Did the Unconventional Monetary Policy of the U.S. Hurt Emerging Markets?

Abstract

Policymakers in emerging markets complained that the unconventional US monetary policy response to the Great Recession hurt their economies. US policymakers responded that the policy was geared toward conditions in the US, and that a strong US economy benefited everyone. Here we evaluate these claims in a two country model of the US and an emerging market country, bombarded by shocks to the net worth of US banks. Our model allows us to calculate a “passive equilibrium” in which US monetary policy does not respond to the shock in any way. Then, we calculate a “self oriented” equilibrium in which quantitative easing is set optimally to maximize US welfare, and a “cooperative” equilibrium in which quantitative easing and the monetary policy in the emerging market country are set to maximize joint welfare. Comparing welfare in these equilibria, we find that the self oriented US monetary policy benefits both countries, and that cooperation brings little further improvements in welfare.

Appendices

Appendix A:: Monetary Policy, Terms of Trade and Real Exchange Rate, and Market Clearing

A.1 Monetary Policy

In the US, the central bank has two independent monetary policy instruments: the net nominal interest rate on reserves $i_{t}^{us}\geq 0$, and the stock of nominal reserves $M_{t}^{us}>0$. In the absence of government bonds, the central bank injects reserves via lump-sum transfers $T_{t}^{us}$. The nominal stock of reserves thus evolves according to

$$ M_{t}^{us}=(1+i_{t-1}^{us})M_{t-1}^{us}+\mathcal{P}_{t}^{us}T_{t}^{us} $$

In the EM, the central bank sets the net nominal interest rate on bank deposit $i_{b,t}^{em}\geq 0$ by following an inflation-targeting Taylor rule with interest-rate smoothing

$$ i_{b,t}^{em}=\rho_{i}i_{b,t-1}^{em}+(1-\rho_{i})\left[\bar{i}_{b}^{em}+\omega_{\Pi}\left( {\Pi}_{t}^{em}-1\right)\right] $$

where $\bar {i}_{b}^{em}$ is the steady state net nominal interest rate, and ${\Pi }_{t}^{em}\equiv \mathit {\frac {\mathcal {P}_{t}^{em}}{\mathcal {P}_{t-1}^{em}}}$ is the gross inflation rate of the consumption good.

A.2 Terms of Trade and Real Exchange Rate

We assume that both US and EM final goods are sold in international markets and the law of one price holds. Let $NOM_{t}^{em}$ denote the EM’s nominal exchange rate, defined as the EM price of the US dollar. Then the law of one price implies

$$ P_{h,t}^{em}=NOM_{t}^{em}P_{f,t}^{us} $$

$$ P_{f,t}^{em}=NOM_{t}^{em}P_{h,t}^{us} $$

where $P_{h,t}^{em}$ is the EM price of the home good (the EM good), $P_{f,t}^{us}$ is the US price of the foreign good (the EM good), $P_{f,t}^{em}$ is the EM price of the foreign good (the US good), and $P_{h,t}^{us}$ is the US price of the home good (the US good).

Let $TOT_{t}^{em}\equiv \frac {P_{f,t}^{em}}{P_{h,t}^{em}}$ be the EM’s terms of trade (analogously, $TOT_{t}^{us}\equiv \frac {P_{f,t}^{us}}{P_{h,t}^{us}}$ is the US’s terms of trade). Then we have

$$ TOT_{t}^{em}=\frac{NOM_{t}^{em}P_{h,t}^{us}}{NOM_{t}^{em}P_{f,t}^{us}}=\frac{P_{h,t}^{us}}{P_{f,t}^{us}}=\frac{1}{TOT_{t}^{us}} $$

Given the definitions of

$$\frac{P_{h,t}^{em}}{\mathcal{P}_{t}^{em}}\equiv\left[\gamma+\left( 1-\gamma\right)\left( TOT_{t}^{em}\right)^{1-\rho}\right]^{\frac{1}{\rho-1}}$$

and

$$\frac{P_{h,t}^{us}}{\mathcal{P}_{t}^{us}}\equiv\left[\gamma+\left( 1-\gamma\right)\left( TOT_{t}^{us}\right)^{1-\rho}\right]^{\frac{1}{\rho-1}}$$

