Adv. Macro 2 Exercises Week 8

Transcription

1 Faculty of Social Sciences Week 8 9th of November 2011 (week 8) Slide 1/29

2 Outline th of November 2011 (week 8) Slide 2/29

3 Outline th of November 2011 (week 8) Slide 3/29

4 We have three exercises today: 3.4: On the housing market. 3.5: On the wage rigidity and menu costs. 3.6: On nominal price rigidity and menu costs. No replay. 9th of November 2011 (week 8) Slide 4/29

5 Outline th of November 2011 (week 8) Slide 5/29

6 I We consider the housing sector in a small open economy and use the following definitions: H t the aggregate stock of houses R t R(H t ) the (real) price of housing services I H t housing investment p t the (real) price of houses ṗ e t the expected change in the (real) price of houses r the risk-free real interest rate given by the global capital market δ the rate of depreciation τ R tax on (imputed) rental income after allowance for depreciation σ tax on property τ r tax on interest rate income 9th of November 2011 (week 8) Slide 6/29

7 II We assume a representative household so that the individual and the aggregate no arbitrage condition for the housing market can be written as R(H t )S t δp t H t τ R (R(H t )S t δp t H t ) σp t H t + ṗ e t H t p t H t = (1 τ r )r where R < 0 and we have that S t = αh t (with α > 0) is the the aggregate housing services at time t. Assuming perfect foresight, ṗ e t = ṗ t this covers the demand side. The supply side can be analyzed with Tobins Q, where q t = p t H t 1 H t = p t because p t H t is the market value of the housing stock and 1 H t is the book or replacement cost value. We get the gross investment relationship as I H t = m (p t )H t m(1) = 0, m > 0 9th of November 2011 (week 8) Slide 7/29

8 a No-arbitrage: R(H t )S t δp t H t τ R (R(H t )S t δp t H t ) σp t H t +ṗt e H t p t H t Investment: It H = m (p t )H t m(1) = 0, m > 0. = (1 τ r )r. a) Interpret the two equations including the parameters and the sign of the derivatives R and m. No-arbitrage: R(H t )S t δp t H t : Imputed net rental income. τ R (R(H t )S t δp t H t ): Taxes paid on imputed net rental income. σp t H t : Property taxes paid. ṗ e t H t : Expected price gain/loss. LHS-nominator: Return of asset. LHS-denominator: Price of asset. RHS: Tax-adjusted risk-less rate of interest. R < 0: Rental price is falling in supply. Investment: m > 0 rising in the price. 9th of November 2011 (week 8) Slide 8/29

9 b No-arbitrage: R(H t )S t δp t H t τ R (R(H t )S t δp t H t ) σp t H t +ṗt e H t p t H t Investment: It H = m (p t )H t m(1) = 0, m > 0. b) Draw the phase diagram in (H,p) space. Answer: = (1 τ r )r. Locus 1: (1 τ R)(αR(H t ) δp t ) σp t +ṗ t p t = (1 τ r )r (using perfect foresight) We get ṗ t = ((1 τ r )r + (1 τ R )δ + σ)p t (1 τ R )αr(h t ) = ap t cr(h t ) From ṗ t = 0 we get 0 = a dp c R dh dp dh = c ṗ t =0 a R < 0, so it is downward sloping. Locus 2: Ḣ t = (m (p t ) δ)h t From ṗ t = 0 we get m (p ) = δ Arrows and full diagram explained on white board. 9th of November 2011 (week 8) Slide 9/29

10 c Figure 1 shows the year-on-year percentage change in p t and I H t /H t in Denmark since c) Does our theory fit with figure 1? Answer: It does not falsify it. 9th of November 2011 (week 8) Slide 10/29

11 d+e I In Denmark, as in a lot of other countries, the housing market has changed somewhat in the last years. In Denmark new loan types in terms of both adjustable-rate loans (from first quarter 2000) and deferred amortization loans (from last quarter 2003) was introduced for the first time and from 2002 on the so-called tax freeze implied that the property tax was frozen measured in nominal kroner (so it was falling in real terms). In an empirical study by the Danish central bank (Nationalbanken) the following figure show the estimated effects. 9th of November 2011 (week 8) Slide 11/29

12 d+e II d) How can we interpret the effects of the new loan types in our model? Does it fit qualitatively with the empirical study? e) How can we interpret the effects of the tax freeze in our model? Does it fit qualitatively with the empirical study? Note: They should both raise the price. 9th of November 2011 (week 8) Slide 12/29

13 d+e III Locus 1: 0 = ((1 τ r )r + (1 τ R )δ + σ)p t (1 τ R )αr(h t ). Locus 2: m (p ) = δ. d) How can we interpret the effects of the new loan types in our model? Does it fit qualitatively with the empirical study? Lower r. For given H we now have a higher p. Locus to the right. e) How can we interpret the effects of the tax freeze in our model? Does it fit qualitatively with the empirical study? Lower σ. For given H we now have a higher p. Locus to the right. Effects drawn on white-board - they fit the empirical study. Note: In the long run p is cost-determined. The hard thing is to find the equilibrium. 9th of November 2011 (week 8) Slide 13/29

14 d+e IV 9th of November 2011 (week 8) Slide 14/29

15 Outline th of November 2011 (week 8) Slide 15/29

16 I We consider an economy with J differentiated types of labor. All workers can perform exactly one type of labor and for all types of labor there is preciselyi workers who can perform only that kind of labor. The workers can therefore be indexed by (i,j) where i = 1,2,3,...,I and j = 1,2,3,...,J. All workers share the same utility function given by U i,j = lnc i,j L1 φ i,j 1 φ φ > 0 where C is consumption and L is hours of labor.the budget constraint is given by W j L PC = 0 where W j is the nominal wage for labor of type j and P is the general price level. 9th of November 2011 (week 8) Slide 16/29

