2.1 Model Setup

We assume an economy with three time periods, t = 0,1, 2, and a unit mass each of two types of agents: bankers and workers. Furthermore, there is a single good that serves both as consumption good and capital.

Bankers In period 0, bankers are born with one unit of the consumption good. They invest a fraction x ϵ [0,1] of it in a project that delivers a risky payoff à in period 1 with a continuously differentiable distribution function G(Ã) over the domain [0, ∞), a density function g(Ã) and an expected value E[Ã] > 1. They hold the remainder (1 - x) in a storage technology with gross return 1.

After the realization of the risky payoff à in period 1, the resulting equity level of bankers is

e = xà +(1 - x)

Consistent with the literature on banking regulation, we will frequently use the term “bank capital” to refer to bank equity e in the following.

In period 1, bankers raise d deposits at a gross deposit rate of r and lend k ≤ d + e to the productive sector of the economy at a gross interest rate R. In period 2, bankers are repaid and value total profits in period 2 according to a linear utility function

π = Rk - rd

Workers Workers are born in period 1 with a large endowment m of consumption goods. They lend an amount d of deposits to bankers at a deposit rate of r and hold the remainder in a storage technology with gross return 1. No arbitrage implies that the deposit rate satisfies r = 1.

In period 2, workers inelastically supply one unit of labor l = 1 at the prevailing market wage w. (The main insights of our framework are unchanged if labor supply is elastic.) Worker utility depends only on their total consumption. For notational simplicity we normalize the expression for worker utility by subtracting the constant m so that

u = wl

Remark: In the described framework, risk markets between bankers and workers are incomplete since workers are born in period 1 after the technology shock à is realized and cannot enter into risk-sharing contracts with bankers in period 0. All the risk xà from investing in the risky technology therefore needs to be borne by bankers. An alternative microfoundation for this market incompleteness would be that obtaining the distribution function G(Ã) requires that bankers exert an unobservable private effort, and insuring against fluctuations in à would destroy their incentives to exert this effort. In practice, bank capital is subject to significant fluctuations, as illustrated in Figure 1, and a large fraction of this risk is not shared with the rest of society.⁴ We will investigate the implications of reducing this market incompleteness below in Section 4.1.

Firms Workers collectively own firms, which are neoclassical and competitive and produce in period 2. Firms rent capital k from bankers at interest rate R at the end of period 1, and hire labor l from workers at wage w in period 2. They seek to maximize profits F (k, l) - wl- Rk, where F(k,l) = Ak^αl^1#x2212;α with α ϵ (0,1). For simplicity, we assume that there is no uncertainty in firms’ production. In equilibrium firms earn zero profits.

The first-order conditions of the firm problem are

Remark 1: In the described setup, we have separated the risk-taking decision x of bankers from the financial intermediation function k by assuming they occur in separate time periods. This simplifies our analysis, but implies that there is no direct contemporaneous benefit to workers if bankers invest more in the risky payoff with higher expected return. We show in appendix A.2 that our results continue to hold if the risk-taking and financial intermediation functions of bankers are intertwined: we assume that the aggregate production function of the economy in both periods 1 and 2 is [Ã_tx_t+1-x_t]F(k_t,l_t) so that workers directly benefit from risk-taking x_t because they receive higher wages in period t.⁵

Remark 2: Our model setup assumes for simplicity that the endowments of labor and savings as well as the firms are owned by the same set of agents which we called workers. Our results would be unchanged if we assigned these ownership claims to separate types of agents since savers earn zero net returns and firms earn zero profits in equilibrium. For example, there could be an additional type of agent called capital owner who own all the savings and firms of the economy.

Timeline

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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Timeline

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Timeline

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2.2 First-Best Allocation

A planner who implements the first-best maximizes aggregate surplus in the economy subject to the resource constraints of the economy,

In period 2, the optimal labor input is l* = 1, and the optimal level of capital investment satisfies $k^{*}={(\alpha A)}^{\frac{1}{1-\alpha }},$ i.e. it equates the marginal return to investment to the return on the storage technology,

We call the resulting output level F (k*, 1) the first-best level of output, or potential output. As we discussed earlier, we assume that m is large so that the resource constraint k ≤ e + m is lax, i.e. there are always sufficient funds available in the economy to invest k* in the absence of market frictions. The marginal product of labor at the first-best level of capital is w* = F_l (k*, 1).

In period 0, the first-best planner chooses the portfolio allocation that maximizes expected bank equity E [e]. Since E[Ã]> 1, she will pick the corner solution x = 1.

