COORDINATION AND EXTERNALITIES 211 1. if ‚ < 1 − Á firms offer an excessive number of vacancies and the equi- librium unemployment rate is below the socially optimal level; 2. if ‚ > 1 − Á wages are excessively high because of the strong bargaining power of workers and this results in an unemployment rate that is above the socially efficient level. Insum,inthemodelofthelabormarketthatwehavedescribedhere we cannot make aprioriconclusions about the efficiency of the equilibrium unemployment rate. Given the complex externalities between the actions of firms and workers, the properties of the matching function and the wage deter- mination mechanism are crucial to determine whether the unemployment rate will be above or below the socially efficient level.  APPENDIX A5: STRATEGIC INTERACTIONS AND MULTIPLIERS This appendix presents a general theoretical structure, based on Cooper and John (1988), which captures the essential elements of the strategic interactions in the models discussed in this chapter. We will discuss the implications of strategic interactions in terms of the multiplicity of equilibria and analyze the welfare properties of these equilibria. Consider a number I of economic agents (i =1, , I ), each of which chooses a value for a variable e i ∈ [0, E ] which represents the agent’s “activity level,” with the objective of maximizing her own payoff Û(e i , e −i , Î i ), where e −i represents (the vector of) activity levels of the other agents and Î i is an exogenous parameter which influences the payoff of agent i.Payoff function Û(·) satisfies the properties Û ii < 0andÛ iÎ > 0. (This last assumption implies that an increase in Î raises the marginal return of activity for the agent.) If all other agents choose a level of activity ¯ e,thepayoff of agent i can be expressed as Û(e i , ¯ e, Î i ) ≡ V(e i , ¯ e). In this case the optimization problem becomes max e i V(e i , ¯ e), (5.A1) from which we derive V 1 (e ∗ i , ¯ e)=0, (5.A2) where V 1 denotes the derivative of V with respect to its first argument, e i . First-order condition (5.A2) defines the optimal response of agent i to the activity level of all other agents: e ∗ i = e ∗ i ( ¯ e). Moreover, using (5.A1), we can also calculate the slope of the reaction curve of agent i: de ∗ i d ¯ e = − V 12 V 11 ≶ 0, if V 12 ≶ 0. (5.A3) By the second-order condition for maximization, we know that V 11 < 0; the sign of the slope is thus determined by the sign of V 12 (e i , ¯ e). In case V 12 > 0, we can 212 COORDINATION AND EXTERNALITIES make a graphical representation of the marginal payoff function V 1 (e i , ¯ e) and of the resulting reaction function e ∗ i ( ¯ e). The left-hand graph in Figure 5.12 illustrates various functions V 1 , corresponding to three different activity levels for the other agents: ¯ e =0, ¯ e = e,and ¯ e = E . Assuming V 1 (0, 0) > 0andV 1 (E , E ) < 0(pointsA and B) guarantees the exis- tence of at least one symmetric decentralized equilibrium in which e = e ∗ i (e), and agent i chooses exactly the same level of activity as all other agents (in this case V 1 (e, e)=0 and V 11 (e, e) < 0). In Figure 5.12 we illustrate the case in which the reaction has a positive slope, and hence V 12 > 0, and in which there is a unique symmetric equilib- rium. In general, if V 12 (e i , ¯ e) > 0 there exists a strategic complementarity between agents: an increase in the activity level of the others increases the marginal return of activity for agent i, who will respond to this by raising her activity level. If, on the other hand, V 12 (e i , ¯ e) < 0, then agents’ actions are strategic substitutes. In this case agent i chooses a lower activity in response to an increase in the activity level of others (as in the case of a Cournot duopoly situation in which producers choose output levels). In the latter case there exists a unique equilibrium, while in the case of strategic complementarity there may be multiple equilibria. Before analyzing the conditions under which this may occur, and before discussing the role of strategic complementarity or