Section Notes 8
Macroeconomic Analysis I
Economics GR5215, Fall 2021(Miller)
November 5, 2021
Akshat Gautam & Susannah Scanlan1
1 Overview
The goal of this note is to cover how we derive the NK IS curve and give an overview of the reduced
form Philips curve that we looked at in lecture. At the end of it we’ll have the three equation NK
model that is commonly used in the literature. The NK IS curve appears in nearly all modern
models, not just NK ones.
2 Model
2.1 Setup
This is going to be a very simple model where we do not really model the production side of the
economy. The model has a representative household that works to produce all the goods in the
economy using just labor, so there is no capital. The households are allowed to save with a risk
free bond and with money. Households derive utility directly from holding money but not bonds,
but bonds pay interest whereas money pays zero interest. The household optimization problem is
going to give us some important conditions for the equilibrium.
The household has the following utility function
U =
∞∑
t=0
βt
[
u(Ct) + Γ
(
Mt
Pt
)
− V (Lt)
]
With the following properties
V ′ > 0, V ′′ > 0
u′ > 0, u′′ < 0 Γ′ > 0, Γ′′ < 0 For this model we’ll just assume CRRA utility for consumption and real money balances u(Ct) = C1−θt 1− θ Γ ( Mt Pt ) = ( Mt Pt )1−ν 1− ν Going into a little more detail. Households gain utility from consumption the same as always since u() is the standard utility function for consumption. In this model we are assuming that households also gain utility from real money holdings. We do not really explain why this is, we just assume it. In particular we assume that households gain utility from money holdings in the 1Thanks to Eddie Shore, Joe Saia, and Miguel Acosta for letting us use their notes with minimal updates. Please send corrections to 1 same way as they do from consumption, so real money holdings is sort of like a second consumption good, thus we give it a CRRA utility function as well (The letter ν is a “nu” pronounced new). To get the budget constraint we’ll assume that the households can save in bonds or money, but that money does not pay any interest. We’ll also assume that the consumption is non-storable and that households derive wage income from their labor. This gives us the following NOMINAL budget constraint WtLt + (1 + it−1)Bt−1 +Mt−1 = PtCt +Bt +Mt Pay careful attention to the time subscripts. Mt is the money holdings that households choose to hold this period, Mt−1 is the money holdings the household chose to hold last period. Bt is the amount of bonds that households buy today to hold into the next period and Bt−1 is the amount that they bought in the previous period. This means that the amount of the wealth that the household will start with in period t is (1 + it−1)Bt−1 +Mt−1. 2.2 Optimization Now we can solve the households optimization problem: max {Ct,Bt,Mt,Lt}∞t=0 ∞∑ t=0 βt [ u(Ct) + Γ ( Mt Pt ) − V (Lt) ] s.t. WtLt + (1 + it−1)Bt−1 +Mt−1 = PtCt +Bt +Mt The Lagrangian is the following L = ∞∑ t=0 βt [ u(Ct) + Γ ( Mt Pt ) − V (Lt) + λt (WtLt + (1 + it−1)Bt−1 +Mt−1 − PtCt −Bt −Mt) ] FOCs: ∂ L ∂ Ct ⇒ C−θt = Ptλt ∂ L ∂ Lt ⇒ V ′(Lt) = Wtλt ∂ L ∂Mt ⇒ 1 Pt ( Mt Pt )−ν = λt − βλt+1 ∂ L ∂ Bt ⇒ 0 = λt − β(1 + it)λt+1 The FOC for bond holdings gives us the usual relationship between lagrange multiplies each period. The FOC for money holdings is similar to bond holdings, except there’s no interest in it. This makes sense since households are willing to give up a little bit of utility tomorrow, from holding interest paying bonds, to hold money which gives them extra utility today. From the consumption and bond FOCs we have the usual Euler equation C−θt C−θt+1 = β 1 + it 1 + πt+1 From the consumption and labor FOC we have that the marginal utility of consumption is equal 2 to the marginal disutility of labor times the real wage rate C−θt = Wt Pt V ′(Lt) We can combine the the Bond and real money holdings FOCs to get the following: βt ( Mt Pt )−ν = Ptλt − Ptλt+1 = Ptλt − Pt λt 1 + it = it 1 + it Ptλt Combining this with the FOC for consumption we have( Mt Pt )−ν = it 1 + it C−θt The intuition for this one is, if the household gives up a dollar of money to purchase consumption and bonds, so that their future consumption is unchanged, they’ll be able to purchase it 1+it units of consumption today giving C−θt extra utility today. If it is very high, then the household can give up one dollar of money today, buy a very small amount of bonds and have the same net worth in the next period, and buy almost a full one unit of consumption today. On the other hand if it is equal to zero then in order to have the same net worth tomorrow, the household would have to buy exactly one unit of bonds today, leaving nothing leftover to buy consumption. 