Lecture 7. The Dynamics of Market Equilibrium. ECON 5118 Macroeconomic Theory Winter Kam Yu Department of Economics Lakehead University

Transcription

1 Lecture 7 The Dynamics of Market Equilibrium ECON 5118 Macroeconomic Theory Winter 2013 Phillips Department of Economics Lakehead University 7.1

2 Outline Phillips Phillips 7.2

3 Market Equilibrium: Two Extreme Assumptions Neoclassical economics asserts that if markets are competitive, price and quantity adjust quickly to achieve equilibrium. Traditional Keynesian economics assumes that some markets adjust slowly, leading to short-run price inflexibility. When prices are sticky, excess demand (e.g. credit crunch) or excess supply (e.g. unemployment) may result. With these market failures, government can play a role in adjusting the economy using fiscal and monetary policies. Real business cycle theory continues to emphasize DGE with flexible prices. Economic fluctuations are generated by productivity shocks. New Keynesian models seek to understand price inflexibility using DGE analysis. Phillips 7.3

4 Features of New Keynesian Models 1 Assumption of the optimization framework 2 Imperfect competition in some markets (goods and labour) 3 Market power gives rise to higher prices and slow adjustment, that is, price inflexibility can be explained by the firms strategic behaviours. Consequences: 1 The speeds and frequencies of price changes are different for different industries, depending on the market structures, nature of exogenous shocks, etc. 2 The heterogeneous changes will have a profound effect on the overall price level. 3 If the output prices of a firm adjust differently from the input prices, it creates real resource costs in the form of production inefficiency. Consequently inflation can be costly. Phillips 7.4

5 Price Changes Average time between price changes Study Country Time Bils et al. U.S. 6 months Blinder et al. U.S. 12 months Rumbler and Vilmunen Euro Area 13 months Phillips 7.5

6 Price Changes Average time between price changes Study Country Time Bils et al. U.S. 6 months Blinder et al. U.S. 12 months Rumbler and Vilmunen Euro Area 13 months Median time of changes between sectors in the U.S. Industry Time (months) Apparel, food, furnishings Transportation 1.9 Entertainment 10.2 Medical services 14.9 All goods 3.2 All services 7.8 All industries 4.3 Phillips 7.5

7 and Wages General observations 1 Price and wage rigidities are temporary. The DGE model is more appropriate in the long run. 2 and wages change on average two or three times a year. 3 Higher inflation causes more frequent changes. 4 Price and wage changes are not synchronized. 5 Frequencies of price changes differ across industries. Changes are more frequent in tradeables than those of nontradeables. Energy and food prices are more volatile. 6 and costs change at different rates at different stages of the business cycle. Costs rise more than prices in late expansion phase, eroding profit margins. Phillips 7.6

8 When Firms Have Market Power Inverse demand function facing a firm: P = D(Q), D (Q) < 0. Production function: Q = F(X 1,..., X n ), where F is increasing and concave. Total cost of production is n C = W i X i. i=1 Each input supplier also has market power so that Phillips W i = W (X i ), W (X i ) >

9 Profit Maximization Problem The firm chooses Q and (X 1,..., X n ) to maximize profit given technology. The Lagrangian is L = D(Q)Q n W (X i )X i + λ[f(x 1,..., X n ) Q]. i=1 The first-order conditions are L Q = P + D (Q)Q λ = 0, L X i = W i W (X i )X i + λ F = 0, X i i = 1,..., n. Phillips 7.8

10 Marginal Revenue and Marginal Cost Eliminate λ from the FOCs, we get P + D (Q)Q = W i + W (X i )X i F/ X i, i = 1,..., n. (1) The left hand side is the marginal revenue and the right hand side is the marginal cost MC = C Q = C X i X i Q = C/ X i Q/ X i = MC i MP i, where MC i is the marginal cost of using one more unit of input i, and MP i is its marginal product. Equation (1) can be written as P + dp Q = MC. (2) dq Recall that the price elasticity of demand is defined as ɛ D = dq P dp Q. Phillips 7.9