substitutability in determining the character- istics of the equilibrium, we must evaluate the problem from the viewpoint of a social planner who implements a Pareto-efficient equilibrium. Figure 5.12. Strategic interactions COORDINATION AND EXTERNALITIES 213 The planner’s problem may be expressed as the maximization of a representative agent’s welfare with respect to the common strategy(activitylevel)ofallagents:the optimum that we are looking for is therefore the symmetric outcome corresponding to a hypothetical cooperative equilibrium.Formally, max e V(e, e), (5.A4) from which we obtain V 1 (e ∗ , e ∗ )+V 2 (e ∗ , e ∗ )=0. (5.A5) Comparing this first-order condition 49 with the condition that is valid in a symmetric decentralized equilibrium (5.A2), we see that the solutions for e ∗ are different if V 2 (e ∗ , e ∗ ) = 0. In general, if V 2 (e i , ¯ e) > (<)0, there are positive (negative) spillovers. The externalities are therefore defined as the impact of a third agent’s activity level on the payoff of an individual. A number of important implications for different features of the possible equilibria follow from this general formulation. 1. Efficiency Whenever there are externalities that affect the symmetric decen- tralized equilibrium, that is when V 2 (e, e) = 0, the decentralized equilibrium is inefficient. In particular, with a positive externality (V 2 (e, e) > 0), there exists a symmetric cooperative equilibrium characterized by a common activity level e  > e. 2. Multiplicity of equilibria As already mentioned, in the case of strategic comple- mentarity (V 12 > 0), an increase in the activity level of the other agents increases the marginal return of activity for agent i ,whichinducesagenti to raise her own activity level. As a result, the reaction function of agents has a positive slope (as in Figure 5.12). Strategic complementarity is a necessary but not a sufficient condition for the existence of multiple (non-cooperative) equilibria. The suf- ficient condition is that de ∗ i /d ¯ e > 1 in a symmetric decentralized equilibrium. If this condition is satisfied, we may have the situation depicted in Figure 5.13, in which there exist three symmetric equilibria. Two of these equilibria (with activity levels e 1 and e 3 ) are stable, since the slope of the reaction curves is less than one at the equilibrium activity levels, while at e 2 the slope of the reaction curve is greater than one. This equilibrium is therefore unstable. 3. Welfare If there exist multiple equilibria, and if at each activity level there are positive externalities (V 2 (e i , ¯ e) > 0 ∀ ¯ e), then the equilibria can be ranked. Those with a higher activity level are associated with a higher level of welfare. Hence, agents may be in an equilibrium in which their welfare is below the level that may be obtained in other equilibria. However, since agents choose the optimal strategy in each of the equilibria, there is no incentive for agents to change ⁴⁹ The second-order condition that we assume to be satisfied is given by V 11 (e ∗ , e ∗ )+2V 12 (e ∗ , e ∗ )+ V 22 (e ∗ , e ∗ ) < 0. Furthermore, in order to ensure the existence of a cooperative equilibrium, we assume that V 1 (0, 0) + V 2 (0, 0) > 0, V 1 (E , E )+V 2 (E , E ) < 0, which is analogous to the restrictions imposed in the decentralized optimization above. 214 COORDINATION AND EXTERNALITIES Figure 5.13. Multiplicity of equilibria their level of activity. The absence of a mechanism to coordinate the actions of individual agents may thus give rise to a “coordination failure,” in which potential welfare gains are not realized because of a lack of private incentives to raise the activity levels. Exercise 52 Show formally that equilibria with a higher ¯ e are associated with a higher level of welfare if V 2 (e i , ¯ e) > 0. (Use the total derivative of function V(·) to derive this result.) 