ECON 401 Advanced Macroeconomics You may consult all course materials and standard Internet resources while working on the exam, but your w

ECON 401

Advanced Macroeconomics

Midterm Exam

Fabio Ghironi University of Washington

April 29, 2020

Instructions:

You have 5 hours to work on this exam. It is worth 100 points, contributing to your overall

score for the course as described in the Syllabus. You may consult all course materials and standard

Internet resources while working on the exam, but your work must be original and you may not

solicit or obtain assistance from or provide assistance to other people for any specific content of the

exam. Activities considered cheating include copying or closely paraphrasing content from websites

and discussing exam questions with other students. All exams will be checked for originality and

copied content, and anyone found cheating will be assigned a zero score for the exam. Read

carefully each step of each problem before you jump into working on it and do not panic if you

cannot complete everything. The exam is intended also to stretch your knowledge by forcing you

to use the tools and information you have acquired to think about some things we have not talked

about in class. I want to see how you think about those things based on what you learned. In

Problems 2 and 3 below, if there is an item that you cannot complete, just take the result in that

item for granted and move on to the next one.

Problem 1: Government Spending in the RBC Model (50 Points)

Consider the RBC model with government spending that you studied in Homework 1.

The representative household maximizes:

Et

∞∑ s=t

βs−t C 1−γ s

1 − γ ,

where 0 < β < 1 and γ > 0, subject to the constraint:

Ct + It + Xt = r̃tKt + wt

in each period. In this constraint, Xt is exogenous lump-sum taxation, which we assume is equal

to government spending. The rest of the notation is as in the slides.

The law of motion for capital is:

Kt+1 = (1 − δ) Kt + It, 0 < δ < 1,

in each period.

The production function is:

Yt = A α t K

1−α t , 0 < α < 1.

You found in the homework that the solution for consumption and capital in the log-linearized

model in which there is no deviation of the exogenous productivity At from its trend path (i.e., it

is at+1 = at = at−1 = 0) and there are only the government spending shocks xt = φxt−1 + εt is

given by:

ct = ηckkt + ηcxxt,

kt+1 = ηkkkt + ηkxxt,

in each period, where the η’s are elasticities that depend on the parameters of the model and the

ratio X̄t/Ȳt, which I told you to treat as exogenously given.

• Use the method of undetermined coeffi cients described in the slides to find the

expressions for ηck, ηcx, ηkk, and ηkx as functions of the underlying parameters

2

and of X̄t/Ȳt.

• Modify the Excel file posted in the Files section of the course Canvas appro-

priately to trace the responses of ct, kt+1, yt, rt, and it to a 1 percent positive

innovation to government spending at time t = 0 with persistence φ = 0.9 and no

other innovations in the following periods. Set the value of X̄t/Ȳt to 0.2. Include

your modified file with your answers. Highlight in red the modifications you make

to my original file.

• What is your intuition for how the variables respond to the government spending

shock? Explain the responses as clearly as you can.

• Do you think the response of consumption would be different if the government

did not finance the increase in government spending today (taken to be time 0)

with the lump-sum tax, but instead financed by issuing debt to be repaid by

taxation in the future? What is your intuition for your answer?

Problem 2: Markups, Distortions, and Optimal Inflation in the New Keynesian

Model (30 Points)

The optimality condition for price setting by wholesaler j in the New Keynesian model with sticky

prices that we studied can be written as:

1

1 − ε

( pt (j)

Pt

)− ε ε−1

Yt + ε

ε − 1

( pt (j)

Pt

)−2ε−1 ε−1

mct (j) Yt − ψ (

pt (j)

pt−1 (j) − 1 )

Pt pt−1 (j)

+ βψ

1 + πt+1

( pt+1 (j)

pt (j) − 1 ) Pt+1 pt (j)

pt+1 (j)

pt (j)

= 0,

where I am using pt (j) to denote the price set by wholesaler j instead of Pjt, mct (j) denotes

the wholesaler’s marginal cost of production, Yt (instead of yt) denotes the output of the final

retail bundle (which has price Pt), and πt+1 is the inflation rate between t and t + 1 (πt+1 ≡

(Pt+1 − Pt) /Pt); ε > 1 is the flexible-price markup, ψ ≥ 0 is the scale parameter for the cost of

adjusting prices, and β is the representative household’s discount factor.