Since a fraction αF (k*, 1) of production is spent on investment, the net social surplus generated in the first-best is

2.3 Financial Constraint

We assume that bankers are subject to a commitment problem to capture the notion that bank capital matters. Specifically, bankers have access to a technology that allows them to divert a fraction (1 - Φ) of their gross revenue, where Φ ϵ [0,1]. By implication depositors can receive repayments on their deposits that constitute at most a fraction ф of the gross revenue of bankers. Anticipating this commitment problem, depositors restrict their supply of deposits to satisfy the constraint

An alternative interpretation of this financial constraint follows the spirit of Holmstrom and Tirole (1998): Suppose that bankers in period 1 can shirk in their monitoring effort, which yields a private benefit of B per unit of period 2 revenue but creates the risk of a bank failure that may occur with probability ≤ and that results in a complete loss. Bankers will refrain from shirking as long as the benefits are less than the costs, or BRk ≤Δ[Rk - rd]. If depositors impose the constraint above for $\varphi =1-\frac{B}{\text{Δ}}$, they can ensure that bankers avoid shirking and the associated risk of bankruptcy.⁶

Furthermore, our model assumes that all credit is used for production so that binding constraints directly reduce supply in the economy. An alternative and complementary assumption would be that credit is required to finance (durable) consumption so that binding constraints reduce demand. In both setups, binding financial constrainst hurt the real economy, with similar redistributive implications.⁷

3.1 Period 1 Equilibrium

We analyze equilibrium in the economy in period 1 for a given level of bank capital e. Employment is always at its optimum level l = 1 since we assumed wages are flexible. The financial constraint is loose if bank equity is sufficiently high so that bankers can intermediate the first-best amount of capital, e ≥ e* = (1 -Φ)k*. In this case, the deposit and lending rates satisfy r = R = 1 and bankers earn zero returns on their lending activity. The wage level is w^* = (1 - α) F (k*, 1). We interpret this situation as “normal times.”

If bank equity is below the threshold e < e* then the financial constraint binds and the financial sector cannot intermediate the first-best level of capital. We interpret this situation as a “credit crunch” or “financial crisis” since the binding financial constraints reduce output below its first-best level. Workers provide deposits up to the constraint d = ΦRk/r, the deposit rate is r = 1, and the lending rate is R = F_k (k, 1). Equilibrium capital investment in the constrained region, denoted by $\widehat{k}(e)$, is implicitly defined by the equation

which has a unique positive solution for any e ≥ 0. Overall, capital investment is given by the expression

Equilibrium k(e) is strictly positive, strictly increasing in e over the domain e ϵ [0,e*) and constant at k* for e ≥ e*. The equilibrium lending rate and the wage level satisfy, respectively,

Let us distinguish aggregate bank equity e, and the equity eⁱ of an individual banker indexed by i. Then we can describe the level of capital intermediated by banker i and the resulting profits by⁸

In equilibrium, eⁱ = e will hold, and we denote the equilibrium profits of the banking sector as a whole as well as the utility of workers by

Total utilitarian surplus in the economy is

Panel 1 of Figure 3 depicts the payoffs of bankers and workers as a function of aggregate bank capital e.⁹ As long as e < e*, capital investment falls short of the first best level. In this region, the welfare of workers and of bankers are strictly increasing concave functions of bank equity. Once bank capital reaches the threshold e^*, the economy achieves the first-best level of investment. Any bank capital beyond this point just reduces the amount of deposits that bankers need to raise, which increases their final payoff in period 2 but does not benefit workers. Beyond the threshold e^*, worker utility therefore remains constant and bank profits increase linearly in e. This generates a non-convexity in the function π (e) at the threshold e^*.

Welfare and marginal value of bank capital e

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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Welfare and marginal value of bank capital e

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Welfare and marginal value of bank capital e

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Our analytical findings on the value of bank capital are consistent with the empirical regularities of financial crises that we depicted in Figure 1 on page 3.

3.2 Marginal Value of Bank Capital

How do changes in bank capital affect output and the distribution of surplus in the economy? If bankers are financially constrained in aggregate, i.e. if e < e^*, then a marginal increase in bank capital e allows bankers to raise more deposits and leads to a greater than one-for-one increase in capital investment k. Applying the implicit function theorem to (2) in the constrained region we find

If bankers are unconstrained, e ≥ e*, then additional bank equity e leaves capital investment unaffected at the first-best level k*; therefore $k^{\prime }(e)=0$.