4. Multipliers Strategic complementarity is necessary and sufficient to guarantee that the aggregate response to an exogenous shock exceeds the response at the individual level; in this case the economy exhibits “multiplier” effects. To clarify this last point, which is of particular relevance for Keynesian models, we will consider the simplified case of two agents with payoff functions defined as V 1 ≡ Û 1 (e 1 , e 2 , Î 1 )andV 2 ≡ Û 2 (e 1 , e 2 , Î 2 ), respectively. All the assumptions about these payoff functions remain valid (in particular, V 1 13 ≡ Û 1 13 > 0). The reaction curves of the two agents are derived from the following first-order conditions: V 1 1 (e ∗ 1 , e ∗ 2 , Î 1 )=0, (5.A6) V 2 2 (e ∗ 1 , e ∗ 2 , Î 2 )=0. (5.A7) We now consider a “shock” to the payoff function of agent 1, namely dÎ 1 > 0, and we derive the effect of this shock on the equilibrium activity levels of the two agents, e ∗ 1 and e ∗ 2 , and on the aggregate level of activity, e ∗ 1 + e ∗ 2 . Taking the total derivative of the above system of first-order conditions (5.A6) and (5.A7), with dÎ 2 = 0, and dividing COORDINATION AND EXTERNALITIES 215 the first equation by V 1 11 and the second by V 2 22 ,wehave: de ∗ 1 +  V 1 12 V 1 11  de ∗ 2 +  V 1 13 V 1 11  dÎ 1 =0,  V 2 21 V 2 22  de ∗ 1 + de ∗ 2 =0. The terms V 1 12 /V 1 11 and V 2 21 /V 2 22 represent the slopes, with opposing signs, of the reaction curves of the agents which we denote by Ò (given that the payoff functions are assumed to be identical, the slope of the reaction curves is also the same). The term V 1 13 /V 1 11 represents the response (again with oppositing signs) of the optimal equilibrium level of agent 1 to a shock Î 1 . In particular, keeping e ∗ 2 constant, we have V 1 1 (e ∗ 1 , e ∗ 2 , Î 1 )=0 ⇒ ∂e ∗ 1 ∂Î 1 = − V 1 13 V 1 11 > 0. We can thus rewrite the system as follows:  1 −Ò −Ò 1  de ∗ 1 de ∗ 2  =  ∂e ∗ 1 ∂Î 1 0  dÎ 1 , which yields the following solution: de ∗ 1 dÎ 1 = 1 1 − Ò 2 ∂e ∗ 1 ∂Î 1 (5.A8) de ∗ 2 dÎ 1 = Ò 1 − Ò 2 ∂e ∗ 1 ∂Î 1 = Ò de ∗ 1 dÎ 1 . (5.A9) Equation (5.A8) gives the total response of agent 1 to a shock Î 1 . This response can also be expressed as de ∗ 1 dÎ 1 = ∂e ∗ 1 ∂Î 1 + Ò de ∗ 2 dÎ 1 . (5.A10) The first term is the “impact” (and thus only partial) response of agent 1 to a shock affecting her payoff function; the second term gives the response of agent 1 that is “induced” by the reaction of the other agent. The condition for the additional induced effect is simply Ò = 0. Moreover, the actual induced effect depends on Ò and de ∗ 2 /dÎ 1 , as in (5.A9), where de ∗ 2 /dÎ 1 has the same sign Ò: positive in case of strategic comple- mentarity and negative in case of substitutability. The induced response of agent 1 is therefore always positive. This leads to a first important conclusion: the interactions between the agents always induce a total (or equilibrium) response that is larger than the impact response. In 216 COORDINATION AND EXTERNALITIES particular, for each Ò =0,wehave de ∗ 1 dÎ 1 > ∂e ∗ 1 ∂Î 1 . For the economy as a whole, the effect of the disturbance is given by d(e ∗ 1 + e ∗ 2 ) dÎ 1 =  1 1 − Ò 2 + Ò 1 − Ò 2  ∂e ∗ 1 ∂Î 1 = 1 1 − Ò ∂e ∗ 1 ∂Î 1 =(1+Ò) de ∗ 1 dÎ 1 . (5.A11) The relative size of the aggregate response compared with the size of the individual response depends on the sign of Ò:ifÒ > 0 (and limiting attention to stable equilibria for which Ò < 1), then aggregate response is bigger than individual response. Strategic complementarity is thus a necessary and sufficient condition for Keynesian multiplier effects. Exercise 53 Determine the type of externality and the nature of the strategic interactions for the simplified case of two agents with payoff function (here expressed for agent 1) V 1 (e 1 , e 2 )=e · 1 e · 2 − e 1 (with 0 < 2· < 1). Furthermore, derive the (sy mmetric) decen- tralized equilibria and compare these with the cooperative (symmetric) equilibrium. REVIEW EXERCISES Exercise 54 Introduce the following assumptions into the model analyzed in Section 5.1: (i) The (stochastic) cost of production c has a uniform distribution defined on [0, 1], so that G(c)=cfor0 ≤ c ≤ 1. (ii) The matching probability is equal to b(e)=b ·e, with parameter b > 0. (a) Determine the dynamic expressions