Toulouse School of Economics, M2 Macroeconomics 1 Professor Franck Portier. Exam Solution

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1 Toulouse Shool of Eonomis, M2 Maroeonomis 1 Professor Frank Portier Exam Solution This is a 3 hours exam. Class slides and any handwritten material are allowed. You must write legibly. I True, False, Unertain? (1/2 of the points) For eah of those statements, say whether it is true, false, or unertain and explain why. You should target around 1 to 15 lines per question. Copying the slides will not bring any points. You should target around 15-2 lines per question. Copying the slides will not bring any points. 1 Figure 1 below display the responses of some maroeonomi variables to a persistent but not permanent Total Fator Produtivity shok in the simple RBC model of the ourse. Give the eonomi mehanisms behind those responses. You may want to use the equations that define a ompetitive equilibrium in that model. 2 Figure 1: Response to a persistent TFP shok Output Consumption Investment Capital Hours worked Labor produtivity (Wages) (a) There are two main eonomi mehanisms that are driving this dynamis: (i) Consumption smoothing : preferenes are onvex so that agents prefer to smooth onsumption over time. Therefore, an inrease in today supply of good will not be all onsumed today, but partly saved. (ii) Consumption/leisure substitution: onsumption and leisure are two normal goods, whose demand inreases when inome inrease or prie derease. The relative prie of leisure is the real wage. Therefore, hanges in the real wake will affet labor supply through an inome and substitution effet. (b) Given those eonomi motives, there are two soures of dynamis in the model: (i) The dynamis of the shok: TFP is assumed to be persistent. A shok today as long lasting effets, although non permanent. (ii) The law of motion of apital: apital an be arried from one period to another, and 1

2 is the way to transform urrent goods into future goods. This is well seen in the ase of the analyti RBC model of the slides, whose solution is y t+1 = αy t + z t+1 + α log αβ z t+1 = z t + ε t+1 where z is the exogenous TFP. Even if z was iid, there would be some persistene on the shoks. () The story of those IRF is therefore: TFP inreases, but not permanently. This reates a wealth effet as existing apital and labor are more produtive eteris paribus. As onsumption and output are normal goods, we should expet both to inrease. This does not happen to leisure beause at the same time leisure beomes more expensive (beause marginal produtivity of labor inreases). This seond effet dominates and labor goes up. Not all this extra prodution (y is inreasing) is going into onsumption, and part is saved. hene, i inreases and so does k. Then all onverges bak to the steady state. (d) Finally, note how pries an deentralize those alloations: the real wage is going up (with Cobb-Douglas tehnology and perfet ompetition, the real wage moves proportionally to average produtivity of labor). The real interest rate (not shown on this graph) is also going up in the short run, so that agents save part of their extra inome. 2 In the Mortensen-Pissarides model, the stok of vaanies is a state variable. True? False? Unertain? False (a) In the dynamial systems that we have studied, endogenous state variables are predetermined (in disrete time) or annot jump (in ontinuous time). They an only be hanged by ations on the ontrol variables. (b) In Mortensen-Pissarides, the stok of vaanies is not a state variable. Atually, it is a ruial assumption that helps slowing the model. There is a large number of potential entrepreneurs waiting to post vaanies. There is free entry so that if the value of a vaany is positive, there will be a disrete entry of news entrepreneurs (the stok of vaanies will jump) so that the value of a vaany is always zero. The stok of vaany is therefore determined by a forward looking behavior (pay a posting ost to get some future benefits) + free entry. 3 Shoks to the apital stok annot be the main soure of flutuations in a Real Business Cyle model. True? False? Unertain? (a) What is the effet of a negative shok to the apital stok? (i) Negative wealth effet:, h. This negative wealth effet generally dominates the substitution one (leisure is heaper) so that in general hours h inrease. Overall, we have nevertheless in general y. (ii) Consumption smoothing : With less apital, marginal produtivity of apital is hight, so that households want to invest more. A feasible path would be to onsume muh less today, invest a lot more so that apital would rebuilt in one period and onsumption would be bak to normal tomorrow. That is not optimal as households like to sloth onsumption. Therefore, investment will inrease smoothly. (b) In that dynamis, and y on the one side and i and h on the other side are negatively orrelated (for the reasonable alibrations). This is at odd with the pattern of orrelations in the business yle, where all these variables ommode positively. Shoks to the apital stok are therefore unlikely to be a story of the business yle. () It is hard to think of reurrent destrution of the apital stok at business yle frequenies in the real life (natural disasters are not a the BC frequeny, fatory fires are not aggregate shoks) 4 What is the role of the transversality ondition is the Ramsey model? (a) Mathematially, the Ramsey model solution is a pair of differential equations in (k, ). With 2