2.3 General Equilibrium This model is a closed economy with no government and no capital, hence no investment so our national accounting identity is Yt = Ct. We also have a representative household so we know that Bt = 0 for each period. Next, we’ll assume that the central bank sets the money supply each period to Mt = M s t . Lastly, we’ll assume that we have a sequence of wages and prices for each period. Using the Euler equation and using the definition of the real interest rate, we have the following: C−θt C−θt+1 = β 1 + it 1 + πt+1 = β(1 + rt) The GE conditon that Ct = Yt gives Y −θt Y −θt+1 = β(1 + rt) Taking Logs −θ log(Yt) + θ log(Yt+1) = log(β) + log(1 + rt) ≈ rt log(Yt) ≈ log(Yt+1)− 1 θ rt This gives us the NK IS curve. In the NK model, output today is a function of output tomorrow and the interest rate, which is similar to the old Keynesian IS curve Yt = C(Yt)+I(rt) where output today is a function of output today and the interest rate. 3 Looking at ( Mt Pt )−ν = it 1 + it C−θt we can substitute in our GE condition for output and rearrange giving Mt Pt = ( it 1 + it )− 1 ν Y θ ν t = ( 1 + it it ) 1 ν Y θ ν t This gives us an LM curve that is just like what we had in the old Keynesian models. Now if we assume that prices and output are fixed within the period, if the central bank sets the money supply to M st then this implies a certain interest rate. This means that the central bank can not independently choose the money supply and interest rate, a value of one implies a value for the other. We’ll come back to this later. It’s important note that these results are very general. All we’ve assumed is that there is a repre- sentative household, no capital, no real government expenditures and a closed economy. We haven’t said anything about the production side of the economy. This means that these results will apply to a wide range of models. The LM curve is also very general, though the exact parameterization will depend on how you model utility. 2.4 Prices So far we haven’t said anything about how prices evolve. This turns out to be tricky to micro-found and it took people a while to do so. For now lets do a hacky version. First remember that output is purely labor driven Yt = F (Lt) and assume that the production function has decreasing returns to scale (F ′′(Lt) < 0). Firms are profits are therefore given by Πt = PtF (Lt)−WtLt. If the labor market is competitive then we have that F ′(Lt) = Wt Pt . Now lets assume a very simple process for setting wages; each period the firms offer a wage equal to the last period’s price level. This gives F ′(Lt) = Pt−1 Pt = 1 1+πt . Because of decreasing returns to scale in F () this means that higher inflation leads to higher output, which makes sense; if the price of your goods goes up a lot today, but your wage does not, then you will higher extra labor and sell more stuff. Note that we have just assumed this model and have not micro-founded it at all. Let’s assume that Yt = F (Lt) = L 1−ξ t , this means that F ′(Lt) = (1− ξ)L −ξ t . If we take logs of the production function then we have log(Yt) = (1− ξ) log(Lt) and taking logs of the first derivate of the production function we have log(F ′(Lt)) = log(1− ξ)− ξ log(Lt). Going back to our model of wages we have F ′(Lt) = 1 1 + πt (1− ξ)L−ξt = 1 1 + πt log(1− ξ)− ξ log(Lt) ≈ −πt Collapsing constant terms into α and dropping the approximation πt = α+ ξ log(Lt) πt = α+ ξ 1− ξ log(Yt) πt = α+ λ log(Yt) 4 This gives us a very crude Philips curve where increases in inflation leads to an increase in output. Obviously when we derived this we made a bunch of assumptions, if we wanted to we could make assumptions that gives us something like this πt − π∗t = λ(log(Yt)− log(Y t)) This says that deviations from the “core” inflation rate cause deviations from the “natural” level of output. Because we have not micro founded this we can not saying anything about what is “core” inflation or the “natural” level of output but you should be aware that this is roughly what the philips curve in a full blown NK model looks like, and deriving it properly will be our focus for the next few lectures. 