for e and c ∗ (repeating the derivation in the main text) under the assumption that y < 1. (b) Find the equilibria for this economy and derive the stability properties of all equilibria with a positive activity level. Exercise 55 Starting from the search model of money analyzed in Section 5.2, suppose that carrying over money from one period to the next now entails a storage cost, c > 0. Under this new assumption, (a) Derive the expected utility for an agent holding a commodity (V C ) and for an agent holding money (V M ), and find the equilibria of the economy. (b) Which of the three equilibria described in the model of Section 5.2 (with c =0) always exists even with c > 0? Under what condition does a pure monetary equilibrium exist? Exercise 56 Assume that the flow cost of a vacancy c and the imputed value of free time z in the model of Section 5.3 are now functions of the wage w (instead of be ing exogenous). COORDINATION AND EXTERNALITIES 217 In particular, assume that the following linear relations hold: c = c 0 w, z = z 0 w. Determine the effectofanincreaseinproductivity(y > 0) on the steady-state equilib- rium. Exercise 57 Consider a permanent negative productivity shock (y < 0) in the match- ing model of Sections 5.3 and 5.4. The shock is realized at date t 1 , but is anticipated by the agents from date t 0 < t 1 onwards. Derive the effect of this shock on the steady-state equilibrium and describe the transitional dynamics of u, v,andË. Exercise 58 Consider the effect of an aggregate shock in the model of strategic interactions for two agents introduced in Appendix A5. That is, consider a variation in the exogenous terms of the payoff functions, so that dÎ 1 = dÎ 2 = dÎ > 0,andderivetheeffect of this shock on the individual and aggregate activit y level.  FURTHER READING The role of externalities between agents that operate in the same market as a source of multiplicity of equilibria is the principal theme in Diamond (1982a). This arti- cle develops the economic implications of the multiplicity of equilibria that have a Keynesian spirit. The monograph by Diamond (1984) analyzes this theme in greater depth, while Diamond and Fudenberg (1989) concentrate on the dynamic aspects of the model. Blanchard and Fischer (1989, chapter 9) offer a compact version of the model that we studied in the first section of this chapter. Moreover, after elaborating on the general theoretical structure to analyze the links between strate- gic interactions, externalities, and multiplicity of equilibria, which we discussed in Appendix A5, Cooper and John (1988) offer an application of Diamond’s model. Rupert et al. (2000) survey the literature on search models of money as a medium of exchange and present extensions of the basic Kiyotaki–Wright framework discussed in Section 5.2. The theory of the decentralized functioning of labor markets, which is based on search externalities and on the process of stochastic matching of workers and firms, reinvestigates a theme that was first developed in the contributions collected in Phelps (1970), namely the process of search and information gathering by workers and its effects on wages. Mortensen (1986) offers an exhaustive review of the contributions in this early strand of literature. Compared with these early contributions, the theory developed in Section 5.3 and onwards concentrates more on the frictions in the matching process. Pissarides (2000) offers a thorough analysis of this strand of the literature. In this literature the base model is extended to include a specification of aggregate demand, which makes the interest rate endogenous, and allows for growth of the labor force, two elements that are not considered in this chapter. Mortensen and Pissarides (1999a, 1999b)provide an up-to-date review of the theoretical contributions and of the relevant empirical evidence. 