Appendix to Chapter 9: The IS-LM/AD-AS Model: A General Framework for Macro Analysis

Transcription

1 Appendix to Chapter 9: The IS-LM/AD-AS Model: A General Framework for Macro Analysis Yulei Luo Econ, HKU November 13, 2017 Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

2 The Labor Market Holding the capital stock K fixed, we can write the production function with output Y as: Y = A (f 1 N 12 ) f 2N 2, (1) where f 1, f 2 > 0. The MPN is MPN = A (f 1 f 2 N). (2) Firms hire labor to the point at which the MPN equals the real wage: w = A (f 1 f 2 ND), (3) where ND is the amount of labor demanded. The supply of labor (NS) is an increasing function of the current and after-tax real wage: NS = n 0 + n w (1 t) w. (4) Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

3 Equilibrium in the labor market In equilibrium, the amounts of labor demanded (ND) and supplied (NS) are equal; their common value is the full-employment level of employment, N: [ f 1 f 2 n 0 w = A 1 + (1 t) Af 2 n w N = n 0 + (1 t) Af 1 n w 1 + (1 t) Af 2 n w ], (5) The full-employment output can therefore be ( Y = A f 1 N 1 ) 2 f 2 2 N, FE line. (6) Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

4 The goods market We start with equations for desired consumption and desired investment. Desirable consumption is where c 0, c r > 0, c Y (0, 1), C d = c 0 + c Y (Y T ) c r r, (7) T = t 0 + ty. (8) Desired investment is where i 0, i r > 0. I d = i 0 i r r, (9) Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

5 Equilibrium in the goods market In the goods market equilibrium, Y = C d + I d + G or (10) S d = I d. (11) Substituting the expressions for desired consumption and investment into the goods market equilibrium: Y = c 0 + c Y (Y t 0 ty ) c r r + i 0 i r r + G [1 (1 t) c Y ] Y = c 0 + i 0 + G c Y t 0 (c r + i r ) r, r = α IS β IS Y, IS curve, (12) where α IS = c 0+i 0 +G c Y t 0 c r +i r and β IS = 1 (1 t)c Y c r +i r. Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

6 The Asset Market Equilibrium In general, the real demand for money depends on real income, Y, and the nominal interest rate on nonmonetary assets, i = r + π e : M d P = l 0 + l Y Y l r (r + π e ). (13) If there are only two types of assets (money and nonmonetary assets), the asset market is in equilibrium when M P = l 0 + l Y Y l r (r + π e ), (14) r = α LM 1 M l r P + β LM Y, LM curve, (15) where α LM = l 0 lr π e and β LM = l Y lr. Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

7 General equilibrium in the IS-LM model We have solved for the general equilibrium levels of the real wage (w), employment ( N ), and output ( Y ) in the labor market. Turning to the goods market: r = α IS β IS Y. (16) Having r and Y, we can find the general equilibrium values of T, C, and I. To find the equilibrium price level, we need to use the money market equilibrium condition: P = M ( ). (17) l 0 + l Y Y l r αis β IS Y + πe Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

8 The AD-AS model The AD curve. Aggregate output demanded at any price level, P, is the amount of output corresponding to the interaction of the IS and LM curves: Y = α IS α LM + (1/l r ) (M/P) β IS + β LM (18) The AS curve. In the SR, firms supply the output demanded at the fixed price level, which is P sr. So the SRAS curve is horizontal line: P = P sr, SRAS. (19) The LRAS curve is a vertical line at the full-employment level of output: Y = Y, LSAS. (20) Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

9 Short-run and long-run equilibrium The SR equilibrium is represented by the interaction of the AD curve and the SRAS curve: Y = α IS α LM + (1/l r ) (M/P sr ) β IS + β LM. (21) The LR equilibrium is represented by the interaction of the AD curve and the LRAS curve: P = M [ ]. (22) l r αlm α IS + (β IS + β LM ) Y Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

10 An numerical example for solving the IS-LM model Consider an economy that is described by the following equations: and C d = (Y T ) 300r (23) T = Y I d = r, L = 0.5Y 500i Y = 2500; G = 600; M = 133, 200; π e = 0.05; P sr = 120. In the SR, the PL is fixed at P sr. Find the SR and LR equilibrium values of Y, P, r, C, I, and i. Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

11 Step 1: Find the IS equation using the goods market equilibrium condition: Y = C d + I d + G. Y = (Y Y ) 300r + ( r) r = Y.IS curve. (24) Step 2: Find the LM equation using the asset market equilibrium condition. 133, 200 P = 0.5Y 500 (r ) r = 0.001Y /P.LM curve. When P = P sr = 120, r = 0.001Y (25) Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

12 Step 3: Find the short-run equilibrium, i.e., find the interaction of the IS and LM curves: Y = 0.001Y 2.27 Y = (26) r = (27) Substituting the equilibrium r and Y into other equations, we can find the equilibrium values for T, C, I, and i. Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

13 Step 4: Find the long-run equilibrium. Use the fact that Y = Y, we can find the equilibrium real interest rate: r = (28) Substituting the equilibrium r and Y into other equations, we can find the equilibrium values for T, C, I, and i. Plug the equilibrium r and Y into the money demand function to obtain the value of real money demand, L: L = 0.5Y 500i = Setting M/P = 1200, we have P = 111. Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14

14 Step 5: Find the equation for the AD curve using the IS-LM curves. The interaction of the IS and LM curves: Y = 0.001Y /P Y = /P. (29) Step 6: Illustrate the use of the AD curve and short-run and long-run AS curves. In the SR, Y = /P sr, Y = In the LR, Y = Y = 2500, P = 111. Luo, Y. (Econ, HKU) ECON2220CB: Intermediate Macro November 13, / 14