11 Price Mark-Up Equation (2) becomes or P + P ɛ D = MC, 1 P = MC 1 + (1/ɛ D ) (9.1) 1 MC i = 1 + (1/ɛ D ) MP i (9.2) The price elasticity of supply of input i is defined as Therefore ɛ i = dx i dw i W i X i. Phillips MC i = W i + dw i dx i X i = W i [1 + (1/ɛ i )]. 7.10

12 The Sum of Market Power Equation (9.2) can be written as P = 1 + (1/ɛ i) W i (9.3) 1 + (1/ɛ D ) MP i Note that if the output market and the input markets are competitive, the output demand curve and input supply curves facing the firm are horizontal, i.e., ɛ D and ɛ i, so that we get the neoclassical result P MP i = W i. In (9.3) if i is labour input, ɛ i < may reflect the bargaining power of the labour union. See Jehle and Reny (2011, pp ) for more discussion on pricing strategy of a monopoly. Phillips 7.11

13 Price Transmission Mechanism The transmission of a change in W i to P depends on, among other things, the elasticity of substitution between the inputs, which is defined as φ ij = d(x j/x i ) MP i /MP j. d(mp i /MP j ) X j /X i Consider the CES production function Q = ( N i=1 α i X ρ i ) 1/ρ, 0 ρ < 1, N α i = 1. It can be shown that the elasticity of substitution between any two inputs is φ = 1 1 ρ. i=1 Phillips 7.12

14 Something About the CES Production Function 1 If ρ 1, φ. It becomes a linear function, with perfect substitution between inputs. 2 If ρ 0, φ 1. The CES becomes a linearly homogeneous Cobb-Douglas function Q = N i=1 X α i i. 3 If ρ, φ 0. It becomes a Leontief function, with perfect complements. The price transmission mechanism from W i to P depends on ρ (see assignment). Conclusion: Speed and amount of price transmission depend on market structure and technology. Phillips 7.13

15 Model assumptions 1 An aggregate firm produces the final good y with a CES technology y = ( N i=1 x ρ i ) 1/ρ, 0 ρ < 1. 2 Each of the N intermediate inputs x i with price p i is produced by a monopoly using only labour input. 3 Households consume the single final good with price P and supply labour n with wage rate W. The aggregate firm s problem is max P x 1,...,x N ( N i=1 x ρ i ) 1/ρ N p i x i. i=1 Phillips 7.14

16 Final- Production The FOCs with respect to the intermediate inputs are ( ) y 1/φ P p i = 0, i = 1,..., N, x i where φ = 1/(1 ρ) is the elasticity of substitution. The demand function for input i is therefore ( ) P φ x i = y. (9.15) Assuming a competitive market, the zero profit condition is N N ( ) P φ Py = p i x i = p i y, p i which gives i=1 P = ( N i=1 p i i=1 p 1 φ i ) 1/(1 φ). (3) Phillips 7.15

17 Intermediate- Production Each intermediate input is produced with a CRTS technology x i = A i n i, i = 1,..., N. Each firm maximizes profit Π i = p i x i Wn i ( ) y 1/φ = P x i Wn i (by (9.15)) x i = Py 1/φ x 1 (1/φ) i Wn i = Py 1/φ (A i n i ) 1 (1/φ) Wn i. The FOC with respect to n i is ( Π i = 1 1 ) Py 1/φ A 1 (1/φ) n i φ i n 1/φ i W = 0. Phillips 7.16

18 and Output of Intermediate Inputs Solving for n i gives the labour demand function [ ] n i = A φ 1 φ 1 P φ i y. φ W The supply function for intermediate good i is x i = A i n i = A φ i By (9.15) the price of input i is ( y p i = P x i [ φ 1 φ P W ] φ y. ) 1/φ = φ W = W. (4) φ 1 A i ρa i Phillips 7.17

19 Efficiency Loss From (9.15) and the production function for x i we have ( ) P φ A i n i = y. p i Total labour demand is therefore N N ( ) 1 P φ n = n i = y, A i i=1 i=1 which can be written as y = vn where p i [ N ( ) ] 1 1 P φ v =. A i i=1 p i Phillips If v < 1, then there is an efficiency loss in producing the final output. 7.18