218 COORDINATION AND EXTERNALITIES In addition to the assumption of bilateral bargaining, which we adopted in Section 5.3, Mortensen and Pissarides (1998a) consider a number of alternative assumptions about wage determination. Moreover, Pissarides (1994) explicitly considers the case of on-the-job search which we excluded from our analysis. Pissarides (1987) develops the dynamics of the search model, studying the path of unemployment and vacancies in the different stages of the business cycle. The paper devotes particular attention to the cyclical variations of u and v around their long-run relationship, illustrated here by the dynamics displayed in Figure 5.11. Bertola and Caballero (1994) and Mortensen and Pissarides (1994) extend the structure of the base model to account for an endogenous job separation rate s . In these contributions job destruction is a conscious decision of employers, and it occurs only if a shock reduces the productivity of a match below some endogenously determined level. This induces an increase in the job destruction rate in cyclical downturns, which is coherent with empirical evidence. The simple Cobb–Douglas formulation for the aggregate matching function with constant returns to scale introduced in Section 5.3 has proved quite useful in interpret- ing the evidence on unemployment and vacancies. Careful empirical analyses of flows in the (American) labor market can be found in Blanchard and Diamond (1989, 1990), Davis and Haltiwanger (1991, 1992) and Davis, Haltiwanger, and Schuh (1996), while Contini et al. (1995) offer a comparative analysis for the European countries. Cross- country empirical estimates of the Beveridge curve have been used by Nickell et al. (2002) to provide a description of the developments of the matching process over the 1960–99 period in the main OECD economies. They find that the Beveridge curve gradually drifted rightwards in all countries from the 1960s to the mid-1980s. In some countries, such as France and Germany, the shift continued in the same direction in the 1990s, whereas in the UK and the USA the curve shifted back towards its original position. Institutional factors affecting search and matching efficiency are responsible for a relevant part of the Beveridge curve shifts. The Beveridge curve for the Euro area in the 1980s and 1990s is analysed in European Central Bank (2002). Both counter- clockwise cyclical swings around the curve of the type discussed in Section 5.4 and shifts of the unemployment–vacancies relation occurred in this period. For example, over 1990–3 unemployment rose and the vacancy rate declined, reflecting the influ- ence of cyclical factors; from 1994 to 1997 the unemployment rate was quite stable in the face of a rising vacancy rate, a shift of the Euro area Beveridge curve that is attributable to structural factors. Not only empirically, but also theoretically, the structure of the labor force, the geographical dispersion of unemployed workers and vacant jobs, and the relevance of long-term unemployment determine the efficiency of a labor market’s matching process. Petrongolo and Pissarides (2001) discuss the theoretical foundations of the matching function and provide an up-to-date survey of the empirical estimates for several countries, and of recent contributions focused on various factors influencing the matching rate. The analysis of the efficiency of decentralized equilibrium in search models is first developed in Diamond (1982b) and Hosios (1990), who derive the efficiency condi- tions obtained in Section 5.5; it is also discussed in Pissarides (2000). In contrast, in a classic paper Lucas and Prescott (1974) develop a competitive search model where the decentralized equilibrium is efficient. COORDINATION AND EXTERNALITIES 219  REFERENCES Bertola, G., and R. J. Caballero (1994) “Cross-Sectional Efficiency and Labour Hoarding in a Matching Model of Unemployment,” Review of Economic Studies, 61, 435–456. Blanchard, O. J., and P. Diamond (1989) “The Beveridge Curve,” Brookings Papers on Economic Activity, no. 1, 1–60. (1990) “The Aggregate Matching Function,” in P. Diamond (ed.), Growth, Productiv- ity, Unemployment, Cambridge, Mass.: MIT Press, 159–201. and S. Fischer (1989) Lectures on Macroeconomics, Cambridge, Mass.: MIT Press. Contini, B., L. Pacelli, M. Filippi, G. Lioni, and R. Revelli (1995) A Study of Job Creation and Job Destruction in Europe, Brussels: Commission of the European Communities. Cooper, R., and A. John (1988) “Coordinating Coordination Failures in Keynesian Models,” Quarterly Journal of Economics, 103, 441–463. Davis, S., and J. Haltiwanger (1991) “Wage Dispersion between and within US Manufacturing Plants, 1963–86,” Brookings Papers on Economic Activity, no. 1, 115–200. (1992) “Gross Job Creation, Gross Job Destruction and Employment Reallocation,” Quarterly Journal of Economics, 107, 819–864. and S. Schuh (1996) Job Creation and Destruction, Cambridge, Mass.: MIT Press. Diamond, P. (1982a) “Aggregate Demand Management in Search Equilibrium,” Journal of Polit- ical Economy, 90, 881–894. (1982b) “Wage Determination and Efficiency in Search Equilibrium,” Review of Economic Studies, 49, 227–247. (1984) A Search-Equilibrium Approach to the Micro Foundations of Macroeconomics, Cambridge, Mass.: MIT Press. and D. Fudenberg (1989) “Rational Expectations Business Cycles in Search Equilibrium,” Journal of Political Economy, 97, 606–619. European Central Bank (2002) “Labour Market Mismatches in Euro Area Countries,” Frankfurt: European Central Bank. Hosios, A. J. (1990) “On the Efficiency of Matching and Related Models of Search and Unem- ployment,” Review of Economic Studies, 57, 279–298. Kiyotaki, N., and R. Wright (1993) “A Search-Theoretic Approach to Monetary Economics,” American Economic Review, 83, 63–77. Lucas, R. E., and E. C. Prescott (1974) “Equilibrium Search and Unemployment,” Journal of Economic Theory, 7, 188–209. Mortensen, D. T. (1986) “Job Search and Labor Market Analysis,” in O. Ashenfelter and R. Layard (eds.), Handbook of Labor Economics, Amsterdam: North-Holland. and C. A. Pissarides (1994) “Job Creation and Job Destruction in the Theory of Unemploy- ment,” Review of Economic Studies, 61, 397–415. (1999a) “New Developments in Models of Search in the Labor Market,” in O. Ashen- felterandD.Card(eds.),Handbook of Labor Economics , vol. 3, Amsterdam: North-Holland. (1999b) “Job Reallocation, Employment Fluctuations and Unemployment,” in J. B. Taylor and M. Woodford (eds.), Handbook of Macroeconomics, Amsterdam: North-Holland. 220 COORDINATION AND EXTERNALITIES Nickell S., L. Nunziata, W. Ochel, and G. Quintini (2002) “The Beveridge Curve, Unemployment and Wages in the OECD from the 1960s to the 1990s,” Centre for Economic Performance Dis- cussion Paper 502; forthcoming in P. Aghion, R. Frydman, J. Stiglitz, and M. Woodford (eds.), Knowledge, Information and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps, Princeton: Princeton University Press. Petrongolo B., and C. A. Pissarides (2001) “Looking into the Black Box: A Survey of the Matching Function,” Journal of Economic Literature, 39, 390–431. Phelps, E. S. (ed.) (1970) Macroeconomic Foundations of Employment and Inflation Theory,New York:W.W.Norton. Pissarides, C. A. (1987) “Search, Wage Bargains and Cycles,” Review of Economic Studies, 54, 473–483. (1994) “Search Unemployment and On-the-Job Search,” Rev iew of Economic Studies, 61, 457–475. (2000) Equilibrium Unemployment Theory, 2nd edn. Cambridge, Mass.: MIT Press. Rupert P., M. Schindler, A. Shevchenko, and R. Wright (2000) “The Search-Theoretic Approach to Monetary Economics: A Primer,” Federal Reserve Bank of Cleveland Economic Review, 36(4), 10–28. [...]... variability perceived by agents Overestimating the unforeseen changes in income may lead to the conclusion that consumption is excessively smooth, even though agents behave as predicted by the rational expectations–permanent income theory Solution to exercise 9 (a) For the assumed utility function, marginal utility is u (c ) = a − bc 0 for c < a/b; for c ≥ a/b, 230 ANSWERS TO EXERCISES As shown in the... > 0, but also when I < 0 As long as installed capital has a positive value, it cannot be optimal for the firm to pay costs in order to scrap it, and the optimal investment flow is never negative The slope at the