3

Now let us make one more change of notation and set:

ε = θ

θ − 1 .

If you replace ε with θ/ (θ − 1) everywhere in the equation above, you get:

− (θ − 1) ( pt (j)

Pt

)−θ Yt + θ

( pt (j)

Pt

)−θ−1 mct (j) Yt − ψ

( pt (j)

pt−1 (j) − 1 )

Pt pt−1 (j)

+ βψ

1 + πt+1

( pt+1 (j)

pt (j) − 1 ) Pt+1 pt (j)

pt+1 (j)

pt (j)

= 0. (1)

• Note that ( pt (j)

Pt

)−θ Yt = yt (j)

by virtue of the demand function facing the wholesaler (where yt (j) denotes the

wholesaler’s output). Use this and algebra to show that equation (1) can be

rewritten as:

pt (j)

Pt =

θ

θ − 1

 1 + ψ(θ − 1) yt (j)

 

( pt(j) pt−1(j)

− 1 )

Pt pt−1(j)

− β 1+πt+1

( pt+1(j) pt(j)

− 1 ) Pt+1 pt(j)

pt+1(j) pt(j)

    −1

mct (j) .

Make sure to write all the steps in your answer.

This equation implies that the wholesaler sets the price of its good as a markup over marginal

cost. When prices are flexible (ψ = 0), the markup boils down to the constant θ/ (θ − 1) (or ε in

the original notation). When prices are sticky (ψ > 0), the markup is time-varying and given by:

θ

θ − 1

{ 1 +

ψ

(θ − 1) yt (j)

[( pt (j)

pt−1 (j) − 1 )

Pt pt−1 (j)

− β

1 + πt+1

( pt+1 (j)

pt (j) − 1 ) Pt+1 pt (j)

pt+1 (j)

pt (j)

]}−1 .

This captures the fact that costs of price adjustment give the wholesalers an incentive to smooth

price changes across periods, absorbing the consequences of shocks in part by letting the markup

component of prices vary. Let us denote the expression for the markup charged by wholesaler j

with µt (j):

µt (j) ≡ θ

θ − 1

{ 1 +

ψ

(θ − 1) yt (j)

[( pt (j)

pt−1 (j) − 1 )

Pt pt−1 (j)

− β

1 + πt+1

( pt+1 (j)

pt (j) − 1 ) Pt+1 pt (j)

pt+1 (j)

pt (j)

]}−1 .

4

Therefore: pt (j)

Pt = µt (j) mct (j) .

Now, if labor is the only factor of production, and technology is such that yt(j) = Ztnt(j), the

wholesaler’s marginal cost is equal to the real wage (wt) divided by productivity:

mct (j) = wt Zt .

• Why is this the expression of marginal cost?

Note that this implies that marginal cost is identical across all the wholesalers. Optimal price

setting becomes: pt (j)

Pt = µt (j)

wt Zt .

However, now consider that all the wholesalers in the economy are symmetric to each other. Other

than the fact that each one of them produces a wholesaler-specific good over which it has monopoly

power, they are all identical. Hence, they will all set the same markup µt (j) = µt and the same

price pt (j) = pt. We thus have: pt Pt

= µtwt. (2)

Finally, consider the production function of the representative retailer:

Yt =

[∫ 1 0 yt (i)

θ−1 θ di

] θ θ−1

.

Since every wholesaler produces the same amount yt (i) = yt, it follows that Yt = yt (the amount

of the final bundle produced by the retailer is equal to the amount of output produced by every

wholesaler). Given the demand function for a wholesaler’s output:

yt =

( pt Pt

)−θ Yt,

it is immediate to prove formally that it has to be Pt = pt.

