# A Step-by-Step Overview of Gertler's and Karadi's "Model of Unconventional Monetary Policy"

Research Paper (undergraduate) 2017 39 Pages

## Excerpt

## Contents

1 Introduction

2 Model Description

2.1 Households

2.2 Financial Intermediaries

2.3 Credit Policy

2.4 Intermediate goods firms

2.5 Capital Producing firms

2.6 Retail Firms

2.7 Resource constraint and government policy

3 Log-linearization and the steady state relations

3.1 The Equilibrium

3.1.1 Equilibrium equations

3.2 Approximation

4 Model Analysis

4.1 The parameters

4.2 Model analysis

5 Discussion

5.1 Impact of the Gertler and Karadi (2011) paper

5.2 Criticism

6 Conclusion

7 Appendix: Taylor Approximations

7.1 Euler - optimal savings consumption . .

7.2 Stochastic Discount Rate

7.3 Arbitrage

7.4 Labour supply

7.5 Optimal leverage ratio

7.6 Growth rate of bank’s net wealth

7.7 Optimal utilization rate

7.8 Production Function

7.9 Net capital

7.10 Capital Accumulation

7.11 Fisher Relation

7.12 Effective Capital

7.13 Wages

## 1 Introduction

In general, the Federal Reserve has three types of instruments in order to conduct monetary policy: setting the discount rate, open market operations, and reserve requirements. The goal of the monetary policy is to maintain stable prices, ultimately this leads to maximum employment and stable economic growth. However, during an economic recession the gen- eral tools are not effective enough to ensure that the Federal Reserve meets its goals. For example, the Fed can lower its Federal Funds Rate to stimulate consumption and to slow down savings. Figure 1 shows the Federal Funds Rate over the last ten years.

Figure 1: Effective Federal Funds Rate 1995-2015. Source: Federal Reserve Economic Data (2016)

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It is clear that the Fed lowered their interest rate substantially during the financial crisis of 2008, nearly reaching its minimum rate of 0.0 percent. This shows that, at this point, altering the discount rate was not effective anymore. The Fed needed to find another way to reach their objectives. At the end of 2007, the Fed started to inject credit into private markets (Gertler and Karadi (2011)). They also started to purchase securities in the open market, next to the usual government bonds. These securities are mostly debt instruments and include, for instance, mortgage backed securities. These actions are often referred to as unconventional monetary policy. Its impact can be easily seen in Figure 2, this figure shows the amount of securities held by the Fed, which increased substantially since the financial crisis of 2008.

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Figure 2: Securities held by Federal Reserve. Source: Federal Reserve Economic Data(2016)

In the academic literature many authors have modelled conventional monetary policy, but these models do not account for disruptions in financial markets, which ultimately would lead to interventions by the Fed. Models that incorporate unconventional monetary policy were only qualitative, until Gertler and Karadi (2011) published their model of unconven- tional monetary policy. Their model includes financial intermediaries which are constrained by an agency problem that limits their leverage ratios. However, the central bank in this model is does not face this agency problem since its risk of defaulting is close to zero. The central bank acts as an intermediary between savers and investors. Thus, during times of financial crises, the central bank can support credit flows. This is exactly the kind of uncon- ventional monetary policy that we want to investigate. Our goal in this paper is to replicate the model of Gertler and Karadi (2011) and to give an elaborate overview of the way in which this model works. Instead of showing the final results, we provide a step-by-step overview of all the necessary calculations and derivations that lead to the ultimate model. Our last objective is to explain the relevance and the impact that the paper of Gertler and Karadi (2011) had.

First, we describe the model with separate sections for the different agents that play a role in the model: households, financial intermediaries, intermediate goods firms, capital producing firms, retail firms, and the government (including the central bank). Secondly, we perform a Taylor approximation and import out model into Matlab. Thirdly, the model is analysed and the results are given. Fourthly, the impact of the Gertler and Karadi (2011) paper is examined. Lastly, the paper is concluded with some final remarks.

## 2 Model Description

This section will focus on explaining the setup of the model and includes a detailed de- scription of the agents and their preference relations. The presented model is based on the standard monetary DSGE models developed by Christiano et al. (2005) and Smets and Wouters (2007), with addition of financial intermediaries. The inclusion of the financial intermediaries allows for observing the effectiveness of government policy in times of crisis.

### 2.1 Households

Within each household there are two types of members: bankers and workers. Workers are responsible for supplying labour, bankers are in charge of financial intermediaries. Both workers and bankers transfer their earnings back to the household, which are either spend on consumption or saved by lending to financial intermediaries or the government. As a consequence, the household preference setting can be described as a standard allocation problem; the choice between consumption, leisure and work.

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Where:

*C t*: consumption at time *t*

1+ *φ L t*: household supply of labour, characterised by constant elasticity of substi- tution (CRRA utility)

*β t*: discount rate

*h*: habit parameter, past consumption has an effect on present consumption, as a result utility function is non monotone in the consumption at any given period

*χ*: relative utility weight of labour in preference relation

Household consumption is limited by their income and savings. Agents are awarded a real wage (*W t*) for each unit of labour they supply. Furthermore, households receive net payouts from ownership of both non-financial and financial firms (Π *t*) and lump sum taxes (*T t*). Financial intermediaries and the government supply short term debt (*B t*), that pays gross return of * R t*. As a result, the households’ budget constraint can be formulated as follows:

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Maximizing household preference (1) over its budget constraint (2) we obtain the following constrained maximisation problem:

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Joining (3a) and (3b), provides the optimum labour supply (4):

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In order to find the optimum savings/consumption relation (5), the problem is further derived w.r.t. short term debt (*B t* +1):

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Once all necessary equilibrium relations (5, 4) for the model household are established we move to the financial intermediaries.