origin of functions in the form I ‚ is zero for all ‚ > 0, and such functions are well defined for I < 0 only when ‚ is an integer If ‚ is an even number, then the sign of ∂G (K , I )/∂ I = x... a negative “interest rate ˙ effect.” If the former dominates, the q = 0 schedule slopes upwards in the (q , y) phase diagram, as in the figure Formally, from (2.35) we get dq dy = ˙ q =0 a1r − (a0 + a1 y)h 1 / h 2 >0 r2 ⇔ a1 > q h1 , h2 ˙ where we used the expression for q = /r which applies along the q = 0 locus This schedule crosses the stationary locus for y from above, since lim y→∞ dq dy =0 ˙ q... steady state (c) As in exercise 10, a quadratic form for G (·) implies that investment is almost costless when it is very small This is not realistic, and Pk represents the market price of capital net of adjustment costs only if ˙ the derivative of adjustment costs is unity at K = 0 Cubic functional ˙ < 0, implying that first-order conditions forms are not convex for K do not identify an optimum ˙ (d) As... firm’s revenues: in the new steady state, the latter is larger but it is more heavily discounted at rate (r + ‰) (c) For the functional form proposed, capital’s marginal productivity is independent of L : ∂ −· ∂Y = √ = 0, ∂K∂L ∂L 2 K and therefore the cost w of factor L has no implications for the firm’s investment policy If instead the mixed second derivative is not zero then, as in Figure 2.8, capital’s... (both discounted at rate Ò) The first is the marginal utility of the stock of durables at the beginning of period t + 1 The second accounts for the additional resources that an increase in the stock of durables makes available for consumption in t + 1 by reducing the need for further purchases, dt+1 These additional resources are measured by (1 − ‰) pt+1 , yielding utility (1 − ‰) pt+1 uc (c t+1 , St+1... consumption in the current period and negatively by consumption in the previous period ANSWERS TO EXERCISES 227 This formulation of utility may capture habit formation behavior: a high level of consumption in period t decreases utility in period t + 1 (but increases period t + 1 marginal utility) Therefore, the agent is induced to increase consumption in period t + 1 This effect is due to a consumption “habit”... subsequent periods In this case there is a dynamic relation between c t+1 and c t , c t−1 and c t−2 , which makes the consumption change c t+1 dependent on lagged values c t and c t−1 Therefore, the orthogonality conditions that hold with separable utility are not valid here 228 ANSWERS TO EXERCISES Solution to exercise 8 (a) The change in permanent income for agents, following version of equation... is no need to save in order to keep the higher level of consumption in the future Solution to exercise 2 We look for a consumption function of the general form c t = r (At + Ht ) = r At + r 1+r ∞ i =0 1 1+r i E t yt+i , as in (1.12) in the main text Given the assumed stochastic process for income, we can compute expectations of future incomes and then the value of human wealth Ht We have ¯ E t yt+1... the quadratic form proposed in the exercise, then the marginal investment cost ∂G (K , I )/∂ I = x · 2I has the same sign as the investment flow I Since the optimal investment flow I ∗ must satisfy the condition x · 2I ∗ = Î, where Î is the marginal value of capital, Î > 0 implies I ∗ > 0 Intuitively, this functional form (whose slope at the origin is zero, rather than unity) implies costs for the firm . Blanchard and Diamond ( 198 9, 199 0), Davis and Haltiwanger ( 199 1, 199 2) and Davis, Haltiwanger, and Schuh ( 199 6), while Contini et al. ( 199 5) offer a comparative analysis for the European countries the interest rate endogenous, and allows for growth of the labor force, two elements that are not considered in this chapter. Mortensen and Pissarides ( 199 9a, 199 9b)provide an up-to-date review of. Studies, 61, 397 –415. ( 199 9a) “New Developments in Models of Search in the Labor Market,” in O. Ashen- felterandD.Card(eds.),Handbook of Labor Economics , vol. 3, Amsterdam: North-Holland. ( 199 9b) “Job