The marginal effect of additional bank capital on total surplus is $s^{\prime }(e)=1+(F_k-1)k^{\prime }(e)$. The first term captures the consumption value of an additional unit of wealth for bankers. The second term captures that bank capital e raises investment in real capital by $k^{\prime }(e)$ which earns a marginal return (F_k - 1).

Looking at the distribution of this additional surplus between bankers and workers we find

The effects of changes in bank equity for the two sectors differs dramatically depending on whether the financial constraint is loose or binding. In the unconstrained region e ≥ e*, the consumption value for bankers is the only benefit of bank capital since $k^{\prime }(e)=0$ and so $w^{\prime }(e)=0$ and ${\pi }^{\prime }(e)=1$. Bank capital is irrelevant for workers and the benefits of additional bank capital accrue entirely to bankers.

By contrast, in the constrained region e < e*, additional equity increases capital intermediation k and output F (k, 1). A fraction (1 - α) of the additional output F_k accrues to workers via increased wages, and a fraction α of the output net of the additional capital input accrues to bankers. These effects are illustrated in Panel 2 of Figure 3.

Technically, the effects of bank capital on wages w (e) and the return on capital R(e) in the constrained region constitute pecuniary externalities. When atomistic bankers choose their optimal equity allocations, they take all prices as given and do not internalize that their collective actions will have general equilibrium effects that move wages and the lending rate. In particular, they do not internalize that equity shortages will hurt workers by pushing down wages and pushing up lending rates.

The decline in wages when e < e* occurs because labor is a production factor that is complementary to capital in the economy’s production technology. The increase in lending rates when e < e* occurs because the financial constraint drives the return to capital investment up to R(e) = F_k((k(e),1)> 1 since not all productive investments can obtain loans. The difference between the lending rate and the deposit rate r = 1 allows bankers to earn a spread R(e)– 1 when the constraint is binding. Observe that this spread plays a useful social role in allocating risk because it signals to bankers that there are extra returns available for carrying capital into states of nature when it is scarce. However, it redistributes from workers to bankers by enabling them to earn a scarcity rent on their capital.

Equity Shortages and Redistribution It is instructive to observe that small shortages of financial sector capital have first order redistributive effects but only second order efficiency effects. In particular, consider an economy in which bank capital is e* so that the unconstrained equilibrium can just be implemented. Assume that we engage in a wealth-neutral reallocation of the wealth of bankers across periods 1 and 2: we take away an infinitesimal amount ε of bank capital from bankers in period 1 so as to tighten their financial constraint and return it to them in period 2. The resulting payoffs for bankers and workers are π(e*−ε) + ε and ω(e*-ε).

Lemma 1 (Redistributive Effects of Equity Shortages) A marginal tightening of the financial constraint around the threshold e* has first-order redistributive effects but only second-order efficiency costs.

Proof. We take the left-sided limit of the derivative of the payoff functions of bankers and workers to find

The effect on total surplus consists of the sum of the two $s^{\prime }={\pi }^{\prime }+{\omega }^{\prime }$ and is zero at a first-order approximation. ◾

Interestingly, a marginal tightening of the constraint imposes losses on workers from lower wages that precisely equal the gains to bankers from higher lending spreads, i.e. the redistribution between workers and bankers occurs at a rate of one-to-one. Conceptually, this is because pecuniary externalities are by their very nature redistributions driven by changes in prices. In our model, when financial constraints reduce the amount of capital intermediated and push down wages, the losses of workers equal the gains to firms. Similarly, when the lending rate rises, the losses to firms equal the gains to bankers. Since firms make zero profits, we can conclude that the losses to workers have to equal the gains to bankers. Intuitively, since bankers are the bottleneck in the economy when the financial constraint binds, they extract surplus from workers in the form of scarcity rents.

3.3 Determination of Period 0 Risk Allocation

An individual banker i takes the lending rate R as given and perceives the constraint on deposits d ≤ϕRk as a simple leverage limit. When a banker is constrained, she perceives the effect of a marginal increase in bank capital eⁱ as increasing her intermediation activity by $k_1(e^i,e)=\frac{1}{1-\varphi R}$, which implies an increase in bank profits by

In period 0, bankers decide what fraction x of their endowment to allocate to the risky project. In the laissez-faire equilibrium, banker i takes the aggregate levels of x and e as given and chooses xⁱ to maximize

At an interior optimum, the optimality condition of bankers is

i.e. the risk-adjusted return on the stochastic payoff à equals the return of the safe storage technology.