3 Three equation Model Now we can take our equations and put them together, writing the central bank’s decisions of the money supply as a policy function. log(Yt) = log(Yt+1)− 1 θ rt πt = π ∗ t + λ(log(Yt)− log(Y t)) Mt Pt = M st (log(Yt)− log(Y t), πt) Pt Mt Pt = ( 1 + it it ) 1 ν Y θ ν t Note that the last two equations are block recursive given Yt and πt. If we know those two variables then real money balances is determined from the policy function, and interest rates are determined from the LM equation. Also note that money does not appear in the Philips or IS equation, just interest rates. This means that we can collapse the last two equations into a single monetary policy equation for the interest rate, and simplify our model considerably. Doing this we have: log(Yt) = log(Yt+1)− 1 θ rt πt = π ∗ t + λ(log(Yt)− log(Y t)) rt = r(log(Yt)− log(Y t), πt) This is the semi micro-founded NK 3 equation model. We’ve micro-founded the household’s problem, giving us the IS equation, and specified a policy rule for monetary policy to make it endogenous. For the Philips curve we just slapped down an equation to dictate how prices evolve, so that’s not micro-funded yet. Note that we also haven’t added any uncertainty to this model. In these models we also usually denote log(Xt) as xt. If we do all this we have 5 yt = Et(yt+1)− 1 θ rt + u IS t πt = Et(π∗t ) + λ(yt − yt) + u P t rt = r(yt − yt, πt) + u MP t which is usually how you’ll see this model presented. Romer has details on different versions of this model using different monetary policy rules and different Philips curves. 3.1 Interpretation Here, we’ll wrap up by discussing the intuition behind each of the equations. Starting with the monetary policy rule, rt = r(yt − yt, πt) + u MP t We say that the central bank sets interest rates depending on output gap “yt − yt” and the rate of inflation. This is based off of how we typically observe modern central banks acting. Next the IS equation, yt = Et(yt+1)− 1 θ rt + u IS t This is all driven by the household’s optimization problem. If the household expects income to be high tomorrow, then they’ll try to consume more today pushing up output today. On the other hand, if the central bank raises the real interest rate, then households try to save more for tomorrow for any given level of income today, pushing down desired consumption today for any given level of income today, but because one person’s consumption spending is another person’s income, this also pushes down incomes today. This decrease in income then restores equilibrium in the bond market by decreasing desired savings. Finally, the Philips curve πt = Et(π∗t ) + λ(yt − yt) + u P t The way we’ve tried to micro-found this Philips curve is by saying, if inflation increases today, firms make more money off each unit of goods that they sell. This increases the amount that they want to sell and increases the amount of labor they want to hire to do so. This pushes up incomes and thus desired consumption today, leading to an increase in output. In this story, it’s the increase in inflation that drives the increase in output. An increase in output does not drive an increase in inflation. Putting all this together we have a story where central banks can affect output but not inflation. If you think carefully through this though, this should seem weird. In our story of the world there’s no place for demand shocks to push up prices. If everyone wakes up today and decides that they want to consume more, and then yt will increase but inflation will stay the same, which is weird. This is indicating that our reasoning for where the Philips curve comes from is incomplete. We’ll get to this shortly in this course, but the modern model’s story for the Philips curve is the following. Some firms set their prices in advance and keep it there for multiple periods, when setting their price they consider where the economy will be in the future. If they believe that prices are going to rise a lot in the next few periods, then they will raise their price a lot today. They also take in to account the demand for their goods today. If demand is high (measured by yt) then firms set a higher price, like we expect from Micro. In this modern story of the world, it’s output that drives 6 inflation not inflation that drives output. 7 Overview Model Setup Optimization General Equilibrium Prices Three equation Model Interpretation