we obtain

$$ TOT_{t}^{em}=\frac{\frac{P_{h,t}^{us}}{\mathcal{P}_{t}^{us}}}{\frac{P_{h,t}^{em}}{\mathcal{P}_{t}^{em}}}\epsilon_{t}^{em}=\left[\frac{\gamma+\left( 1-\gamma\right)\left( TOT_{t}^{us}\right)^{1-\rho}}{\gamma+\left( 1-\gamma\right)\left( TOT_{t}^{em}\right)^{1-\rho}}\right]^{\frac{1}{\rho-1}}\epsilon_{t}^{em} $$

$$ {\Pi}_{t}^{em}=\pi_{t}^{em}\left[\frac{\gamma+\left( 1-\gamma\right)\left( TOT_{t}^{em}\right)^{1-\rho}}{\gamma+\left( 1-\gamma\right)\left( TOT_{t-1}^{em}\right)^{1-\rho}}\right]^{\frac{1}{1-\rho}} $$

$$ {\Pi}_{t}^{us}=\pi_{t}^{us}\left[\frac{\gamma+\left( 1-\gamma\right)\left( TOT_{t}^{us}\right)^{1-\rho}}{\gamma+\left( 1-\gamma\right)\left( TOT_{t-1}^{us}\right)^{1-\rho}}\right]^{\frac{1}{1-\rho}} $$

where ${\Pi }_{t}^{em}=\frac {\mathcal {P}_{t}^{em}}{\mathcal {P}_{t-1}^{em}}$, $\pi _{t}^{em}=\frac {P_{h,t}^{em}}{P_{h,t-1}^{em}}$, ${\Pi }_{t}^{us}=\frac {\mathcal {P}_{t}^{us}}{\mathcal {P}_{t-1}^{us}}$, $\pi _{t}^{us}=\frac {P_{h,t}^{us}}{P_{h,t-1}^{us}}$, and $\epsilon _{t}^{em}\equiv \frac {NOM_{t}^{em}\mathcal {P}_{t}^{us}}{\mathcal {P}_{t}^{em}}$ is the EM’s real exchange rate.

A.3 Market Clearing

US Market Clearing –

The final good is divided between consumption, investment, cost of banking, and export. The goods market clearing condition is thus given by

$$ Y_{t}^{us}=\left( \frac{P_{h,t}^{us}}{\mathcal{P}_{t}^{us}}\right)^{-\rho}\gamma\left\{ c_{t}^{us}+\left[1 + \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{us}}{I^{us}}-1\right)^{2}\right]I_{t}^{us}+{\varGamma}\left( q_{t}^{us}K_{t}^{us}+l_{t}^{em},m_{t}^{us}\right)\right\} +{\Xi}_{t}^{us} $$

where

$$ {\Xi}_{t}^{us}=\left( \frac{P_{f,t}^{em}}{\mathcal{P}_{t}^{em}}\right)^{-\rho}\left( 1-\gamma\right)\left\{ c_{t}^{em}+\left[1 + \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{em}}{I^{em}}-1\right)^{2}\right]I_{t}^{em}+\left( \frac{\kappa}{2}\right){x_{t}^{2}}q_{t}^{em}K_{t}^{em}\right\} $$

The real trade balance is defined as

$$ \begin{array}{@{}rcl@{}} && TB_{t}^{us}{=}\left( \frac{P_{h,t}^{us}}{\mathcal{P}_{t}^{us}}\right){\Xi}_{t}^{us}-\left( \frac{P_{f,t}^{us}}{\mathcal{P}_{t}^{us}}\right)^{1{-}\rho}\left( 1-\gamma\right)\left\{ c_{t}^{us}+\left[1 {+} \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{us}}{I^{us}}-1\right)^{2}\right]\right.\\ &&\qquad\qquad\times\left.{\vphantom{\left[1 {+} \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{us}}{I^{us}}-1\right)^{2}\right]}} I_{t}^{us}{+}{\varGamma}\left( q_{t}^{us}K_{t}^{us}+l_{t}^{em},m_{t}^{us}\right)\right\} \end{array} $$

