Solutions to Problem Set 4 Macro II (14.452)

Solutions to Problem Set 4 Macro II (14.452) Francisco A. Gallego 05/11 1 Money as a Factor of Production (Dornbusch and Frenkel, 1973) The shortcut used by Dornbusch and Frenkel to introduce money in the economy is that they assume that output available for consumption is equal to a fraction of production net of capital maintenance (G (k; 1)), where the fraction is an increasing function of real money balances: (1 L (m)) G (k; 1) L(:) satis es these properties: L(1) = 0, L(0) = 1, L 0 () < 0, L 00 () > 0. The households maximize: 1Z Max U(c t ) exp( t)dt s:t: 0 c = (1 L (m)) G (k; 1) + x m k m where c,m; k, and x are consumption, real money balances, capital holdings, and government transfers. 1. Derive the FOCs of this problem. Some people had some trouble with this problem. There are many ways of solving the problem, but I will use a "detailed" method because using intuition you could simplify it a lot. First, notice that we have two states (capital and money) and only one dynamic accumulation equation. To get a second accumulation equation let s de ne a slack variable: q = m. Now, let s de ne the current value Hamiltonian as: H c = U(c t ) + 1 [(1 L (m)) G (k; 1) + x m c q] + 2 q FOCs: @H c @c = 0, U c(c t ) = 1 1

@H c @k = 1 + 1, 1 ((1 @H c @q = 0, 2 = 1 L (m)) G k (k; 1)) = 1 + 1 @H m @m = 2 + 2, 2 ( L m (m) G (k; 1) ) = 2 + 2 Plus the transversality conditions Intuition: The rst is standard. The second tells us that the shadow value of accumulating money and the shadow value of accumulating capital has to be the same. This should be quite intuitive. More importantly, this result also allows us to simplify the Hamiltonian and wrote it as follows: H c = U(c t ) + [(1 L (m)) G (k; 1) + x m c] So just using one accumulation equation is correct in this case. The third and fourth conditions are also standard. 2. Characterize the steady state. Is money neutral? Superneutral? In steady state = 0. Therefore, the last two FOCs in steady state imply that: (1 L (m)) G k (k; 1) = (1) L m (m) G (k; 1) = + (2) And we also know that in steady state consumption is equal to: c = (1 L (m)) G (k; 1) To see if money is superneutral I will di erentiate (1) and (2) with respect to : @m @ L mg k + @k @ (1 L (m)) G kk (k; 1) = 0 (3) @m @ L mmg @k @ L mg k = 1 (4) Using these two equations you can solve for @m @ @m @ = (1 L)G kk @k L m G {z k @ } >0 @k and @ : (5) <0 z } { @k @ = L m G k <0 z } { (1 L)G kk GL mm (L m G k ) 2 7 0 (6) {z } >0 2

Therefore, in general, in ation has an e ect on k (there is the case when <0 <0 z } { z } { L m G k = (1 L)G kk GL mm ) and m moves in the same direction as k. Finally, notice that: @c @ = @m @ L mg+ @k @ (1 L)G k = (1 L)G kkl m G @k L m G {z k @ + @k @ (1 L)G k {z } } >0 >0 Therefore, c moves in the same direction as m and k: Hence, money is not superneutral. What about neutrality? If M (nominal money) increases P will increase proportionally and m will be constant. Therefore, money is neutral. 3. What are the basic di erences of this approach with respect to including money in the utility function or a cash in advance constraint? In this case we get non-superneutrality in contrast to the CIA and MIU models. The basic di erence here is that money is a factor of production (or strictly speaking, a factor needed to convert output into consumption) and therefore its cost () matters for the real equilibrium In the other cases, money is not needed for production. 2 Nominal Rigidities This question asks you to extend the analysis on nominal rigidities developed in class. Take the Blanchard and Kiyotaki (1987) yeomen farmers model with nominal rigidities. The model assumes that unexpected changes in money are small. 1. If you take a second-order approximation around the initial equilibrium, you can compute the loss in pro ts that comes from not adjusting the price level of each rm in response to a proportional change in nominal money dm M, given the other prices do not adjust. Expressed as a ratio of initial pro ts this loss is: L = 1=2 ( 1)2 ( 1) 1 + ( 1) 2 dm M Interpret this condition. What is the role of and? Use the diagram that resumes the model to discuss this point. This condition tells you that (i) there is only a second-order loss if money increases.? @L @ > 0 3