3 the notations of the slides, k t = Lf( k t L ) (δ + g)k t t ċ t = σ(f ( k t ) δ ρ) g. t L Figure 2: Phase diagram of the Ramsey model d/dt= dk/dt= k Figure 1: The Saddle-Path From initial onditions (k, ), the pair of differential equations uniquely determines one trajetory (that we an see using the phase diagram). But if k is given is not as it an jump at time. The transversality ondition lim t + e ρt µ t K t = with µ t = Ct α 1 is another boundary ondition. The two boundary onditions plus the pair of differential equations uniquely define a trajetory. (b) From an eonomi point of view, the TC is an optimality ondition. It states that the value of the apital stok at plus infinity has to be zero. It is best to understand this as the limit of a model with finite horizon T as T goes to infinity. The TC states that one should not leave apital for the period after the end of the world. Why isn t it optimal? Beause anther feasible path that would inrease onsumption at the last period and leave no apital would bring more utility. () Graphially, the TC is ruling out trajetories with too low that would take the eonomy to a long run path with zero onsumption. Note: The TC lim t + e ρt µ t K t = is an optimality ondition while the No-Ponzi ondition lim t + e ρt µ t K t is a onstraint imposed to the agents, not a optimality ondition. II The Harrod-Domar s model (1/3 of the points) The model of Harrod (1939) and Domar (1949) an be seen as a version of the Solow model but with limited substitution between aptal and labor. It produes a result of instability of balaned growth. This problem solves the Harrod-Domar s model. Time is ontinuous. Aggregate prodution funtion is ( K(t) Y (t) = min v, L(t) α ), ν >, α >. (1) Y (t), K(t) and L(t) are, respetively, aggregate output, the apital stok, and employment. 3

4 The saving rate is exogenous and assumed to be < s < 1; labor fore grows exogenously at rate n L. The other equations of the model are therefore: S(t) = sy (t) (2) I(t) = δk(t) + K(t) (3) I(t) = S(t) (4) L(t) L(t) = n L (5) 1 Draw the isoquants of the prodution funtion. Derive expressions for Y/K, Y/L, and K/L under the assumption that both prodution fators are fully employed. What happens if the atual K/L is less than v/α? What if it is larger than v/α? Figure 3: Isoquant of the prodution funtion in the Harrod-Domar model There is full employment of both fators when Y = 1 and Y = 1, whih gives K = v. In that ase, K v L α L α alloations are exatly at the kink of the isoquants. If K < v, then there is unused labor. L α If K > v, then there is unused apital. L α 2 Show that, in order to maintain full employment of apital in the model, output and investment must grow at the so-alled warranted rate of growth whih is equal to (s δv)/v. There is full employment of apital when Y = K. This implies Ẏ = 1 K. Furthermore, I = sy v v implies I = sẏ. Now use K = I δk, replae K and K using the two previous equations to obtain Ẏ Y = s δv I sv I. Y must grow at rate s δv sv ( warranted rate of growth ) to maintain full employment of apital. 4

5 3 Show that, in order to maintain full employment of labour in the model, output and investment must grow at the so-alled natural rate of growth whih is equal to n L. There s full employment of labor if Y = 1 L. Differening gives α Ẏ Y = L L = n L. Y must grow at rate n L ( natural rate of growth ) to maintain full employment of labor. 4 Derive the ondition under whih the eonomy grows with full employment of both fators of prodution. This is alled the Harrod-Domar s ondition. What happens if (s δv)/v is above or below n L? The ondition for full employment of both fators is n L = s δv. sv This is a ondition of parameters, and no eonomi fores is making sure that it holds. Therefore, balanes growth is a knife-edge ase, meaning that it holds only for that speifi onfiguration of parameters. If n L > (s δv)/v, there is an inreasing unemployment of labor, If n L < (s δv)/v, there is an inreasing unemployment of apital. 5 Show that a Harrod-Domar s like ondition also appears in the Solow s model. Hint: The Solow model is desribed by equations (2) to (5) plus (1) with a general neolassial prodution funtion F (K, L). Write the model is per apita terms and derive a ondition that involves K/L and n L at the steady state. In Solow, we have k = sf(k) (δ + n L )k, where k = K/L and f(k) = y = Y/L. At the steady state, k =, so that sy = (δ + n L )k, or equivalently s y δ = n k L. Unisng the notation y = 1, we k v obtain n L = s δv. sv 6 Explain why the Solow s model does not suffer from the instability (or knife-edge stability) of the Harrod-Domar s model. The differene between the two models is that in Solow, v = k is not a parameter, but an endogenous y variable. It is indeed dereasing returns of f that pins down the steady state level of k that makes sure that the ondition holds. III The Diamond s oonut model (1/3 of the points) Let s onsider the following environment. Time is ontinuous. There is a ontinuum of agents of mass one on an island. The island is omposed of a forest and a beah. The forest is where oonut trees grow, while the beah is where people meet. Coonuts are the only good in the eonomy. Eah agent has a linear utility u(t). Consuming one oonut gives utility y >, while utility is when the agent is not onsuming. Agents disount utility at rate r (whih is also the risk free rate as 5