20 Is v < 1? Assuming that A i = 1 for i = 1,..., N, then from (4) we have p i = W /ρ. Using (3), P = [ N i=1 ( ) ] 1/(1 φ) W 1 φ ( ) W = N 1/(1 φ). ρ ρ Therefore v becomes ( ) φ N v = N 1/(1 φ) (W /ρ) W /ρ i=1 = N 1/(1 φ) 1 Phillips Depending on ρ, the range of φ is (0, ). Therefore v < 1 for 0 < φ < 1 (in the short run) but v > 1 for φ >

21 Effect on Inflation To see the effect of price changes in the intermediate inputs on the price of the final good, we put back the time subscript t in (3): P t = ( N i=1 p 1 φ i,t ) 1/(1 φ). Suppose that in period t + 1 inflation rates for all the N inputs are the same at π. Then P t+1 = ( N ) 1/(1 φ) [ ] 1 φ (1 + π)pi,t i=1 = (1 + π) ( N i=1 p 1 φ i,t ) 1/(1 φ) = (1 + π)p t. Phillips Therefore P t and p i,t have the same inflation rate. 7.20

22 Why Adjust Slowly We have seen that producers in different sectors adjust their prices at different frequencies. The general price level displays various degree of inertia, depending on the source of change. Various models exist to capture the sources and processes of price changes: 1 The Taylor staggered wage contract model emphasizes the adjustment process in the labour market. 2 The Calvo staggered pricing model assumes that price changes is a random process. 3 The optimal dynamic adjustment model maintains that agents optimize the speed of price adjustment. Phillips 7.21

23 Variables in the Definitions: Price: P t = general price level p t = log P t π t = p t = inflation Wage: W t = average wage rate Wt N = new wage contract made in period t w t = log W t, wt N = log Wt N Mark-up by firms: v t. Marginal product of labour: z t = log(mp L ) in period t. Phillips 7.22

24 Assumptions of the 1 Price is a markup over marginal cost and the markup may be time-varying and affected in the short-run mainly by the wage rate: p t = w t + v t. (9.16) 2 The wage rate at any point in time is an average of wage contracts that were set in the last period but are still in force, and of those set in the current period: w t = 1 2 (w t N + wt 1 N ). (9.17) 3 When they were first set, wage contracts were profit maximizing and reflected the prevailing marginal product of labor and the expected future price level: W = P MP L, or Phillips w N t = 1 2 (p t + E t p t+1 ) + z t. (9.18) 7.23

25 Inflation Dynamics of the Putting (9.16), (9.17), and (9.18) together gives p t = 1 {[ ] (p t + E t p t+1 ) + z t [ ]} (p t 1 + E t 1 p t ) + z t 1 + v t. Solving for p t p t 1 gives the inflation rate in period t as π t = E t π t+1 + 2(z t + z t 1 ) + 4v t + η t, (9.19) where η t = E t 1 p t p t. Assuming expectations are rational, E t 1 η t = 0. Equation (9.19) is a first-order difference equation in π t, with forward-looking solution π t = E t s=0 4(z t+s + v t+s ) + 2z t 1 + η t. (9.20) Phillips 7.24

26 Implications 1 An example Period z t v t η t π t t t t For a permanent increase in z t or v t, equation (9.20) implies that inflation will increase without bound. Therefore the model only works when z t and v t are constrained to zero in the long run. Phillips 7.25

27 An Example of Staggered Contracts Phillips 7.26

28 Assumptions of the 1 There are many identical firms in the market. 2 Each firm has a probability of ρ that it can adjust the price of its product in any period. 3 Consequently in period t + s there is a probability of (1 ρ) s that its price is still p t. 4 When a firm have the opportunity to change its price it chooses a p # t to minimize [β(1 ρ)] s E t [p # t pt+s], s=0 where pt+s is the profit maximizing price in period t + s. Phillips 7.27

29 Optimal Price The FOC of the firm s problem with respect to p # t is ] γ s E t [p # t pt+s = 0, s=0 where γ = β(1 ρ). Once p # t is chosen, p # t 1 γ s=0 γ s E t p t+s = 0, so that p # t = (1 γ) γ s E t pt+s. (9.21) s=0 This can be written recursively as Phillips p # t = (1 γ)p t + γe t p # t