and L = 0 if = 1. Make sure that you understand that 1? @L @ > 0 To see clearly the role of and, notice that the equilibrium for any monopolist is given by:p i (1 + 1 " ) = MC i, where " is the elasticity of demand facing the monopolist. In this case this condition becomes: P i P = ( + 1 )Y 1 i = ( 1 )Y 1 i (7) This equation tell us that an increase in should increase the loss of not adjusting prices. Why? Because the marginal cost is steeper and as M increases production. What about? If increases, the demand becomes more elastic, and, therefore the demand is going to increase by more given an increase in M and therefore prices need to be adjusted faster (Notice that @L @ is proportional to ( 1); thus, this re ects that the e ect of on the loss has to do with the marginal cost if = 1, there is no e ect of on the loss). The diagram that summarizes the model is Figure 8.1 in BF, pp. 381. 2. What if = 1? Do we need some friction (e.g. menu costs) to get nominal rigidities? Why? In this case L = 0. So there is no loss of not adjusting. Let s consider again equation (7) to see what s going on. If = 1, marginal cost is at and given the demand for the good we get that an increase in the demand for good Y i does not a ect the marginal cost of producing and any level of production can be an equilibrium. Thus, we get price rigidities without any friction. [Aside. Some people were concerned by the fact that in equilibrium if = 1: 1 = ( 1 ) ) = 1. To avoid this relationship we can extend the utility function in the lecture notes and put a constant in front of 1 Y i, as BF does. In that case if = 1: 1 = ( 1 d )] 3. Now, let s move to a related but di erent question. If other producers change their prices after a nominal demand increase, is it more or less likely that an individual rm will want to adjust its price too? Assume that in response to a change in nominal money, the price level adjusts by a fraction k of that change: dp = k P dm M In this case, the second-order private loss of not adjusting individual prices given an increase in nominal money dm M and 8, is (again as a ratio to initial (8) 4

revenues): L = 1=2 ( 1) fk [1 + ( 1) ( 1)] + ( 1)g2 1 + ( 1) 2 dm M Interpret this condition. What is the e ect of k on the loss? Interpret. Plot L as a function of k (given dm M ). In this case an increase in k is increases the loss of not adjusting and the increase is convex. The relevant diagram is Figure 8.2 in BF, pp. 386. If k = 0 we are back to the previous case. If k = 1, L = 1=2 ( 1) (1 + ( 1)) dm M The intuition is quite simple, as the e ect of the monetary increase on current prices gets bigger, the loss of not adjusting each rm prices is bigger 2. 4. Now, let s extend the model and assume that there is a "menu cost" of c if a rm changes prices. What happens if c is really big (i.e. c is bigger that the loss of not-adjusting when k = 1)? what if c is really small (i.e. c is less that the loss of not-adjusting when k = 0)? What if c is in between (discuss stability and uniqueness of the equilibrium/equilibria)? The relevant gure is The relevant diagram is Figure 8.2 in BF, pp. 386. We have three potential equilibria: If c < L(0), the economy converges to k = 1 unique an stable. If c > L(1), the economy converges to k = 0 unique an stable. This equilibrium is This equilibrium is If L(0) < c < L(1), there exists a potential equilibrium such that 0 < k < 1 and L(k ) = c This equilibrium is unstable, however; change k by a small number and the economy diverges from this equilibrium. 5. Use these results to discuss potential changes in the e ect of money on output as technology evolves. Basically technology should decrease c: This will make more likely to converge to an equilibrium with k = 1. In this case certainly an increase in M has no real e ects. Another way in which technology may a ect this problem is by increasing ;in the sense that there is more competition (of course, this is a short cut). If so, the loss of not adjusting prices increases, as we have already discussed. 3 (Optional) A simple 3-equation model 1 Consider the following economy: 1 This exercise is based on "Should Monetary Policy Respond Strongly to Output Gaps?" by Bennett McCallum (American Economic Review P.P. (91-2): 258-262) 5