6 utility is linear) and the ost of olleting a oonut is (t). Agents intertemporal expeted utility, as evaluated at date T, is therefore V (T ) = E T e rt( u(t) (t) ) dt T Prodution is done by limbing at oonut trees, who are of various height. Agents without oonuts walk in the forest and look for trees. They are said to be unemployed. When they are unemployed, they randomly bump into oonut trees. The proess of finding a tree is Poisson with rate a. The ost (t) of olleting one oonut is i.i.d, with df G(), defined over [, ], with >. One annot ollet or arry more than one oonut. There is a taboo on this island, aording to whih one annot eat self-olleted oonuts. Therefore, one a oonut is olleted, an agent has to walk on the beah arrying her oonut, and she will randomly meet another agent with a oonut. When walking in searh for a trade partner, agents are said to be employed. When a meeting happens, the pair of agents that meets exhange their oonuts (one against one) and eat them. We denote by e(t) (employment) the measure of agents that are holding a oonut and looking for a partner. The proess of meeting someone else on the beah is Poisson with rate b(e). It is assumed that b (e) >. The life of an agent an therefore be desribed as follows: one agents have a oonut, they simply walk on the beah until they meet another agent with a oonut, and trade. Without a oonut, they walk in the forest until they run into the oonut tree, and then they deide whether to ollet the oonut given the ost. We assume that agents strategy is a stationary poliy p(, e) whih determines a probability of olleting a oonut given its ost [, ] as also a funtion of the urrent measure of traders looking for partners, e [, 1]: p : [, ] [, 1] [, 1] A symmetri equilibrium is a poliy p that is the best response to itself, in the sense that when all other agents use p, it is optimal for eah of them to do so as well. We will assume that in suh an environment, the optimal poliy is a reservation ost poliy, i.e., p(e(t), ) = 1 for all (t) and p(e(t), ) == otherwise (where we write (t) instead of (e(t)) to simplify notation). We denote V E (t) the value of holding a oonut and looking for a trade partner (being employed) and V U (t) the value of looking for a tree (being unemployed). 1 Why don t we have to are about the prie of a oonut in a math? Why does it simplify the analysis? There is only one good in the eonomy (oonuts) and by assumption one annot arry more than one oonut. Therefore, trades are always one-for-one. There is no room to split the surplus of a math in a different way (no money, no finanial laims, e...). The prie of ontents is therefore irrelevant. The fat that we do not need to medal the way the surplus is shared within a math simplifies the analysis (we don t need a bargaining theory). 2 How to justify b (e) >? Explain why this assumption is alled thik market externality? Why externality? b (e) > : It is easier to bump into someone on the beah if a lot of inhabitants are walking on the beah. It is an externality beause no one is taking that effet into aount when deiding to limb at a tree. 6

7 3 Write the arbitrage equation that gives the flow value of being employed. Hint: by arbitrage, the flow value of being employed should be equal to the expeted gain in value of meeting a partner plus the time hange of V E (t): ( flow value rv E (t) = probability of a meeting utility of onsumption + ) hange in value from employment to unemployment (1) + V E (t) The equation writes rv E (t) = b(e(t)) ( ) y + V U (t) V E (t) + V E (t) 4 Explain why the flow value of unemployment is given by rv U (t) = a (t) (V E (t) V U (t) )dg() + V U (t) (2) The probability of moving from unemployment to employment is the probability of finding a tree a times the probability of deiding to limb to ollet the oonut. This happens only is is lower that (t). In that ase, the gain in value is V E (t) V U (t). 5 Express the threshold (t) as a funtion of V E (t) and V U (t). (t) is suh that an individual is indifferent between olleting the oonut and beoming employed (value = V E (t) (t)) and staying unemployed (value = V U (t)). Hene, we have (t) = V E (t) V U (t). 6 Given the reservation ost poliy, write the law of motion of employment e, whih is here the equivalent of the Beveridge urve in Mortensen-Pissarides: ė(t) = a(1 e(t))g( (t)) }{{} b(e(t))e(t) }{{} flow from unemp. to emp. flow from emp. to unemp. (3) 7 For the rest of the problem, we restrit to steady states of this eonomy, denoted (e, ). From equations (1) and (2) and the answer to question 5, derive a first impliit relation (A) between and e. Show that (A) defines a positively sloped lous in the plane (e, ) when y >. Explain why this ondition y > must hold. Admit that this lous = A (e) is a onave funtion. Given that = V E V U, (1) writes Plug it bak in (1) to get V E = b(e)(y ). r V U = + b(e) y. r 7