### 2.2 Financial Intermediaries

The financial intermediaries obtain deposits from households and lend these to non-financial firms; so they make sure that the money flows from savers to investors. They hold long term assets which are paid for with short term liabilities, these liabilities are larger than their capital, so they create leverage. This leverage leads to agency costs, which is discussed later on. Furthermore, investments banks as well as commercial banks are part of the financial intermediaries

The intermediary balance sheet is given by:

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*Q _{t}*: relative price of each claim

*S _{jt}*: quantity of financial claims on non-financial firms that the intermediary holds

*N _{jt}*: amount of wealth intermediary

*j*has at the end of period t

B _{jt} +1: deposits the intermediary obtains from the households

The intermediary’s debt consist of all of the deposits from the households * B jt* +1, which pay the real gross return * R t* +1. The amount of wealth of the intermediary * N jt* can be thought of as their equity capital. Lastly, the financial intermediaries earn the stochastic return * R kt* 1 on their assets. The amount of wealth at period *t* + 1 equals the earnings on assets minus the interest payments to the households.

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We rearrange (6) to obtain:

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Combining (7) and (8) yields:

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From 9 it can be concluded that the growth in equity, above the riskless return * R t* +1, depends on the risk premium * R kt* +1 *− R t* +1 and the total amount of assets * Q t S jt*. The banker applies a stochastic discount at time * t* to earnings at time * t* + * i*, we denote this discount by *β i* Λ *t,t* + *i*. The banker will only fund assets whenever the discounted return is higher than the discounted cost of borrowing. Intermediary assets earn the stochastic return * R kt* +1, and bankers pay * R t* +1 to the households which is thus the cost of borrowing. So,

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As long as 10 holds it is profitable for the banker to keep building assets. Ultimately, the banker wants to maximise expected terminal wealth, which is given by:

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The banker cannot keep on borrowing funds to increase his assets so we have to put a limit on its borrowing ability. The following moral hazard problem is introduced: at the beginning of the period the banker can divert the fraction *λ* of his available funds from the project and transfer these back to his household. So it is possible for the depositors to force the intermediary into bankruptcy and recover the remaining fraction of the assets, 1 *− λ*. For lenders to be willing to supply funds, the possible loss from diverting assets should be bigger than the gain from diverting these assets:

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We can express *V jt* as:

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With:

[Abbildung in dieser Leseprobe nicht enthalten]gross growth rate in assets between * t* and * t* + * i*

[Abbildung in dieser Leseprobe nicht enthalten]: gross growth rate of net worth *N jt *

*ν t*: the expected discounted marginal gain to the banker of expanding assets *Q t S jt* by a unit, holding *N jt* constant

*η t*: the expected discounted value of having another unit of *N jt*, holding *S jt* constant Financial intermediaries will keep on borrowing until the marginal gain of expanding assets, *ν t*, becomes zero. However, the intermediary is constrained by the agency problem and the amount that it can borrow is dependent on its equity capital. One of the most important differences between the financial intermediaries and the central bank is that the central bank is not constrained by this problem.

We can combine (12) and (13) to obtain:

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When this constraint is binding:

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Where *φ* is the private leverage ratio, or the ratio of privately intermediated assets to equity. (15) confirms that the borrowing capacity of the intermediary depends on its equity capital. Furthermore, when *N jt >* 0, the constraint (14) is only binding if 0 *< ν t < λ*. Since *ν t* is greater than zero, it is profitable for the intermediary to increase assets. As *ν t* increases, the opportunity cost to the banker from the risk of defaulting increases. Thus, the households will tolerate a higher leverage ratio whenever *ν t* increases.

Using (15), the evolution of the banker’s net worth can be described as:

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From this it follows that:

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The components of *φ t* do not depend on firm-specific factors, * j*, so total intermediary demand for assets can be determined by summing across individual demands, this gives:

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With:

*S t*: aggregate quantity of intermediary assets *N t*: aggregate intermediary capital

During times of crisis the wealth of the intermediaries, *N t*,will sharply decline, this results in a lower demand for assets. We can separate *N t* into two parts; the net worth of existing bankers, *N et*, and the net worth of new bankers, *N nt*. So,

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Furthermore, only the fraction *θ* of bankers at t-1 survive until period t. From (16) it follows that:

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Note that * N et* is most affected by fluctuations in * R kt* and that the leverage ratio *φ t* increases this impact on * N et*. In section 1, the households, it was mentioned that exiting bankers provide start up capital to new bankers. This start up capital is equal to a fraction of the asset value that the exiting banker intermediated in his final period. So, the start up capital depends on the scale at which the exiting banker was operating. The total final period assets of exiting bankers equals (1 *− θ*) *Q t S t −* 1. Each period the household transfers the fraction[Abbildung in dieser Leseprobe nicht enthalten] ofthisvaluetothenewbankers.Summingtheseupgives:

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Combining (20), (21), and (22) results in:

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Note that the steady state leverage ratio,[Abbildung in dieser Leseprobe nicht enthalten],isdependenton *ω*.

### 2.3 Credit Policy

The total value of intermediated assets consists of privately and governmental intermediated assets:

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The central bank acts as an intermediary between the households and non-financial firms; households can buy treasury bills or other types of government debt and obtain the riskless rate *R t* +1 as return. Non-financial firms can borrow money from the central bank and pay interest, *R kt* +1. This intermediation by the government comes at a certain efficiency cost of *τ* per unit. These costs emerge because the central bank has to search for valuable private sector investments, for non-financial firms it is also a reflection of the costs of obtaining funds through government debt. However, the government will always honor its debt and government debt is thus risk-free. Therefore, there are no agency costs involved for the operations of the central bank, unlike the financial intermediaries.

Suppose that the government is willing to fund a fraction *ψ t* of intermediaries assets, thus:

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