ηπ +αµ αηπ e +αηπ η +α +1 Exercise: How do π and π e affect π? Provide some economic intuition (not just the comparative static results). π π η η +α +1 > 0 π π e α η +α +1 > 0 Exercise: Design an expansionary monetary policy, and show that it is effective in increasing g and decreasing u? If there is no such policy, explain why not? u µ η +α +1 < 0 6 g µ α η +α +1 > 0

7 Remark: You can solve the same model by eliminating u and m from the system. This defines two relations; one for the demand side of the economy and the other for the supply side. Use the first two equations of the model to define the supply side relation between g and π as g g 0 α(π π e ) (Supply) Next, use the last two equations to define the demand side relation between g and π: Rewriting these equations, we have π π +g µ η(π π ) ( ) ( ) 1 1 g+π e g 0 α α (Demand) (Supply) π g +µ η(π π ) (Demand) These two relations on (π, g)-plane can be thought of the-rate-of-change versions of short-run AS and AD curves. Exercise: Draw a figure that shows these two relations, and indicate the equilibrium point. 7