The reserve market clearing condition is

$$ m_{t}^{us}=\frac{M_{t}^{us}}{\mathcal{P}_{t}^{us}} $$

EM Market Clearing –

The final good is divided between consumption, investment, cost of foreign borrowing, and export. The goods market clearing condition is thus given by

$$ Y_{t}^{em}{=}\left( \frac{P_{h,t}^{em}}{\mathcal{P}_{t}^{em}}\right)^{{-}\rho}\gamma\left\{ c_{t}^{em}{+}\left[1 {+} \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{em}}{I^{em}}{-}1\right)^{2}\right]I_{t}^{em}+\left( \frac{\kappa}{2}\right){x_{t}^{2}}q_{t}^{em}K_{t}^{em}\right\} +{\Xi}_{t}^{em} $$

where

$$ {\Xi}_{t}^{em}=\left( \frac{P_{f,t}^{us}}{\mathcal{P}_{t}^{us}}\right)^{-\rho}\left( 1-\gamma\right)\left\{ c_{t}^{us}+\left[1 + \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{us}}{I^{us}}-1\right)^{2}\right]I_{t}^{us}+{\varGamma}\left( q_{t}^{us}K_{t}^{us}+l_{t}^{em},m_{t}^{us}\right)\right\} $$

The real trade balance is

$$ \begin{array}{@{}rcl@{}} &&TB_{t}^{em}=\left( \frac{P_{h,t}^{em}}{\mathcal{P}_{t}^{em}}\right){\Xi}_{t}^{em}-\left( \frac{P_{f,t}^{em}}{\mathcal{P}_{t}^{em}}\right)^{1-\rho}\left( 1-\gamma\right)\left\{ c_{t}^{em}+\left[1 + \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{em}}{I^{em}}-1\right)^{2}\right]\right.\\ &&\qquad\qquad\left.{\vphantom{c_{t}^{em}+\left[1 + \frac{\kappa_{I}}{2}\left( \frac{I_{t}^{em}}{I^{em}}-1\right)^{2}\right]}}\times I_{t}^{em}+\left( \frac{\kappa}{2}\right){x_{t}^{2}}q_{t}^{em}K_{t}^{em}\right\} \end{array} $$

The real current account is given by

$$ CA_{t}^{em}=TB_{t}^{em}+\epsilon_{t}^{em}\left[\left( R_{d,t}^{us}-1\right)d_{t-1}^{em}-\left( R_{l,t}^{us}-1\right)l_{t-1}^{em}\right] $$

The balance of payments implies that the real current account equals net foreign capital outflow

$$ CA_{t}^{em}=\epsilon_{t}^{em}\left[\left( d_{t}^{em}-d_{t-1}^{em}\right)-\left( l_{t}^{em}-l_{t-1}^{em}\right)\right] $$

Appendix B: The Banker’s Optimization Problem

US Banker’s Problem –

Given that bankers pay dividends to their household only when they exit, a banker j in period t maximizes the expected discounted value of its terminal dividends. Switching to the recursive formulation for the franchise value of bank j at the end of period t, the banker’s problem can be written as

$$ V_{jt}^{us}=\max_{\left\{K_{jt}^{us}, l_{jt}^{em}, m_{jt}^{us}, d_{jt}^{us}, d_{jt}^{em}\right\} }E_{t}\left\{ {\varLambda}_{t,t+1}^{us}\left[\left( 1-\sigma^{us}\right)n_{jt+1}^{us}+\sigma^{us}V_{jt+1}^{us}\right]\right\} $$

(29)

subject to the balance sheet condition and the capital requirement

$$ q_{t}^{us}K_{jt}^{us}+l{}_{jt}^{em}+m_{jt}^{us}+{\varGamma}(q_{t}^{us}K_{jt}^{us}+l_{jt}^{em}, m_{jt}^{us})=d_{jt}^{us}+d_{jt}^{em}+n_{jt}^{us} $$

(30)

$$ n_{jt}^{us}=R_{k,t}^{us}q_{t-1}^{us}K_{jt-1}^{us}+R_{l,t}^{us}l_{jt-1}^{em}+R_{m,t}^{us}m_{jt-1}^{us}-R_{d,t}^{us}d_{jt-1}^{us}-R_{d,t}^{us}d_{jt-1}^{em} $$

