Course name: Money and Banking. This is a problem set. I uploaded a word doc file that includes a problem, and a pdf file that has class material.
Unit 1: Inflation Money Demand We’ll start out with a model of ‘money demand’ and solve for the price level via equilibrium in the money market. P = price level Y = real GDP M = Nominal money supply v = velocity = PY/M Velocity refers to the number of times a unit of currency moves through the economy in a transaction. The basic idea is that nominal GDP (P x Y) = M x v. Why must this be true? Remember GDP is a flow variable that measures transactions that occur within a geographic unit within a given period of time. Unit 1: Inflation May, 24 2022 7 / 58 Money Demand We’ll start by assuming monetary neutrality. That is, real variables (e.g. real GDP) is unaffected by the money supply. In fact, we’ll assume that Y is exogenous and constant. Why is this an OK assumption when studying a hyper-inflation? Money supply, M, is also exogenous and controlled directly/perfectly by the monetary authority. Is this realistic? We won’t assume anything about P...that’s what we’re trying to explain (changes in P = inflation!) ... but we still need to model velocity, v. How we decide to model v is the main way in which different models/explanations of inflation differ. Unit 1: Inflation May, 24 2022 8 / 58 Inflation in our model We get inflation (growth in P) using the money market equilibrium condition: πt+1 = Pt+1 Pt = Mt+1 Mt vt+1 vt So inflation in the money demand model can either come from increases in the money supply or increases in velocity. The Quantity Theory of Money assumes that velocity is constant, so changes in M feed directly into changes in P and therefore money growth is the only explanation for growing prices. Changes in velocity will be the way inflation expectations enter this model. Unit 1: Inflation May, 24 2022 9 / 58 Where are the firms? Wait...but who actually changes prices in the real economy? What’s actually happening here? This will become much more clear when we study the New-Keynesian model. For now, you can think of a doubling in M or velocity as consumers coming to the firm with double the purchasing power (say, within a single day!) and because the firm can’t meet this new demand, it has no choice but to double it’s prices. Unit 1: Inflation May, 24 2022 10 / 58 Nominal interest rates and money demand Velocity is likely not perfectly constant, even in the short run. How might nominal interest rates affect velocity? When nominal interest rates increase, the opportunity cost of holding liquid assets (money) increases. Why? When the opportunity cost of holding money balances goes up, this incentivizes consumers to hold smaller money balances, meaning each unit of money they do hold must work harder (circulate more). So velocity increases. So we can write, vt(it) Unit 1: Inflation May, 24 2022 11 / 58 Recap: nominal interest rate and inflation What is the relationship between the nominal interest rate and inflation expectations? If a lender charges a nominal rate of 4% and inflation ends up being 2%, what real rate are they actually receiving? But lenders don’t know what inflation will be beforehand! So if a lender wants to receive a real rate r, and they expect inflation to be π, what nominal rate should they demand? it+1 = rt+1 + Et [πt+1] Therefore, velocity is increasing in Et [πt+1] → vt+1(Et [πt+1]) Unit 1: Inflation May, 24 2022 12 / 58 Money supply vs. inflation expectations Now we’re ready to consider the two competing explanations for hyper-inflations. 1 + πt+1 = Mt+1 Mt vt+1(Et+1[πt+2]) vt(Et [πt+1]) But how do we model expectations about inflation? Two methods: 1 Adaptive expectations (backward looking, agents use previous periods’ data) 2 Rational expectations (agents use all available information) Unit 1: Inflation May, 24 2022 13 / 58 Adaptive expectations Adaptive expectations are relatively easy to understand: Et [πt+1] = αt−1πt−1 + αt−2πt−2 + ... Where ∑ αt = 1 (i.e. a weighted average of previous periods’ inflation rates) Intuition: People in the economy look to the recent past to form their expectations about inflation. Usually, we think that αt → 0 as t → ∞ Unit 1: Inflation May, 24 2022 14 / 58 Rational Expectations Rational Expectations means that your expectations are model consistent (not systematically wrong) in the sense that you take all available information. This does not necessarily imply perfect foresight: i.e. Et [xt+1] ̸= xt+1 in every period. Sometimes we think people are genuinely ‘shocked’ or surprised. However, modeling