17 II The J different types of workers all have their own union which is a monopolist and takes the marginal rate of substitution of its representative member as the cost of the labour services when they negotiate wage contracts. The unions perceive that labor demand is given by L d j = ( Wj W ) ε L e J where L e is the aggregate demand for labor. Assume that the unions are subject to some form of menu costs in their wage setting. 9th of November 2011 (week 8) Slide 17/29

18 a Denote the optimum wage as W j so that U U ( W j,w,mrs L/C,j,P,L e). a) Explain intuitively why the the maximization problems of the unions can be written as maxu j W j ( Wj P MRS L/C,j ) L d j I U ( W j,p,mrs L/C,j,W,L e) Answer: The parenthesis is MB MC measured in real money terms and then we multiply this with employment for a worker of type J. 9th of November 2011 (week 8) Slide 18/29

19 b Denote the optimum wage as W j so that U U ( W j,w,mrs L/C,j,P,L e). b) Draw the graph of U ( W i,p,mrs L/C,j,W,L e) for given P, MRS L/C,j, W and L e in the neighborhood of U. Answer: Maximum in U. Higher W j lowers employment too much compared with the gain from a higher wage. Lower W j does not increase employment enough to compensate for the lower wage. 9th of November 2011 (week 8) Slide 19/29

20 c Assume that a negative demand shock hits the economy so that L e falls. c) Reduce the expression du(w j,w,mrs L/C,j,P,L e ) U(W j,w,mrs L/C,j,P,L e ) W j W j L e = dl e + U(W j,w,mrs L/C,j,P,L e ) L e further. Comment. Answer: W j = L e 0 due to the envelope theorem (in an interior optimum, the total derivative of a maximized function wrt. a parameter is equal to the partial derivative wrt. that parameter). 9th of November 2011 (week 8) Slide 20/29

21 d Assume that a negative demand shock hits the economy so that L e falls. d) Illustrate graphically why the nominal wage may remain constant if the change in L e is small enough. Discuss. Answer: Shown on the white board - same idea has in figure 19.2, p Further effect: The above assumes that the firms do not immediately lower their prices in response to the negative demand shock. Lower prices means a higher real wage, which should make it even more profitable for the unions to lower their wages. But if prices are sticky then this effect disappears. If both prices and wages are very sticky, we should expect to see large changes in GDP and employment without changes in the real wage. 9th of November 2011 (week 8) Slide 21/29

22 e d) Discuss the evidence of nominal and/or real wage rigidity. Answer: Nominal wage rigidity: Change in quantities rather than prices/wages. Real wage rigidity: Change in quantities rather than relative prices. Conclusion: Strong evidence of real wage rigidity. 9th of November 2011 (week 8) Slide 22/29

23 Outline th of November 2011 (week 8) Slide 23/29

24 I Nokumura and Steinsson conclude the followining in their widely-cited 2008-paper Five Facts about prices: A reevaluation of menu cost models : 1 For consumer prices, the median frequency of nonsale price change is roughly half of what it is including sales (9 12% per month versus 19 20% per month for identical items; 11 13% per month versus 21 22% per month including product substitutions). The median frequency of price change for finished-goods producer prices is comparable to that of consumer prices excluding sales. 2 One-third of nonsale price changes are price decreases. 3 The frequency of price increases covaries strongly with inflation, whereas the frequency of price decreases and the size of price increases and price decreases do not. 4 The frequency of price change is highly seasonal: it is highest in the first quarter and then declines. 5 We find no evidence of upwardsloping hazard functions of price changes for individual products. Note on fact 5: If prices become more likely to change the longer they have remained unchanged, the hazard function of price changes is upward sloping. d) Discuss these empirical facts in relation to the menu cost theory. 9th of November 2011 (week 8) Slide 24/29

25 II d) Discuss these empirical facts in relation to the menu cost theory. Ad 1) The duration of non-sale prices are roughly 9-10 month. Including sales the price duration is only about 5 months. Normally sale prices changes are temporary. Fits with a menu cost theory. Ad 2+3) Fits rather well with a menu cost theory. Positive inflation-trend should make price increases more likely than decreases. High inflation should imply more price increases. Ad 4) Hard to reconcile with a menu cost theory, because menu costs should not be seasonal - but maybe there could be some psycological factors. Ad 5) Hard to reconcile with a menu cost theory, because we should move longer and longer away from the initial maximum. 9th of November 2011 (week 8) Slide 25/29

26 Outline th of November 2011 (week 8) Slide 26/29

27 Analyzing the housing market using q-theory. Demand determined in the short run. Supply determined in the long run. The effect of new loan types and the tax freeze can be analyzed qualitatively in the model. The theory generally fits the empirical evidence. Wage and price rigidity: Small menu costs can explain nominal wage (and price rigidity). The analysis uses the envelope theorem and is proved graphically. Combined nominal wage and price rigidity results in real wage rigidity. There is strong evidence of real wage rigidity. Prices are rather sticky Non-sale prices have a duration of 9-10 months. The seasonality of price changes is not easily explained by a menu cost theory. The hazard rate of price changes is independent of the time since the last price change. 9th of November 2011 (week 8) Slide 27/29

28 Outline th of November 2011 (week 8) Slide 28/29

29 New Keynesian definition of output gap: Deviation from flexible price/wage equilibrium. Figure from Smets, Wouters and Gali ( unpublished) 9th of November 2011 (week 8) Slide 29/29