y t = b 0 + b 1 (i t E t p t+1 ) + E t y t+1 + v t p t = p e t+1 + (y t y t ) + u t i t = (1 3 ) [r + p t + 1 (p t ) + 2 (y t y)] + 3 i t 1 + e t where i is a one-period interest rate, p and y are logs of the price level and output, with y the natural-rate value of y. is some in ation target. E t z t+j is the expectation of z t+j conditional on information available in t. v; u; and e are exogenous shocks, which may have an AR(1) representation. In addition, y follows: y t+1 = y t + " t+1 1. How can you derive the rst two equations from rst-principles? If so, can you relate the parameters of these equations to some "deep" parameters? The rst equation can be derived from rst principles, it could be understood as a log-linear version of the Euler equation. In that case, b 1 is the elasticity of intertemporal substitution. The second equation is a pricesetting equation quite similar to the one derived in the Calvo model. See the notes for topic 9. There is a di erence, though, because the coe cient in front of expected in ation is not 1. As you may suspect is the discount factor. The intuition is that price setters discount the future and, therefore, they use their subjective discount rate. For a formal derivation of this see Gali and Gertler, 1999, JME. 2. What about the third equation? How is it called in the paper? How can you relate this equation to the LM equation? When and why may you prefer to include an equation like this instead of an equation for money? The third equation is a monetary policy rule (the now famous Taylor rule, called after John Taylor). It basically says that the Central Bank reacts positively to deviations of in ation from a target and to the output gap. In addition, there is monetary policy inertia. This equation is closely related to the LM. Basically, central banks can control the money supply or the interest rate. In environments where there is a lot of money demand stability, central banks tend to prefer xing the nominal interest rate because the interest rate a ects expenditure and money supply only a ects aggregate demand, in the short run, through its e ects on the nominal interest rate. To gain more intuition, notice that the money market equilibrium should look like m t p t = y t ci t. Therefore, if there is no uncertainty, you could write the Taylor rule as a rule for m The problem emerges if you add a disturbance term to the money demand. 6

Now, we are going to do some numerical exercises using this simple model. To do so, we are going to use slightly modi ed versions of some MATLAB programs written by Matteo Iacoviello from Boston College. These programs use the Uhlig method (that we already have used in our previous problem set) to solve the dynamic system. Follow the instructions in the tutorial le, which is on the class web page. The code called mc is the one you have to modify if you want to change the parameters of the three equation model (right now it has the values for the basic speci cation in McCallum, 2001). 3. Use the programs and the basic speci cation of the paper to simulate the response of the economy (i.e. output, prices, interest rate) to the four shocks of the economy (v; u; ", and e) the program produces a gure like Figure 1 in the paper, you are not going to get exactly the same results because the method is slightly di erent that what McCallum does) Basis case in McCallum (2001) 4. Interpret the results. What are the main puzzles you observe. We have four shocks: 7

The IS shock. This shock increases the nominal rate and there is persistence because of the smoothing interest rate term. The increase in output is short lived and there is also a small increase in in ation on impact that reverts in the next periods, as the interest rate decreases and the output gap is even slightly negative. The supply shock. This shock produces an increase in output, a decrease in the nominal interest rate, and a decrease in in ation. This shock is more persistence mechanically because we are assuming so. The in ation shock. This produces an increase in the real interest rate, a temporary increase in in ation, and a decrease in output (basically related to the increase in the real interest rate). A monetary shock. This produces a big decrease in output and a decrease in in ation. There is more inertia here because monetary policy has a lot of persistence. Therefore, we do not see much inertia in the IRFs of in ation and output especially after IS and in ation shocks. The only place where we get more persistence is in the case of supply shock, but we get this only by assuming that these shocks are more persistent. 5. Now, let s try to solve at least some of the puzzles. McCallum proposes some solutions in the paper and there may others you may want to propose. Manipulate the parameters in the program and show how to improve upon the basic speci cation. Justify and interpret your results. McCallum proposes to introduce habit formation to solve for the little inertia we observe after an IS shock. Let s use his parameter and see what happens (he implements this by adding a lagged term to the IS equation). 8