(31)

$$ n_{jt}^{us}\geq\theta^{us}\left( q_{t}^{us}K_{jt}^{us}+l_{jt}^{em}\right) $$

where the functional form of the bank’s cost of making loans is given by

$$ {\varGamma}\left( q_{t}^{us}K_{jt}^{us}+l_{jt}^{em},m_{jt}^{us}\right)=\kappa_{m}^{us}\left( \frac{q_{t}^{us}K_{jt}^{us}+l_{jt}^{em}}{m_{jt}^{us}}\right)^{\varsigma_{m}^{us}}\left( q_{t}^{us}K_{jt}^{us}+l_{jt}^{em}\right) $$

with $\kappa _{m}^{us},\varsigma _{m}^{us}>0$. To solve this problem, we first guess that the value function is linear in net worth as

$$ V_{jt}^{us}=\psi_{t}^{us}n_{jt}^{us} $$

Then we can write the value function (29) as

$$ \psi_{t}^{us}\equiv\frac{V_{jt}^{us}}{n_{jt}^{us}}=E_{t}\left\{ {\varLambda}_{t,t+1}^{us}\left[\left( 1-\sigma^{us}+\sigma^{us}\psi_{t+1}^{us}\right)\frac{n_{jt+1}^{us}}{n_{jt}^{us}}\right]\right\} $$

Using the definitions of leverage multiple $\phi _{jt}^{us}=\frac {q_{t}^{us}K_{jt}^{us}+l_{jt}^{em}}{n_{jt}^{us}}$, ratio of loans made to EM banks to net worth $\phi _{ljt}^{us}=\frac {l_{jt}^{em}}{n_{jt}^{us}}$, and ratio of reserves to bank assets $x_{jt}^{us}=\frac {m_{jt}^{us}}{q_{t}^{us}K_{jt}^{us}+l_{jt}^{em}}$, and the balance sheet condition (30) and (31), we have

$$ \begin{array}{@{}rcl@{}} &&\frac{n_{jt+1}^{us}}{n_{jt}^{us}}=\left( R_{k,t+1}^{us}-R_{d,t+1}^{us}\right)\phi_{jt}^{us}+\left( R_{l,t+1}^{us}-R_{k,t+1}^{us}\right)\phi_{ljt}^{us}\\ &&\qquad\qquad+\left( R_{m,t+1}^{us}-R_{d,t+1}^{us}\right)\phi_{jt}^{us}x_{jt}^{us}+\left[1-\kappa_{m}^{us}(x_{jt}^{us})^{-\varsigma_{m}^{us}}\phi_{jt}^{us}\right]R_{d,t+1}^{us} \end{array} $$

Then the banker’s problem can be rewritten as

$$ \psi_{t}^{us}{=}\max_{\left\{ \phi_{jt}^{us}, \phi_{ljt}^{us}, x_{jt}^{us}\right\} }\left\{ \mu_{t}^{us}\phi_{jt}^{us}+\mu_{lt}^{us}\phi_{ljt}^{us}+\mu_{mt}^{us}\phi_{jt}^{us}x_{jt}^{us}+\left[1-\kappa_{m}^{us}(x_{jt}^{us})^{-\varsigma_{m}^{us}}\phi_{jt}^{us}\right]\upsilon_{t}^{us}\right\} $$

subject to

$$ 1\geq\theta^{us}\phi_{jt}^{us} $$

(32)

where

$$ \mu_{t}^{us}=E_{t}\left[{\Omega}_{t+1}^{us}\left( R_{k,t+1}^{us}-R_{d,t+1}^{us}\right)\right] $$

$$ \mu_{lt}^{us}=E_{t}\left[{\Omega}_{t+1}^{us}\left( R_{l,t+1}^{us}-R_{k,t+1}^{us}\right)\right] $$

$$ \mu_{mt}^{us}=E_{t}\left[{\Omega}_{t+1}^{us}\left( R_{m,t+1}^{us}-R_{d,t+1}^{us}\right)\right] $$

$$ \upsilon_{t}^{us}=E_{t}\left( {\Omega}_{t+1}^{us}R_{d,t+1}^{us}\right) $$

$$ {\Omega}_{t+1}^{us}={\varLambda}_{t,t+1}^{us}\left( 1-\sigma^{us}+\sigma^{us}\psi_{t+1}^{us}\right) $$

Let $\lambda _{cjt}^{us}$ be the Lagrangian multiplier for the capital requirement (32). Then using the Lagrangian

$$ \mathcal{L=}\mu_{t}^{us}\phi_{jt}^{us}+\mu_{lt}^{us}\phi_{ljt}^{us}+\mu_{mt}^{us}\phi_{jt}^{us}x_{jt}^{us}+\left[1-\kappa_{m}^{us}(x_{jt}^{us})^{-\varsigma_{m}^{us}}\phi_{jt}^{us}\right]\upsilon_{t}^{us}+\lambda_{cjt}^{us}\left( 1-\theta^{us}\phi_{jt}^{us}\right) $$