30 General Price Level Using (9.21), the general price level is p t = ρp # t + (1 ρ)p t 1 (9.22) = ρ(1 γ) γ s E t pt+s + (1 ρ)p t 1 s=0 Inflation is given by π t = ρ(1 γ) γ s E t pt+s ρp t 1 s=0 = ρ(1 γ) γ s E t pt+s ρ(1 γ) γ s p t 1 = ρ(1 γ) s=0 s=0 γ s [E t pt+s p t 1 ] (5) s=0 Phillips 7.29

31 General Inflation Equation (5) can be written recursively as (exercise) π t = 1 1 ργ [ρ(1 γ)(p t p t 1 ) + γe t π t+1 ]. (9.23) Therefore the inflation rate is a linear combination of the desirable price change p t p t 1 and the expected inflation for the next period. In the steady state p t = p and π t = 0. Phillips 7.30

32 Modification of the Basic If firms cannot change their prices, they can index their current price change to the past inflation rate. Equation (9.22) becomes Inflation is given by p t = ρp # t + (1 ρ)(π t 1 + p t 1 ). π t = ρ(p # t p t 1 ) + (1 ρ)π t 1. Hence inflation is a weighted mean of the optimal price change and past inflation. Using equation (9.21), it can be shown that (exercise) inflation is backward and forward looking: Phillips π t = ρ(1 γ)(p t p t 1 ) + γe t π t+1 + (1 γ)(1 ρ)π t γ 2ργ 7.31

33 Assumptions 1 It is costly for firms to deviate from the static profit maximizing price p t. 2 Price changes, p t, on the other hand, is also costly. The two costs are captured by the cost function C t = β s E t [α(p t+s p t+s ) 2 + ( p t+s ) 2], α > 0. s=0 The FOC for this cost minimization problem is C [ ] t = 2E t β s { α(pt+s p t+s ) + p t+s } β s+1 p t+s+1 p t+s = 0. Phillips 7.32

34 Inflation Dynamics Setting s = 0 gives p t = α(p t p t ) + βe t p t+1. (9.24) Adding and subtracting p t 1 to p t p t and solving for p t p t 1 gives π t = α 1 + α (p t p t 1 ) + β 1 + α E tπ t+1. (9.25) As in the Calvo model the inflation rate is a linear combination of the desirable price change p t p t 1 and the expected inflation for the next period. In the steady state p t = p and π t = E t π t+1. Phillips 7.33

35 A General Formulation A general formulation that captures the above models is π t = απ t + βe t π t+1, α > 0, 0 < β < 1, (9.26) where π t = p t p t 1. To see that the price is sticky, rewrite (9.26) as or p t = α(p t p t 1 ) + βe t p t+1, (9.27) βe t p t+1 + (1 + β)p t (1 α)p t 1 = αp t. (9.28) Therefore the static profit maximizing price p t not only affect the current price but also past and expected future price. A complete solution can be obtained by solving (9.28) as a second-order difference equation. Phillips 7.34

36 remainder of the period changes in import prices will have had little or no effect on the rate of change of wage rates. The Original Phillips (1958) U 0 cf. 0 0~~~~ cl) E? 2-0? -2. ~~~~~~~~~~* $ Unemptoyment, %. Fig,t A scatter diagram of the rate of change of wage rates and the percentage unemployment for the years is shown in Figure 1. During this time there were 61 fairly regular trade cycles with an average period of about 8 years. Scatter diagrams for the years of each trade cycle are shown in Figures 2 to 8. Each dot in the diagrams represents a year, the average rate of change of money wage rates The Keynesian model of inflation: π t = α βu t. (9.33) Phillips 7.35

37 It Didn t Work Quite Well Later Expectations-augmented Phillips curve: π t = E t π t+1 β(u t u n t ), (9.34) Phillips where ut n is the non-accelerating inflation rate of unemployment (NAIRU). 7.36

38 Expectations-Augmented Phillips In equation (9.34), trade-off between inflation and unemployment only works in the short run, in the long run unemployment will return to u n t, then π t = E t π t+1, any rate of inflation will satisfy the equation. Phillips 7.37