• Write this proof.

5

Given Pt = pt, it follows that the optimal price setting equation (2) implies:

wt = Zt µt .

Now, the first-order condition for optimal labor supply implies:

U1−n (ct, 1 − nt) Uc (ct, 1 − nt)

= wt,

where ct is consumption in period t, and 1 − nt is leisure.

With flexible prices and perfect competition, it would be wt = Zt (the marginal product of

labor), and the amount of labor employed would be determined by:

U1−n (ct, 1 − nt) Uc (ct, 1 − nt)

= Zt.

In our economy with monopoly power and sticky prices, the real wage is lower than the marginal

product of labor, and the amount of labor employed by the economy is such that:

U1−n (ct, 1 − nt) Uc (ct, 1 − nt)

= Zt µt ,

• What are the distortions that affect this condition?

• Suppose you are the central banker and you can commit the economy to a choice

of inflation rate. What inflation rate would you choose and why?

Problem 3: The New Keynesian Phillips Curve (20 Points)

Once you impose symmetry across wholesalers in the optimality condition for wholesaler price

setting in the New Keynesian model we studied, you have the following relation between real

marginal cost (mct), output (yt), and inflation in periods t and t + 1 (respectively, πt and πt+1):

1

1 − ε (1 − εmct) yt − ψπt (1 + πt) + βψπt+1 (1 + πt+1) = 0,

where the notation is the same as in Problem 2. Sanjay Chugh refers to this equation as the New

Keynesian Phillips Curve (NKPC). Most scholars think of the NKPC as the equation that you find

after log-linearizing this equation and imposing equilibrium conditions on mct. This question asks

6

you to explore the implications of this equation for the behavior of markups.

Start by recalling from Problem 2 that, in equilibrium, it has to be:

mct = 1

µt ,

where µt is the markup charged by every wholesaler. Therefore, the NKPC equation above can be

rewritten as: 1

1 − ε

( 1 −

ε

µt

) yt − ψπt (1 + πt) + βψπt+1 (1 + πt+1) = 0. (3)

Denote the steady-state levels of variables by dropping the time subscript.

• Suppose that the steady-state inflation rate is zero. Use equation (3) to show

that the steady-state markup µ is such that µ = ε.

In addition to assuming that π = 0, assume also that the steady-state level of output is equal

to 1: y = 1. Use hats to denote percentage deviations from the steady state, so that:

ŷt ≡ dyt y

= dyt,

µ̂t ≡ dµt µ

= dµt ε .

In the case of inflation, focus on gross inflation, but note, however, that our assumption that π = 0

implies:

π̂t ≡ d (1 + πt)

1 + π = d (1 + πt) = dπt.

• Log-linearize equation (3) and write the resulting equation with only µ̂t on the

left-hand side as a linear function of only π̂t and π̂t+1 on the right-hand side.

• What happens to the markup if current inflation increases?

• Given the observation that real wages in the United States are pro-cyclical or

a-cyclical in response to demand expansions (i.e., if aggregate demand rises, real

wages rise or stay flat), can you explain why a countercyclical markup was a de-

sirable feature of New Keynesian models? (Hint: Remember that the real wage,

wt, is equal to the marginal product of labor– MPL– under perfect competition,

but this is no longer the case when we have monopolistic competition, and, as you

7

reviewed in Problem 2, the wedge between wt and MPLt created by the markup

becomes time-varying when prices are sticky.)

8

ECON 401

Advanced Macroeconomics

Topic 1

The Stochastic Growth Model

Fabio Ghironi

University of Washington

Introduction

• This course explores modern theories of macroeconomic fluctuations.

• The goal will be to take you as close as possible to understanding how many macroecono- mists at academic and policy institutions think about business cycles and policy questions,

including the crisis created by COVID-19, by studying a sequence of models.