The first order conditions with respect to $\left \{ \phi _{jt}^{us}, \phi _{ljt}^{us}, x_{jt}^{us}\right \}$ imply

$$ \mu_{t}^{us}+\mu_{mt}^{us}x_{jt}^{us}-\kappa_{m}^{us}(x_{jt}^{us})^{-\varsigma_{m}^{us}}\upsilon_{t}^{us}=\theta^{us}\lambda_{cjt}^{us} $$

(33)

$$ \mu_{lt}^{us}=0 $$

$$ \mu_{mt}^{us}=-\kappa_{m}^{us}\varsigma_{m}^{us}(x_{jt}^{us})^{-\left( \varsigma_{m}^{us}+1\right)}\upsilon_{t}^{us} $$

(34)

From Eq. 34, we can determine the ratio of reserves to assets of the bank as

$$ x_{jt}^{us}=\left( -\frac{\mu_{mt}^{us}}{\kappa_{m}^{us}\varsigma_{m}^{us}\upsilon_{t}^{us}}\right)^{-\frac{1}{\varsigma_{m}^{us}+1}} $$

(35)

Combining (33) and (34) yields

$$ \lambda_{cjt}^{us}=\frac{\mu_{t}^{us}-\kappa_{m}^{us}\left( \varsigma_{m}^{us}+1\right)(x_{jt}^{us})^{-\varsigma_{m}^{us}}\upsilon_{t}^{us}}{\theta^{us}} $$

(36)

Given that $\mu _{t}^{us},\mu _{mt}^{us}$ and $\upsilon _{t}^{us}$ are not bank-specific, Eqs. 35 and 36 imply that $x_{jt}^{us}=x_{t}^{us}$ and $\lambda _{cjt}^{us}=\lambda _{ct}^{us}$. When the capital requirement is binding, we have

$$ \phi_{jt}^{us}=\frac{1}{\theta^{us}} $$

This implies that the leverage multiple $\phi _{jt}^{us}=\phi _{t}^{us}$ is common across banks. Accordingly, the aggregate assets in banking sector can be expressed in terms of the aggregate net worth

$$ q_{t}^{us}K_{t}^{us}+l_{t}^{em}=\phi_{t}^{us}n_{t}^{us} $$

An equation of motion for the aggregate net worth $n_{t}^{us}$ is the sum of the net worth of existing bankers $n_{et}^{us}$, and the net worth of entering bankers $n_{nt}^{us}$

$$ n_{t}^{us}=n_{et}^{us}+n_{nt}^{us} $$

The net worth of existing bankers equals the earnings on assets and reserves held last period net the costs of liabilities, multiplied by the fraction of bankers that survive until present period σ^us

$$ n_{et}^{us}=\sigma^{us}\left( R_{k,t}^{us}q_{t-1}^{us}K_{t-1}^{us}+R_{l,t}^{us}l_{t-1}^{em}+R_{m,t}^{us}m_{t-1}^{us}-R_{d,t}^{us}d_{t-1}^{us}-R_{d,t}^{us}d_{t-1}^{em}\right) $$

The household transfers to each new banker, a fraction $\frac {\xi ^{us}}{\left (1-\sigma ^{us}\right )}$ of the assets and reserves of exiting bankers in present period, $\left (1-\sigma ^{us}\right )\left (R_{k,t}^{us}q_{t-1}^{us}K_{t-1}^{us}+R_{l,t}^{us}l_{t-1}^{em}+R_{m,t}^{us}m_{t-1}^{us}\right )$. The aggregate net worth of entering bankers is thus as follows

$$ n_{nt}^{us}=\xi^{us}\left( R_{k,t}^{us}q_{t-1}^{us}K_{t-1}^{us}+R_{l,t}^{us}l_{t-1}^{em}+R_{m,t}^{us}m_{t-1}^{us}\right) $$