39 Lessons Learned The simple negative relation between inflation and unemployment holds when the government is not aware of it. The Lucas critique kicks in when central banks tried to exploit the relation in the 1970s and 80s. The parameters α and β in equation (9.33) no longer remain constant when households form rational expectations on the central bank s behaviours. Starting in the early 1990s some central banks began to shift their attention to inflation targeting. The great recession revived the traditional Keynesian ideas of liquidity trap and quantitative easing. Read chapter 13 for more discussions on monetary policy. Phillips 7.38

40 NAIRU Even the NAIRU is not constant Phillips Question: Can we link inflation to unemployment anymore? 7.39

41 New Keynesian Inflation Equation Assumptions: optimal pricing, imperfect competition, price stickiness, (recall equation (9.25)) π t = ( 1 β 1 α ) π + α 1 α (p t p t ) + β 1 α E tπ t+1. (9.35) In the long run p t = p t so that π t = π. Using the results in the imperfect competition model, the profit maximizing price p t is a mark-up µ over marginal cost. In log form, p t = µ t + mc t. Since in (9.1) the mark-up is increasing in ɛ D, we can set µ t ɛ D,t. Consider a single input factor labour only, (9.3) implies that (lower case variables mean log transformed.) mc t = ν t + w t mp t, where ν t ɛ i,t is the labour mark-up. Phillips 7.40

42 Optimal Price Assuming a Cobb-Douglas production function (in log form) y t = a t + φn t, 0 < φ < 1, the log of marginal product of labour is mp t = log φ + y t n t = log φ + a t φ 1 φ φ y t. (9.36) Putting everything together the optimal price is p t = log φ + µ t + ν t + w t a t φ + 1 φ φ y t. (9.37) Deviation of the optimal price from the actual price is Phillips p t p t = log φ+µ t +ν t a t φ + 1 φ φ y t +(w t p t ). (9.38) 7.41

43 Inflation Equation Putting p t p t in (9.38) into (9.35) gives π t = [( 1 β 1 α + ) π α ] 1 α log φ + β 1 α E tπ t+1 α(1 φ) (1 α)φ y t + α 1 α (w t p t ) α (1 α)φ a t + α 1 α (µ t + ν t ) (9.39) We also want to express π t in terms of deviation of all the other variables from their steady-state values, when p t = pt. Equation (9.38) implies that 0 = log φ + µ t + ν t a t φ + 1 φ φ y t + (w t p t ). Phillips 7.42

44 Inflation Equation Subtracting the last equation from (9.38) gives p t p t = µ t ν t + ãt φ 1 φ φ ỹt ( w t p t ), (9.40) where µ t = µ t µ t, ν t = ν t ν t, and so on. Putting p t p t in (9.40) into (9.35) gives π t = + α 1 α ( 1 β ) π + β 1 α E tπ t+1 1 α [ µ t ν t + ãt φ 1 φ ] φ ỹt ( w t p t ). (9.41) Phillips 7.43

45 Inflation Increases When... Therefore inflation will increase when µ t > µ t : price mark-up exceeds its equilibrium level, ν t > νt : labour mark-up is above its equilibrium level, a t < at : there is a negative technology shock, y t > yt : output is above its capacity, w t p t > wt pt : real wage rate is above its equilibrium level. Phillips 7.44

46 Inflation Increases When... Therefore inflation will increase when µ t > µ t : price mark-up exceeds its equilibrium level, ν t > νt : labour mark-up is above its equilibrium level, a t < at : there is a negative technology shock, y t > yt : output is above its capacity, w t p t > wt pt : real wage rate is above its equilibrium level. 1 The inflation equations (9.39) and (9.41) are more complicated than the previous Phillips curves. 2 In practice we can extend them to include other input factors likes capital, energy, materials, etc. 3 In the multi-factor case, each single input factor may have a smaller impact on inflation due to substitution. Phillips 7.44

47 The In equation (9.36), we can instead express the marginal product in terms of labour, mp t = log φ + a t (1 φ)n t. The inflation equation (9.41) becomes ( π t = 1 β ) π + β 1 α 1 α E tπ t+1 + α 1 α [ µ t ν t + ã t (1 φ)ñ t ( w t p t )]. (9.42) The term ñ t = n t n t can be interpreted as unemployment in period t. Therefore (9.42) is an extended version of the expectations-augmented Phillips curve. Phillips 7.45