The Tragedy of the Commons and Inflation Bias in the Euro Area

Abstract

Central bank credit has expanded dramatically in some of the Euro Area member countries since the beginning of the financial crisis. This paper makes two contributions to understand this stylized fact. First, we discuss a simple model of monetary policy that includes (i) a credit channel and (ii) a common pool problem in a monetary union. We illustrate that the interaction of the two elements leads to an inflation bias that is independent of the standard time-inconsistency bias. Secondly, we present an institutional analysis that is consistent with the view of fragmented monetary policy and empirical evidence that illustrates the heterogeneity of central bank credit expansion.

Appendices

Appendix A: The Interaction with a Barro-Gordon Time Inconsistency Problem

In this appendix we analyze the interaction between the two new elements–a credit channel and a common pool problem–with the standard time inconsistency problem that derives from the Phillips curve trade-off in a Barro-Gordon setting. We show that the results presented above are independent from this other classical inflation bias in the literature.

1.1 A Single Country

We keep the notation as above, and add the expectation about future inflation denoted by π ^e. The utility function and output in a single country are as follows:

$$ U\left(y,\pi \right)=\lambda \left(y-\overline{y_n}\right)-\frac{1}{2}{\pi}^2 $$

(6)

$$ y=\overline{y_n}+\alpha \left(\pi -{\pi}^e\right)+{y}_c\left(\Delta d\right) $$

(7)

with $\frac{\partial \pi }{\partial \Delta d}>0$ and Δm = Δd. Furthermore, we assume that agents are characterized by rational expectations.

Lemma 2

In a single country with a Barro-Gordon time inconsistency problem and a credit channel of monetary policy, the inflation bias is: $\lambda \left(\alpha +\frac{\partial y}{\partial \Delta m}\right)$.

Proof

$\mathrm{ARG}\ {\mathrm{MAX}}_{\Delta m}U=\lambda \left(\alpha \left(\Delta m-{\pi}^e\right)\right)+{y}_c\left(\Delta m\right)-\frac{1}{2}{\left(\Delta m\right)}^2;\frac{\partial U}{\partial \Delta m}=\lambda \left(\alpha +\frac{\partial y}{\partial \Delta m}\right)-\Delta m=0;\Delta {m}^{*}=\lambda \left(\alpha +\frac{\partial y}{\partial \Delta m}\right).$

The optimal inflation $\Delta {m}^{*}=\lambda \left(\alpha +\frac{\partial y}{\partial \Delta m}\right)$ is larger than zero, and larger than the standard Barro-Gordon result, which is Δm* = λα in the simple setting. The existence of a credit channel adds a further motive to conduct expansionary monetary policy.

1.2 Currency Union

Now consider, again, the same setup for a currency union. Utility and output functions are given as follows:

$$ U\left({y}_i,{\pi}_i\right)=\lambda \left({y}_i-\overline{y_n}\right)-\frac{1}{2}{\pi}_i^2 $$

(8)

$$ {y}_i=\overline{y_n}+\alpha \left({\pi}_i-{\pi}^e\right)+{y}_c\left(\Delta {d}_i\right) $$

(9)

with $\frac{\partial {y}_i}{\partial \Delta {d}_i}>0$. We make the same assumptions as above, namely, π _i  = π, and $\pi =\frac{1}{n}{\displaystyle {\sum}_{i=1}^n\Delta {m}_i}$, as well as Δm _i  = Δd _i .

Proposition 2

In a currency union with a Barro-Gordon time inconsistency problem and a credit channel of monetary policy, the inflation bias is larger than in a single country.

Proof

$\mathrm{ARG}\ {\mathrm{MAX}}_{\Delta {m}_i}{U}_i=\lambda \left(\alpha \left(\frac{1}{n}{\displaystyle {\sum}_{i=1}^n\Delta {m}_i-{\pi}^e}\right)+{y}_c\left(\Delta {m}_i\right)\right)-\frac{1}{2}{\left(\frac{1}{n}{\displaystyle {\sum}_{i=1}^n\Delta {m}_i}\right)}^2;\frac{\partial {U}_i}{\partial \Delta {m}_i}=\frac{1}{n^2}\left({\displaystyle {\sum}_{\begin{array}{c}\hfill j=1\hfill \\ {}\hfill j\ne i\hfill \end{array}}^n\Delta {m}_j-{n}^2\frac{\partial {y}_i}{\partial \Delta {m}_i}\lambda - n\alpha \lambda +\Delta {m}_i}\right)=0;\Delta {m}_i^{\ast }= n\lambda \left(\alpha +n\frac{\partial {y}_i}{\partial \Delta {m}_i}\right)-{\displaystyle {\sum}_{\begin{array}{c}\hfill j=1\hfill \\ {}\hfill j\ne i\hfill \end{array}}^n\Delta {m}_j}; In\ symmetric\ equilibrium:\Delta {m}^{\ast }=\lambda \left(\alpha +n\frac{\partial y}{\partial \Delta m}\right);\lambda \left(\alpha +n\frac{\partial y}{\partial \Delta m}\right)-\lambda \left(\alpha +\frac{\partial y}{\partial \Delta m}\right)=\frac{\partial y}{\partial \Delta m}\lambda \left(n-1\right)>0.$

The inflation in equilibrium will be $\Delta {m}^{\ast }=\lambda \left(\alpha +n\frac{\partial y}{\partial \Delta m}\right)$. Note that the original Barro-Gordon inflation bias is unaffected by our extensions. When comparing the optimal inflation rate in the currency union of the main paper (without Barro-Gordon) and the appendix, we get exactly the standard inflation bias explaining the difference:

Corollary

The tragedy of the commons does not affect the Barro-Gordon time inconsistency bias.

Proof

In a currency union without time inconsistency problem: $\Delta {m}_i^{\ast }= n\lambda \frac{\partial {y}_i}{\partial \Delta {m}_i};$ in a currency union with time inconsistency problem: $\Delta {m}^{\ast }=\lambda \left(\alpha +n\frac{\partial y}{\partial \Delta m}\right); n\lambda \frac{\partial {y}_i}{\partial \Delta {m}_i}-\lambda \left(\alpha +n\frac{\partial y}{\partial \Delta m}\right)=\lambda \alpha .$

The intuition for this corollary can be illustrated by analyzing the effect of a currency union on the marginal cost and benefit from inflation. As the benefits from inflation in the Barro-Gordon model derive from the impact of inflation on wages, the currency union will not affect the trade-off between the output and inflation. Printing more money will be associated with the average cost in terms of inflation, as above. But it will also lead only to the average benefit. As both are aligned, there is no additional incentive for printing money to make use of the Phillips-curve trade off.

Appendix B: Country-level Figures

Fig. 4

a-b central bank credit and unemployment

Fig. 5

a-b central bank credit and total bank lending

Appendix C: Monetary Policy of Major Central Banks

Fig. 6

Monetary expansion, unemployment and inflation

Appendix D: Data Sources

Table 1 Data sources