This chapter studies how the introduction of a general class of production func tion, namely a CES, which is able to match the empirical evidence on the substi tutability between labour and capital, changes the local determinacy analysis of a neoclassical economy where fiscal policy follows a balanced-budget rule (BBR),
I obtain two sets of results. Firstly, from a theoretical point of view, I show analytically that the degree of substitutability between production factors is a key ingredient in understanding the (de)stabilising properties of a BBR. In this respect I contribute to the theoretical literature of both the BBR and the CES production function (see among others Cuo and Harrison 2004, 2008, Ciannarit- sarou, 2007, Liimemann, 2008, Cuo and Lansing, 2009). Secondly, from a policy point of view, when I calibrate the model consistently with the empirical evi dence, i.e. elasticity of substitution below unity, the instability problems that affect the Schmitt-Crohè and Uribe model are sensibly reduced for the US, the EU and the UK. In this respect, our results contribute to the current policy debate and open the door to the plausibility of adopting a BBR.
4. A ggregate S ta b ility and B a la n ced -B n d g et R u les 172
A p p en d ix A - T h e N orm alisation P rocedure
As stated before, I am using a normalised CES production function. The mean ing of the normalisation is that the class of CES production functions needs a common benchmark point. Since the elasticity of substitution is defined as a point of elasticity, I need to fix the benchmark values for the level of production, factor inputs and for the marginal rate of substitution or equivalently for the per-capita production, capital deepening and factor income share.
Therefore, I need to recalibrate the parameters B and a each time th at the elasticity of substitution a is varied in order to have the factor shares and the steady state allocations constant. In particular, when a is varied, I want to maintain the value of B equal to its CD counterpart. Furthermore, the value of
a is set such that the steady state capital income share equal to 0.7.
For our analysis I use as reference point the normalised quantities for the CD case, i.e. a = 1 and B = 1.
In order to achieve this I follow Cuo and Lansing (2009) where the parameters take the following expressions:
y C D
^ = 7--- :--- i w
(4-19)
4. A g g reg a te S ta b ility and B a la n ced -B n d g et R u les 173
where and are the steady state values of capital and labour of a CD production function. Eq. (4.18) can be easily derived following Klump and de La Crandville (2000) and Klump and Saam (2008). They show th at the value n of a generic CES production function can be normalized in the following way:
where Ko, Hq represents the values at the chosen normalization point and ckq is defined as:
where represents the calibrated value of the capital share at the normalization point. Civen that I normalize the quantities for the CD case, I have th at ai = â, Ko = and Hq = This implies th at ao = â. From this, it is
straightforward to recover the values of the normalized parameter a.
A p p en d ix B - T h e Log-Linearised S y stem
In order to study the local determinacy, I log-linearise the structural equations around the non-stochastic steady state. Here I present the log-linearised system. A lower case variable identifies Ct = log (§■). identifies the log-linearised labour tax rate. Wt is the log-linearisation real wage and ji^ the log-linearised real interest rate.
4. A g g reg a te S ta b ility and B a la n ced -B u d g et R u les 174 —rh Ct — (4.22) 0 — Wt ht (4.23) y c kt = -j^yt — hkt 4— — Ct (4.24) Ct = ypLt (4.25) ht ~ ^ iVt kt) (4.26) '^t = ~ {Vt — ht) (4.27) Vt — 01 ^ 4- (1 — a) ^ ht (4.28) where -0 = If I substitute (4.18) and (4.19) into the previous expression I obtain:
4. A g g reg a te S ta b ility and B a la n ced -B u d g et R ules 175
A p p en d ix C - D eterm in acy A nalysis
The model can be reduced to one involving only two dynamic variables, namely, consumption (non-predetermined) and capital (predetermined), as:
Ct Ct = J kt kt (4.29) where: J = a + T ^ —a r ^ —l I rv—rr'T^
The system is determinate if and only if:
Det ( J) < 0
While the system displays indeterminacy if and only if:
Det (J) > 0 n Trace (J) < 0 (4.30)
i.e. both eigenvalues of J are negative, and it is unstable if and only if:
Det (J) > 0 n Trace {J) > 0 i.e. both eigenvalues of J are positive.
4. A g g reg a te S ta b ility and B a la n ced -B u d g et R u les 176
P r o o f o f P ro p o sitio n s 1 an d 2
Determinant and trace of J are respectively:
(d - «)
- 2 (l - 5^ ) r-* + (l - 5^ ) )
Trace (J) = iP + 5) {l - ar'^) - {p + 8) {{l - r>^) (1 - â)) _ ^
a — <7T"
Sign o f th e D eterm in a n t
The sign of the determinant is the product of the sign of and (1 — a) (t^)'
2 (i -
’■'* + (l - “4?) ■
Part 1: Sign of
S + p (err — a ) cr
Part2: Sign of (1 - a) - 2 ( l - + ( l - : See Steady State section in the main text, i.e:
(1 — ü) (r^)^—2 ^1 — ^ r^ + ^ 1 — > 0 if and only if r^ G [0,r^*)
Furthermore note th at if r^ = 0.5, (1 — ü) (r^)^—2 r^+ ^1 — > 0. Putting Part 1 and part 2 together one obtains th at the determinant is posi
4. A gg reg a te S ta b ility and B a la n ced -B u d g et R u les 177 tive: i / r ' * * > - e O’ o else T G , 1)) o cr Sign o f th e Trace
The trace of J is:
Trace ( J) = (P + ^) (l ~ - ( , + ^) ( (l - r^) ( 1 - 5 ) ) _ ^
a — O T ^
It is easy to show that the conditions for Trace ( J) < 0 are:
i f a < + P O’ elseif cr > 1 + — — — ^ - < \ = p cr per — (1 — a;) (d + p) elseif cr < Ô —> Trace ( J) > 0 V G [0,1]
R e su lt 1: D eter m in a cy
The system as defined in (4.29) is determinate if and only if Det (J) < 0, hence:
i f O’ G [ü, +oo) = > G [0, min ( “ j ) U G (max , 1]
4. A g g reg a te S ta b ility and B a la n ced -B u d g et R ules 178
R esu lt 2: In d e te rm in a cy -In sta b ility
Define e = • The following four cases fully categorise all the model outcomes:
a) I f (7 e and — < the model is indeterminate if E ( — )
cr Vcr /
and determinate if ^ .