The stochastic discount factor π₁ in this expression is given by equation (3) and is strictly declining in e as long as e < e* and constant at 1 otherwise. Observe that each banker i perceives his stochastic discount factor as independent of his choices of eⁱ and xⁱ. However, in a symmetric equilibrium, eⁱ = e as well as xⁱ = x have to hold, and equilibrium is given by the level of x and the resulting realizations e = Ãx+(1-x) such that the optimality condition (5) is satisfied. As long as E[Ã] > 1, the optimal allocation to the risky project satisfies x > 0. If the expected return is sufficiently high, equilibrium is given by the corner solution x = 1. Otherwise it is uniquely pinned down by the optimality condition (5).

Denote by x^LF the fraction of their initial assets that bankers allocate to the risky project in the laissez faire equilibrium. The resulting levels of welfare for workers and entrepreneurs are Π^LF=E[π(1-x^LF+Ãx^LF)] and W^LF=E[ω(1-x^LF+Ãx^LF)].

For a given risky portfolio allocation x, we define by Ā (x) the threshold of à above which bank capital e is sufficiently high to support the first-best level of production. We can express this function as

Well-Capitalized Banking System If e* ≤ 1, then the safe return is sufficient to avoid the financial constraint and the first-best level of capital intermediation k^* would be reached for sure with a perfectly safe portfolio x = 0. We can interpret this case as an economy in which the financial sector is sufficiently capitalized to intermediate the first-best amount of capital without any extra risk-taking. In that case, we can interpret the risky project à as a diversion from the main intermediation business of banks, e.g. a diversification from retail banking into investment banking, or loans by US banks to Latin American governments that offer extra returns at extra risk.

For the case of e* ≤ 1, bankers find it optimal to choose x^LF > 1*#x2212;e*, i.e. they take on sufficient risk so that the financial constraint is binding at least for low realizations of the risky return so that Ā(x) > 0. This is because the expected return on the risky project dominates the safe return, and bankers perceive the cost of being marginally constrained as second-order. We also observe that for e* < 1, the function Ā(x) is strictly increasing from Ā(1 #x2212; e*)=0 to Ā(1) = e*, i.e. more risk-taking makes it more likely that the financial sector becomes constrained.

Under-Capitalized Banking System If e* > 1, then the economy would be constrained if bankers invest all their endowment in the safe return. We can interpret this as an economy where banks are systematically undercapitalized and risk-taking helps them to mitigate these constraints. In that case, the function Ā(x) is strictly decreasing from Ā(1#x2212;e*)=0 to Ā(1)=e*, i.e. more risk-taking makes it more likely that the financial sector becomes constrained.

4.1 Market Incompleteness and the Distributive Conflict

To pinpoint why there is a distributive conflict over the level of risk-taking, it is instructive to analyze the role of the two financial market imperfections in our results. First, assume that we remove the financial constraint on bankers in period 1. In that case, the profits/losses of bankers do not affect how much capital can be intermediated to the real economy and workers are indifferent about the level of risk-taking - bank capital does not generate any pecuniary externalities. In such an economy, financial risk-taking and financial intermediation are two orthogonal activities and we find that x^W = x^B = x^FB = 1, i.e. the distributive conflict disappears.

Secondly, assume that we introduce a complete insurance market in period 0 in which bankers and workers can share the risk associated with the technology Ã, but we keep the financial constraint in period 1. In that case, workers will insure bankers against any capital shortfalls so that bankers can invest in the risky technology without imposing negative externalities on the real economy. By implication all agents are happy to invest the first-best amount x^W = x^B = x^FB = 1 in the risky techology, and the distributive conflict again disappears.

More generally, introducing a risk market in period 0 puts a formal price on risk-taking and, if both sets of agents can participate in this market, it implies that bankers and workers will agree on a common price of risk. Loosely speaking, this provides workers with a channel through which they can transmit their risk preferences to bankers.

An interesting special case in which financial markets in period 0 are effectively complete is a ‘Marxist’ two sector framework in which bankers/capitalists own all the capital and households/workers own all the labor in the economy (i.e. there are no desposits d = 0 and no storage). By implication, bankers invest all their equity into real capital k = e. Given a Cobb-Douglas production technology, the two sectors earn constant fractions of aggregate output so that π(e) = αF(e,1) and ω(e) = (1#x2212;α)F(e,1) for e = Ãx+(1#x2212;x). As long as the two sectors have preferences with identical relative risk aversion (in our benchmark model both have zero risk-aversion), the optimal level of risk-taking for capitalists simultaneously maximizes total surplus and worker welfare:

Bank capital still imposes pecuniary externalities on wages in this ‘Marxist’ setting, but the pecuniary externalities under a Cobb-Douglas technology guarantee that both sets of agents obtain constant fractions of output, replicating the allocation under perfect risk-sharing. (Analytically, the constant capital and labor shares drop out of the optimization problem.) Again, there is no distributive conflict.