• We will start from the stochastic growth model (also known as real business cycle—RBC— model), in which fluctuations are the result of random shocks to technology and economic

outcomes are efficient.

1

Introduction, Continued

• In this course, the word efficiency has a very precise meaning:

– The market economy is efficient when the outcome it generates is the same as the

outcome that would be chosen by a benevolent social planner in charge of allocating

resources.

• The market economy is efficient in the RBC model: A benevolent planner who acts to maximize social welfare would not do better than the market.

• This implies that this is a model in which there is no role for policy to improve on market outcomes.

2

Introduction, Continued

• We will study the RBC model not because we believe that it is an accurate, realistic theory of how the macroeconomy works, but because it is a useful starting point to become familiar

with concepts, tools, and techniques that we will use many times throughout the quarter.

• We will then introduce a number of more realistic features into our framework: monopoly power, nominal rigidity, financial market frictions, labor market imperfections, producer entry

dynamics, heterogeneity across agents, and more.

• These features will imply that the economy we model is no longer efficient: Policy can improve outcomes relative to the market.

• We will conclude the course with an example of how the tools we study can be used to analyze the COVID-19 crisis.

3

Introduction, Continued

• The tools used in the RBC model became the foundation of the mainstream framework for studying macroeconomic fluctuations in the 1980s, starting with seminal work by Finn

Kydland of U.C. Santa Barbara and Edward Prescott of Arizona State University published

in Econometrica in 1982.

• The model studies fluctuations of the economy around its growth trend (business cycles) triggered by unexpected, random shocks, assuming that agents in the economy act to

optimize intertemporal objective functions under rational expectations about the future.

• In its standard versions, the analysis assumes that shocks generate departures from trend that disappear over time: For instance, an unexpected improvement in technology causes

the economy’s GDP to rise above trend for some time, but eventually the effect of the shock

disappears, and the economy is back chugging along its trend growth path.

• The figure in the next slide shows the behavior of U.S. GDP since 1947. It gives you an illustration of situations in which the standard approach can work (much of the time) but also

situations in which it will do very poorly (the aftermath of the Great Recession that followed

the Global Financial Crisis of 2007-08).

– U.S. GDP appears to have returned to its post-Great Recession trend in the aftermath of the COVID-19 recession, but the standard solution technique for many models that we will learn (log-linearization) may have problems because of the size of this shock.

4

B ill io n s o f C h ai n ed 2 01 2 D o lla rs

1950 1960 1970 1980 1990 2000 2010 2020 2,000

4,000

6,000

8,000

10,000

12,000

14,000

16,000

18,000

20,000

Real Gross Domestic ProductReal Gross Domestic Product

Shaded areas indicate U.S. recessions. Source: U.S. Bureau of Economic Analysis fred.stlouisfed.org

Introduction, Continued

• As we were saying, market outcomes are efficient in the basic RBC model.

• This happens because there are no distortions (or frictions) that would cause a benevolent social planner to choose outcomes that differ from the market ones.

• But the usefulness of the model does not depend the assumption that business cycles are triggered by technology shocks or that of an efficient model-economy.

• Many scholars have studied the consequences of other types of shocks and of departures from an efficient environment by modifying the model and the approach to it in the directions

we will talk about.

5

Introduction, Continued

• We will introduce distortions, obstacles to the smooth functioning of markets, realistic features that will allow us to tackle questions that the basic RBC setup cannot address—

including issues that have taken center stage in discussions on macroeconomics since the

Great Recession.

• Studying the RBC model will prepare us to study those more realistic models, and it will help us understand exactly when and why market outcomes in those models are not efficient, and

therefore when and why there is a role for policy in addressing those suboptimal outcomes.

– Put differently, understanding the outcomes in an efficient model-economy helps us

understand the mechanisms through which distortions lead to inefficient outcomes when

we introduce realistic features in our models, and when and how policy action can be

optimal.

6

Solving Models

• In studying the RBC model, we will pay special attention to the procedure for solving it.