b) I f a e
P and — > the model is unstable if
E ^r^*,min and determinate if ^ min .
c) I f a E ^1 H ^ ,+ o o ^ and e > r^*, the model is indeterminate if E ( “ 5'^^*)
and determinate if ^ •
d) I f a E 4— ---- —, +0 0^ and e < the model is indeterminate if E ( “ 3^)
C hapter 5
C oncluding R em arks
This thesis examined various aspects of DSGE modeling and its empirical and policy applications.
The first chapter addressed several important issues that are being studied in the macro-financial literature. In particular, it deals with the importance of financial frictions along the business cycle. This chapter sets out a closed econ omy New Keynesian DSGE model with two distinct sets of financial frictions. The first affects the relationship between banks and depositors where the level of deposits is endogenously determined and depends on the net worth of the fi nancial institution. This is modeled along the lines of Gertler and Karadi (2011) and Gertler and Kiyotaki (2011). The second one affects the firm-bank relation ship. In this case the amount of bank lending is determined endogenously and depends on the net worth of the entrepreneur. On the modeling side, the novel feature is how this last friction is modeled. The literature assumed asymmetric
5. C onclu d ing R em arks 180
information introduced by Townsed (1979) and formalized in a DSGE frame work by Bernanke, Gertler and Gilchrist (1999). In my framework I assume perfect information, in the sense th at the bank can full information about the creditworthiness of the borrower through complete information of the balance sheet.
I compared the empirical properties of this model with two models: a model with a single financial friction on, the banking side and a plain vanilla New Keynesian model. On this side the novelty resides in the fact that I compared the models in environments with financial and non financial data. In both cases, the result provided strong evidence when an additional friction is included in the model. Moreover, variance decomposition highlighted that the capital quality shock dominates the investment shock. This is an important result since it highlights the fact th at financial shocks are the key driver of the economy.
However, this framework does not consider a third friction th at could have interesting business cycle implication: the collateral constraint in the household sector. As highlighted by lacoviello (2005) a collateral constraint tied to hous ing values has strong empirical and policy implication. An interesting extension of this exercise would be to include this extra friction. This comparison be tween three would complete the analysis of Brzoza-Brzezina & Kolasa (2012) and Brzoza-Brzezina et al. (2013). In this series of papers, the authors analyse the empirical properties of models with different financial frictions. However
5. C onclud ing R em arks 181
they make a “horizontal” comparison between models in the sense that they compare models with a single financial friction only. My approach would be to make a “vertical” comparison in the sense th at I would examine the empirical properties of models that have a set of additional financial frictions.
The second chapter examined how macroprudential policy can mitigate the effects of shocks and in particular of shocks that have a financial origin. To this extend I developed an open economy New Keynesian model with banking frictions modeled after Gertler and Karadi (2011). The key novelty is how the macroprudential policy is modeled. I assumed th at banks are charged with a penalty when they deviate from a given value of the leverage ratio. Furthermore this given leverage ratio changes with the economic conditions as in the proposed Basel III regulation. The results highlighted the effectiveness of macropruden tial policies in mitigating the effects of the shocks and in particular of financial shock. Moreover, I showed that, in terms of welfare loss, an economy with macroprudential regulation and a Taylor rule that responds to credit growth is preferred.Currently, together with the Asia-Pacific division of the International Monetary Fund, I am working in extending this framework to include more fi nancial frictions that open the door for the study of the interactions of different macroprudential policy instruments. Moreover, we will compute the optimal pol icy with estimated values of the shocks in order to have an empirical foundation for the analysis of different Taylor rules. In addition to that, the analysis will
5. C onclud ing R em ark s 182
consider different exchange rate regimes to create a toolkit that can be applied to a broad range of countries.
Finally, the last chapter studied how the indeterminacy problem studied by Schmitt-Grohe and Uribe (1997) can be solved. By generalizing the Cobb- Douglas production function to a Constant Elasticity of Substitution production function I showed th at the indeterminacy region reduces drastically. Further more I showed that the likelihood for the United States, the United Kingdom and Europe to be in the indeterminate area is sensibly reduced.
At the moment I am working on an interesting and promising extension of this framework th at will be soon published in the International Monetary Fund working paper series. This project aims to study the indeterminacy problem caused by interest rate policy using the framework of Benhabib and Eusepi (2005). Following my chapter the framework considers balanced-budget rules and it also allows for public debt as in Linnemann (2006), Schmitt-Grohe and Uribe (2007) and Kurozumi (2010).
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