The aggregate net worth of bankers is then given by

$$ n_{t}^{us}=\left( \sigma^{us}+\xi^{us}\right)\left( R_{k,t}^{us}q_{t-1}^{us}K_{t-1}^{us}+R_{l,t}^{us}l_{t-1}^{em}+R_{m,t}^{us}m_{t-1}^{us}\right)-\sigma^{us}R_{d,t}^{us}\left( d_{t-1}^{us}+d_{t-1}^{em}\right) $$

EM Banker’s Problem –

In the EM, a banker j’s optimization problem is given by

$$ V_{jt}^{em}=\max_{\left\{K_{jt}^{em}, b_{jt}^{em}, l_{jt}^{em}\right\} }E_{t}\left\{ {\varLambda}_{t,t+1}^{em}\left[\left( 1-\sigma^{em}\right)n_{jt+1}^{em}+\sigma^{em}V_{jt+1}^{em}\right]\right\} $$

subject to the balance sheet condition and the incentive compatibility constraint

$$ q_{t}^{em}K_{jt}^{em}\left[1+\left( \frac{\kappa}{2}\right)x_{jt}^{2}\right]=b_{jt}^{em}+\epsilon_{t}^{em}l_{jt}^{em}+n_{jt}^{em} $$

(37)

$$ n_{jt}^{em}=R_{k,t}^{em}q_{t-1}^{em}K_{jt-1}^{em}-R_{b,t}^{em}b_{jt-1}^{em}-\epsilon_{t}^{em}R_{l,t}^{us}l_{jt-1}^{em} $$

(38)

$$ V_{jt}^{em}\geq\theta^{em}(x_{jt})q_{t}^{em}K_{jt}^{em} $$

where

$$ x_{jt}=\frac{\epsilon_{t}^{em}l_{jt}^{em}}{q_{t}^{em}K_{jt}^{em}} $$

$$ \theta^{em}(x_{jt})=\vartheta_{0}^{em}exp(-\vartheta_{1}^{em}x_{jt}) $$

To solve the problem, we first guess that the value function is linear in bank net worth as

$$ V_{jt}^{em}=\psi_{t}^{em}n_{jt}^{em} $$

Using the definition of leverage multiple $\phi _{jt}^{em}=\frac {q_{t}^{em}K_{jt}^{em}}{n_{jt}^{em}}$ and the balance sheet condition (37) and (38), we can rewrite the banker’s problem as

$$ \psi_{t}^{em}=\max_{\left\{ \phi_{jt}^{em},x_{jt}^{em}\right\} }\left\{ \left( \mu_{t}^{em}+\mu_{lt}^{em}x_{jt}\right)\phi_{jt}^{em}+\left[1-\left( \frac{\kappa}{2}\right)x_{jt}^{2}\phi_{jt}^{em}\right]\upsilon_{t}^{em}\right\} $$

(39)

subject to

$$ \psi_{t}^{em}\geq\theta^{em}(x_{jt})\phi_{jt}^{em}=\vartheta_{0}^{em}exp(-\vartheta_{1}^{em}x_{jt})\phi_{jt}^{em} $$

(40)

where

$$ \mu_{t}^{em}=E_{t}\left[{\Omega}_{t+1}^{em}\left( R_{k,t+1}^{em}-R_{b,t+1}^{em}\right)\right] $$

$$ \mu_{lt}^{em}=E_{t}\left[{\Omega}_{t+1}^{em}\left( R_{b,t+1}^{em}-\frac{\epsilon_{t+1}^{em}}{\epsilon_{t}^{em}}R_{l,t+1}^{us}\right)\right] $$

$$ \upsilon_{t}^{em}=E_{t}\left( {\Omega}_{t+1}^{em}R_{b,t+1}^{em}\right) $$

$$ {\Omega}_{t+1}^{em}={\varLambda}_{t,t+1}^{em}\left( 1-\sigma^{em}+\sigma^{em}\psi_{t+1}^{em}\right) $$

Let $\lambda _{cjt}^{em}$ be the Lagrangian multiplier for the incentive compatibility constraint (40). Then using the Lagrangian

$$ \begin{array}{@{}rcl@{}} \mathcal{L} & =&\psi_{t}^{em}+\lambda_{cjt}^{em}\left[\psi_{t}^{em}-{{\vartheta}}_{0}^{em}exp(-{\vartheta}_{1}^{em}x_{jt})\phi_{jt}^{em}\right]\\ & =&\left( 1+\lambda_{cjt}^{em}\right)\left[\left( \mu_{t}^{em}+\mu_{lt}^{em}x_{jt}\right)\phi_{jt}^{em}+\left( 1-\left( \frac{\kappa}{2}\right)x_{jt}^{2}\phi_{jt}^{em}\right)\upsilon_{t}^{em}\right]-\lambda_{cjt}^{em}{\vartheta}_{0}^{em}exp(-{\vartheta}_{1}^{em}x_{jt})\phi_{jt}^{em} \end{array} $$