• The difficulty in solving the model is a fundamental non-linearity that arises from the interaction between multiplicative elements, such as Cobb-Douglas production, and additive

elements, such as capital accumulation and depreciation.

• This non-linearity makes it impossible, in general, to solve the model without resorting to some kind of approximation.

– The only case in which this problem does not arise is when capital depreciates fully in

one period and agents utility from consumption takes the logarithmic form.

– This is a very special, unrealistic combination of assumptions.

7

Solving Models, Continued

• The solution method that we will study for the more general scenario is called log- linearization.

• It starts from the model’s optimality conditions and budget constraints and transforms them into a system of log-linear expectational difference equations in which all endogenous

variables are function of the capital stock and of the exogenous shocks that cause

fluctuations.

• Variables in the log-linearized model measure percentage deviations of original variables from their trend (or steady-state) levels.

– We will use the words trend and steady state interchangeably, with the understanding

that underlying variables are constant in steady state only if long-run growth is zero,

otherwise they are moving at their trend-growth rate.

• The approximated model can then be solved using a method known as the method of undetermined coefficients.

• An advantage of this solution method over alternatives is that it can be applied also to models in which the market outcome is not efficient.

8

Solving Models, Continued

• There are plenty of situations in which you would not want to log-linearize your model (and therefore assume that your variables always display a tendency to return to the steady state

around which you approximated the original, non-linear model).

– There are cases in which log-linearization is used also in non-stationary environments

(scenarios in which the economy does not eventually return to its steady state, or trend,

after shocks), but this is appropriate only for specific types of exercises.

• Log-linearization limits the range of questions you can study, or it can yield very misleading conclusions.

• For example, log-linearization cannot handle phenomena like the Great Recession and the years that followed and situations in which accounting for nonlinearity (like the zero—or

effective—lower bound on central bank policy interest rates) is necessary to understand

what is happening.

• But log-linearization is still used to work on many other interesting, important questions and understanding how it works also helps us understand its limitations and why alternative,

more complicated techniques become necessary in other scenarios.

9

Households in the Basic RBC Model

• Consider an economy populated by a large number of identical, infinitely lived households, all subject to the same uncertainty.

• At time t, the representative household maximizes its expected intertemporal utility from t to the infinite future, discounting utility in future periods according to a discount factor β:

Et

[ ∞∑ s=t

βs−tu(Cs)

] = Et

( ∞∑ s=t

βs−t C1−γs 1 −γ

) , 0 < β < 1, (1)

where Et denotes the expectation based on the information available at time t, ∑∞

s=t is the

summation operator for time that goes from the current period (t) all the way to infinite, and

Cs is consumption in period s (s = t, …,∞).

• We assume that this expectation is rational, i.e., the household uses optimally all the information that is available to it.

– Much macroeconomic literature studies the consequences of departures from rationality.

It is one of the many topics that, unfortunately, we do not have time to study. Michael

Woodford of Columbia University has been doing very interesting work in this area

recently.

10

The Intertemporal Utility Function

• The expression

Et

( ∞∑ s=t

βs−t C1−γs 1 −γ

) is a compact way of writing

Et

( C 1−γ t

1 −γ + β

C 1−γ t+1

1 −γ + β2

C 1−γ t+2

1 −γ + … + β∞

C 1−γ t+∞

1 −γ

) .

• Discounting by β captures the idea that households care about utility from current consumption more than they care about utility from future consumption.

– Very interesting literature has explored the consequences of different forms of discount-

ing. See, for instance, work by David Laibson of Harvard University.

11

The Intertemporal Utility Function, Continued

• γ > 0 is the coefficient of relative risk aversion:

– It measures the attitude of our representative household toward risk (uncertainty).

– If γ were equal to zero—linear utility—the household would not care about risk. It would

be perfectly indifferent between a certain level of consumption and an uncertain one.

– If you are not familiar with the concept of risk aversion, you find more information in

Appendix A.

12

The Intertemporal Utility Function, Continued

• Let us define the parameter σ ≡ 1 γ .