randomness/shocks is difficult. How can we capture the “idea” of a shock without worrying about modeling probability distributions over events? A simple tool we can use is called an ‘MIT shock’ - we write down a ‘deterministic’ or ‘perfect foresight’ model where the (ex ante) probability of a shock is essentially 0...but then a shock occurs! Therefore it makes sense for agents to act as if the shock won’t happen, and only change their behavior once it does. Unit 1: Inflation May, 24 2022 15 / 58 Rational vs. Adaptive What does RE get us? Bottom line: It allows us to capture the idea that news about the future can form expectations, not just past experience (adaptive expectations) To see the difference between the two types of expectations, let’s see how they operate in the model of inflation outlined above: Let Mt+1/Mt = 1 + µt+1 ... the growth in the money supply Suppose the government has been increasing the money supply at a steady (low) rate, µ forever (or, if you like, as long as anyone can remember). Before period t (so for periods, ... t-3, t-2, t-1), the probability of this changing was extremely low (we’ll just say it’s essentially 0). All of a sudden, a (super low probability) change in the money supply occurs! They start printing money faster (µ̃ > µ) Unit 1: Inflation May, 24 2022 16 / 58 Rational vs. Adaptive Because money growth has been steady forever, it makes sense that (in both models) inflation expectations - and therefore inflation - has just been determined by money growth for t-1, t-2, t-3 etc... so πt−1 = πt−2 = πt−3... = µ Why does this make sense? 1 + πt−1 = Mt−1 Mt−2 vt−1(Et−1[πt ]) vt−2(Et−2[πt−1]) = 1 + µ At time t, the government announces that Mt+1 = (1 + µ̃)Mt , and money will grow at µ̃ from then on. Question: What does each model of expectations predict about the path of inflation? Unit 1: Inflation May, 24 2022 17 / 58 Adaptive Expectations 1 + πt = Mt Mt−1 vt(Et [πt+1]) vt−1(Et−1[πt ]) = 1 + µ To make it easy, let’s do a simple case where αt−1 = 1 and αt−j = 0 for all j. So Et [πt+1] = πt−1 So what are Et−1[πt ] and Et [πt+1]? Just µ! Therefore, what is 1 + πt? πt = µ because the increase in M has been announced but hasn’t occurred yet! What about 1 + πt+1? 1 + πt+1 = (1 + µ̃) vt+1(Et+1[πt+2]) vt(Et [πt+1]) = 1 + µ̃ Unit 1: Inflation May, 24 2022 18 / 58 Rational Expectations Solving the RE version is a bit more complicated ... Et−1[πt ] = µ just as in the AE case. Why? But what about Et [πt+1]? Remember that, starting in t+1, there will be no more shocks, so Et+1[πt+1] = πt+2 (it’s a perfect foresight model after t+1) To solve this, we’ll use a ‘guess and verify’ method. Unit 1: Inflation May, 24 2022 19 / 58 Guess and Verify We’ll guess that πt+j = µ̃ for j ≥ 1 Then Et+j [πt+j+1] = µ̃ 1 + πt+j = (1 + µ̃) vt+j(Et+j [πt+j+1]) vt+j−1(Et+j−1[πt+j ]) = 1 + µ̃ Success! So now we have everything we need to solve for πt 1 + πt = (1 + µ) vt(Et [πt+1]) vt−1(Et−1[πt ]) = (1 + µ) vt+1(µ̃) vt(µ) > 1 + µ Unit 1: Inflation May, 24 2022 20 / 58 What’s the intuition? In the AE case, no one at time t anticipates the effect of the increase in the money supply at t+1. Their expectations are firmly routed in the past, meaning there is no increase in inflation until t+1 when the monetary increase actually occurs. However, in the RE case, agents take into account news about future money growth before the growth actually happens. In this way, inflation can actually precede money growth and become a ‘self-fulfilling prophecy’. Unit 1: Inflation May, 24 2022 21 / 58 Weimer Germany Source: Uribe, Martin. Lecture: Hyper-inflations and Their Ends. (2017) Unit 1: Inflation May, 24 2022 22 / 58 Seignoirage Why do governments do this?! What does it mean to ‘monetize’ government spending? Suppose an economy is only capable of producing Ȳ . If the government wants to re-apportion some of that output (Y’) - they could just tax their population: T=Y’. Why might this be difficult? Suppose instead that the government prints M units of money and uses this new money to buy Y ′ instead. How much will they have to print? Because the new money will increase the price level, they’ll have to print more than Y’ in order to siphon off Y’ in real output. Unit 1: Inflation May, 24 2022 23 / 58 Seignoirage Revenue How to solve for M’, the amount of new money they’ll have to print to get Y’? Note that the following must be true: M0 +M ′ Pnew = Ȳ 1 v M ′ Pnew = Y ′ 1 v We have 2 equations and 2 unknowns (M’ and Pnew) So M0Pnew = (Ȳ − Y ′) 1v → vM0 (Ȳ−Y ′) = Pnew Finally, M ′ = Y ′ 1v ∗ Pnew Unit 1: Inflation May, 24 2022 24 / 58