The first order conditions with respect to $\left \{ \phi _{jt}^{em},x_{jt}\right \}$ imply

$$ \left( 1+\lambda_{cjt}^{em}\right)\left[\left( \mu_{t}^{em}+\mu_{lt}^{em}x_{jt}\right)-\left( \frac{\kappa}{2}\right)x_{jt}^{2}\upsilon_{t}^{em}\right]=\lambda_{cjt}^{em}{\vartheta}_{0}^{em}exp(-{\vartheta}_{1}^{em}x_{jt}) $$

(41)

$$ \left( 1+\lambda_{cjt}^{em}\right)\left( \kappa x_{jt}\upsilon_{t}^{em}-\mu_{lt}^{em}\right)=\lambda_{cjt}^{em}{\vartheta}_{1}^{em}{\vartheta}_{0}^{em}exp(-{\vartheta}_{1}^{em}x_{jt}) $$

(42)

Combining (41) and (42) yields

$$ x_{jt}=\frac{\mu_{lt}^{em}}{\kappa\upsilon_{t}^{em}}-\frac{1}{{\vartheta}_{1}^{em}}+\sqrt{\left( \frac{\mu_{lt}^{em}}{\kappa\upsilon_{t}^{em}}\right)^{2}+\left( \frac{1}{{\vartheta}_{1}^{em}}\right)^{2}+2\frac{\mu_{t}^{em}}{\kappa\upsilon_{t}^{em}}} $$

Given that $\mu _{t}^{em},\mu _{lt}^{em}$ and $\upsilon _{t}^{em}$ are not bank-specific, x_jt = x_t. When the incentive constraint is binding, from Eqs. 39 and 40, we have

$$ \psi_{t}^{em}=\theta^{em}(x_{t})\phi_{jt}^{em} $$

$$ \phi_{jt}^{em}=\frac{\upsilon_{t}^{em}}{\theta^{em}(x_{t})+\left( \frac{\kappa}{2}\right){x_{t}^{2}}\upsilon_{t}^{em}-\left( \mu_{t}^{em}+\mu_{lt}^{em}x_{t}\right)} $$

These imply that $\phi _{jt}^{em}=\phi _{t}^{em}$. Accordingly,

$$ q_{t}^{em}K_{t}^{em}=\phi_{t}^{em}n_{t}^{em} $$

Analogous to the US banks, an equation of motion for the aggregate net worth is the sum of the net worth of existing bankers $n_{et}^{em}$, and the net worth of entering bankers $n_{nt}^{em}$

$$ n_{t}^{em}=n_{et}^{em}+n_{nt}^{em} $$

The net worth of existing bankers equals the earnings on assets held in last period net the costs of liabilities, multiplied by σ^em

$$ n_{et}^{em}=\sigma^{em}\left( R_{k,t}^{em}q_{t-1}^{em}K_{t-1}^{em}-R_{b,t}^{em}b_{t-1}^{em}-\epsilon_{t}^{em}R_{l,t}^{us}l_{t-1}^{em}\right) $$

The net worth of entering bankers is a fraction $\frac {\xi ^{em}}{\left (1-\sigma ^{em}\right )}$ of the assets of exiting bankers in present period, $\left (1-\sigma ^{em}\right )\left (R_{k,t}^{em}q_{t-1}^{em}K_{t-1}^{em}\right )$. Thus,

$$ n_{nt}^{em}=\xi^{em}\left( R_{k,t}^{em}q_{t-1}^{em}K_{t-1}^{em}\right) $$

The aggregate net worth is then given by

$$ n_{t}^{em}=\left( \sigma^{em}+\xi^{em}\right)\left( R_{k,t}^{em}q_{t-1}^{em}K_{t-1}^{em}\right)-\sigma^{em}\left( R_{b,t}^{em}b_{t-1}^{em}+\epsilon_{t}^{em}R_{l,t}^{us}l_{t-1}^{em}\right) $$