• This is known as the elasticity of intertemporal substitution.

– As we shall see, it measures the responsiveness of consumption to interest rate

changes: the willingness of agents to postpone or anticipate consumption across periods

in response to movement in interest rates.

• Tight connection between attitude toward uncertainty (γ) and toward intertemporal substitution (σ) is an undesirable feature of the model when studying important questions

(for instance, related to asset pricing).

• Larry Epstein of Boston University and Stanley Zin of NYU developed a framework that unties risk aversion from intertemporal substitution. Their work, published in the Journal of

Political Economy in 1989, became widely used to address important questions.

• We will stick to the basic framework, keeping in mind that it has significant limitations (see Appendix A for an example).

13

Capital Accumulation and Labor Supply

• Households can accumulate a single asset, homogeneous physical capital, Kt and

Kt+1 = (1 − δ)Kt + It, 0 < δ ≤ 1 (2)

where Kt is the capital stock with which the household begins period t, It is investment in

period t, δ is the rate of capital depreciation, and Kt+1 is the stock of capital with which the

household will begin period t + 1.

• Each household supplies a fixed amount of labor (Nt = 1) in each period in a perfectly competitive labor market.

14

The Budget Constraint

• The representative household’s consumption is constrained by:

Ct + It = r̃tKt + wt, (3)

where r̃t is the rental rate the household receives from the firms that rent its capital in a

perfectly competitive rental market, and wt is the wage (Nt = 1 implies wtNt = wt = labor

income).

– This is the household’s period budget constraint: A constraint like this applies to every

period (to every s, for s = 0, …,∞).

• (2) and (3) imply that the household’s budget constraint can be rewritten as:

Ct + Kt+1 = (1 + r̃t − δ)Kt + wt. (4)

• The problem of the household is to maximize (1) subject to (4).

• How do we solve such intertemporal optimization problem?

15

Solving the Household’s Problem: Intuitive Approach

• Let us start with in intuitive approach.

• Suppose we have a household that must decide what to do with 1 dollar in the current period (t): it can use it to buy consumption in t (let us assume that 1 unit of consumption costs 1

dollar) or it can invest it in an asset that will generate the uncertain gross return Rt+1 at time

t + 1.

• Consider what the two possible choices do for the household:

• If it uses the dollar to buy consumption, it obtains the benefit given by the increment in utility from consuming an extra unit of consumption today—the marginal utility of consumption:

u′(Ct).

• If it invests the dollar, it will receive the return Rt+1 at time t + 1. In terms of the utility increment generated by the extra consumption this allows the household to do at t + 1, this

translates into u′(Ct+1)Rt+1.

16

Notation Digression

• When we are dealing with functions of only one variable, we will denote the first derivative by using a superscript “′” and the second derivative by using a superscript “′′.”

• When we are dealing with functions of more than one variable, we will denote the first partial derivative with respect to a variable by having that variable indicated once as subscript and

the second derivative by having the variable indicated twice as subscript.

– Example: The first derivatives of the function f(x,y) with respect to x and to y are

denoted fx(x,y) and fy(x,y), respectively, and the second derivatives are fxx(x,y) and

fyy(x,y) (I am omitting cross derivatives, assuming things are clear).

• An alternative way of indicating partial first derivatives that may appear in these course slides will be to use numerical subscripts referring to the variable with respect to which we

are taking the derivative.

– Example: The first derivatives of the function f(x,y) with respect to x and to y are

denoted f1(x,y) and f2(x,y), respectively, and the second derivatives are f11(x,y) and

f22(x,y) (again omitting cross derivatives).

• Do not use “′” or “′′” superscripts when denoting derivatives of functions of more than one variable in this course.

• Why? Because f ′(x,y) does not tell anyone with respect to what variable the derivative is taken!

17

Solving the Household’s Problem: Intuitive Approach, Continued

• But the household does not know Ct+1and Rt+1 with certainty at time t, wh

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