Assuming habit formation Results are interesting because we get more persistence in output and bigger e ects on output after most shocks. However, in ationary shocks still produce too little in ation inertia. Thus, let s try a second (incremental) x. I can assume that in ationary shocks have a lot of inertia. I know this is not a very elegant way of solving the problem, but this is what the program allows me to do. A more interesting extension could be to modify the in ation equation by adding lagged in ation and, as Olivier mentioned in class yesterday, I will assume that the weight on expected in ation is 1/3 and the weighted on lagged in ation is 2/3 In any case, let s see what happens. 9

Adding habit formation and inertia to in ation shocks It looks like we get more inertia, but still it does not look like we can replicate the empirical IRFs. 4 (Optional) In ation targeting and the liquidity trap (Final 2003, Q2) 1. Consider the following economy: (y t by t ) = a (i t t r) t+1 = t + b (y t by t ) 1. Suppose the central bank wants to achieve an in ation rate of. Show that there is an equilibrium with nominal interest rate i, y = by and =. Derive i as a function of r and. From the second equation we get that: = +b (y t by t ) ) y t = by t 10

Thus, using this result in the rst equation you get: 0 = a (i r) ) i = + r 2. Suppose that the central bank xes the nominal interest rate at i. Show that the economy (output and in ation) is unstable. Explain in words. Plug i into the rst equation: (y t by t ) = a ( t ) Plug this equation into the equation for t+1 : t+1 = t ba ( t ), t+1 = t (1 + ab) ba So if in ation increase today, it will increase by more than one tomorrow and in ation will explode. As in ation has an e ect on output, will also be unstable. 3. Suppose that instead of xing the nominal interest rate, the central bank adopts the following rule: (i t i ) = c ( t ) Derive the condition on c so the economy is stable. Explain in words. Let s rewrite the interest rate rule as follows (using the value for i we already found): i t = + r + c ( t ) Plug this into the equation for (y t by t ) and you get that (y t by t ) = a ( + c ( t ) t ) Plug this into the in ation equation: t+1 = t ab ( + c ( t ) t ), t+1 = t (1 abc + ab) + d where d is a constant. We want: abc + ab < 0, c > 1 Why? Just look at the interest rate rule and notice that the only way of stabilize in ation after facing an in ation shock is to increase the nominal interest rate more than proportionally. I.e. that s the only way of increasing the real interest rate. We want this to stabilize the economy because that is the only way of getting a negative output gap, which is going to a ect negatively in ation. 11

4. Assume the economy is at a state where i t = 0, and t < r (so has zero nominal interest rates, and su ciently high de ation). Describe the dynamics of the economy. What can the monetary authority do? In this case (i t t r) < 0 ) (y t by t ) < 0 ) ( t+1 t ) < 0 ) (i t+1 t+1 r) << 0 ) (y t by t ) << 0:::So the economy is going to have to stay with a negative output gap and a decreasing in ation forever. Monetary policy cannot do much here because i cannot be less than 0! 5. Is the answer to (4) too pessimistic?what modi cations might you want to make to the equations above, and how might they modify your answer to (4)? The basic problem here is that you need to create in ation from somewhere else. To do so, you may add an equation to get an independent e ect of some other variable on (y t by t ) ; to allow an e ect of scal policy for instance. This is going to create in ation, which can put the economy in the region with negative real interest rates. Another possibility if to forget about monetary policy and import in ation from above, e.g. dollarization. 12