8 (2) writes whih is equivalent to Put everything together to get Fully differentiate (A) to get rv U = a ( )dg()), rv U = a G( ) a dg(). = b(e )y + a dg() r + b(e ) + ag( ). (A) (r + b(e ) + ag( ))d = b (e )(y )de. The slope d /de is positive if y >. It has to be the ase as if the threshold ost of grabbing a oonut was higher than the gain from eating it instantaneously, no one would ollet it and wait to meet someone to trade. It is admitted that this lous is onave. Figure 4: The (A) lous 8 From (3), derive a seond impliit relation (B) between and e. Show that (B) defines a positively sloped lous in the plane (e, ). Denote = B (e) At the steady states, (3) writes Fully differentiate (B) to get e = ag( ) ag( ) + b(e ). (ag( ) + b(e ) + e b (e ))de = ((1 e )ag ( ))d. The slope is positive. 8

9 Figure 5: The (B) lous 9 A steady state is an intersetion of A (e) and B (e). Explain why (, ) is always a steady state. When e =, and with the assumption that b() =, there is no hane to meet a trading partner, so that no one will ollet a oonut, even when the ost is the minimum one. 1 Draw (A) and (B) in a ase of two interior steady states. Explain in words the reason of multipliity. Figure 6: Multiple steady states See Figure 6. There an be many interior steady states for the following reason. If e is large, it is optimal to lim high trees to ollet oonuts, and e is indeed large. If e is small, one would have to wait a long time 9

10 before meeting a trading partner on the beah. Therefore, it is not worth limbing high trees, and e is indeed small. 11 Can you Pareto rank an interior steady state and the zero steady state? Explain what is going on. Take an interior steady state. For all osts <, we have a positive utility from being employed, as V E ( ) > V (u) +. Therefore, all employed workers have a positive value. As unemployed bear no osts and have a positive probability of beoming employed, they also have a positive value. As all the agents have a zero value at the zero ativity steady state, any interior steady state Pareto dominates the zero ativity one. 12 Is the level of e suboptimally low in any interior steady state? Hint: the algebra is quite tedious. Restriting to steady state welfare omparison, you would need to assume a situation in whih everybody ommits to use a poliy p suh that p(, e ) = 1 for all trees with + δ for δ positive and small. Denoting V U (δ) and V E (δ) the values in suh an eonomy, you would need to show that V U (δ = ) and V E (δ = ) are both positive. Unless you have a lot of time left, just explain what is the answer likely to be. Intuition: agents do not internalize the fat that they inrease the probability to trade for the others when they deide to grab a oonut. Therefore, for a tree of height, the private value of olleting a oonut is exatly zero but the soial one is positive, as olleting that oonut inreases b. In more details: imagine that a planner fores all agents to aept using a poliy suh that all + δ are aepted, where δ > and small. Let us ompare steady states (in full generality, one should inlude the transition). First, denoting g = G, assuming that g( ) exist and letting e = e (δ), equation (B) gives de dδ = δ= ag( )b(e ) (ag( ) + b(e )) 2. The value of agents searhing for trees in steady state is now: rv u (δ) = a +δ Totally differentiating this with respet to δ, we obtain?moreover, in steady state: Using ( ), we have (V E (δ) V U (δ) )dg(). V U(δ = ) = ag( )V E (δ = ). ( ) r + ag( ) V E (δ) = b(e (δ))(y + V U (δ). r + b(e (δ)) V E(δ = ) = b (e )(y + V U ) (r + b(e )) 2 + V U(δ = ) = V E(δ = ) = b (e )(y + V U ) + ag( )V E (δ = ) (r + b(e )) 2 r + ag( ) = b (e )(r + ag( ))(y + V U ) > r(r + b(e )) 2 whih is stritly positive given the thik market externalities, i.e., b (e) >. Moreover, this implies from ( ) that V U (δ = ) >. Consequently, a small inrease in (i.e., in the reservation ost 1

11 of agents looking for oonuts) will result in an inrease in the values of both types of agents. Therefore, none of the equilibria are onstrained Pareto effiient. Some type of subsidy to searh would improve the welfare of all agents. This is a onsequene of the thik market externalities. 11