By contrast, in our benchmark framework with the two market imperfections reintroduced, the negative pecuniary externalities only occur on the downside. Once bank capital exceeds the threshold where financial constraints are loose, it is irrelevant and has no further effects on workers. The distributive conflict is therefore generated by the combination of the occasionally binding financial constraints and the lack of risk-sharing between bankers and workers.

4.2 Financial Regulation

We interpret financial regulation in our framework as policy measures that affect risk-taking x and therefore move the economy along the Pareto frontier.¹⁰ The unregulated equilibrium - in the absence of any other market distortions - is represented by the laissez-faire equilibrium x^LF on the frontier.

The two simplest forms of financial regulation of risk-taking are:

1. Regulators may impose a ceiling on the risk-taking of individual bankers such that $x^i\le \overline{x}$. Such a ceiling will be binding if $\overline{x}<x^{LF}$. This type of regulation closely corresponds to capital adequacy regulations as it limits the amount of risk-taking per dollar of bank equity.

2. Regulators may impose a tax τ^x on risk-taking xⁱ so as to modify the optimality condition for the risk-return tradeoff of bankers to E[π₁.(Ã−τ^x−1)] = 0. Such a tax can implement any level of risk-taking x ϵ [0,1]. For simplicity, we assume that the tax revenue is rebated to bankers in lump-sum fashion.

Financial regulators can implement any risk allocation $\overline{x}\le x^{LF}$. by imposing $\overline{x}$ as a ceiling on risk-taking or by imposing an equivalent tax on risk-taking τ^x ≥0. The distributive implications are straightforward:

Corollary 4 (Redistributive Effects of Financial Regulation) Tightening regulation by lowering $\overline{x}$ or raising τ^x increases worker welfare and reduces banker welfare for any $\overline{x}\in [x^W,x^{LF}]$.

Conversely, financial deregulation increases the ceiling $\overline{x}$ and redistributes from workers to bankers.

Scope for Pareto-Improving Deregulation An interesting question is whether there exists a mechanism for Pareto-improving deregulation if we add further instruments to the toolkit of policymakers in addition to the regulatory measures on x that we described in Corollary 4. Such a mechanism would need to use some of the gains from deregulation obtained by bankers to compensate workers for the losses they suffer during credit crunches.

Consider first a planner who provides an uncontingent lump-sum transfer from bankers to compensate workers for the losses from deregulation. The marginal benefit to workers is $1-E[w^{\prime }(e)]$ if the transfer is given in period 1 or $1-\varphi E[w^{\prime }(e)]$ if it is given in period 2, i.e. workers would obtain a direct marginal benefit of 1 in all states of nature, but in constrained states they would be hurt by a tightening of the financial constraint which reduces their wages by $w^{’}(e)$ if given in period 1 or a fraction ϕ therefore if it is given in period 2, since the transfer reduces the capital or the pledgeable income of bankers in period 2. Both types of uncontingent transfers entail efficiency costs from tightening the constraints on bankers. Compensating workers with an uncontingent payment without imposing these efficiency costs would require that the planner has superior enforcement capabilities to extract payments in excess of the financial constraint (1).

Alternatively, consider a planner who provides compensatory transfers to workers contingent on states of nature in which bankers are unconstrained, i.e. in states in which they make high profits from the risky technology Ã. This would not impose any efficiency costs but would require that the planner can engage in state-contingent transactions that are not available via private markets in our model. (It can be argued that this type of transfer corresponds to proportional or progressive profit taxation.)

In short, the planner only has efficient compensation mechanisms if she can get around at least one of the two financial market imperfections in our framework, i.e. if she can mitigate either the financial constraint (1) or the incompleteness of risk markets.

If the planner cannot improve on these market imperfections and/or if transfers require distortionary taxation, then the scope for Pareto-improving deregulation is more limited as the redistributive benefit of any transfer has to be weighed against the cost of the distortion introduced. In general, this creates a constrained Pareto frontier along which the trade-off between the welfare of the two agents is significantly less favorable, i.e. a Pareto frontier that is enveloped by the frontier depicted in Figure 4.

5.1 Asymmetric Compensation Schemes

It is frequently argued that managers of financial institutions may have incentives to increase risk-taking because of asymmetric compensation schemes that reward them them for higher risk and that this may have played an important role in the build-up of risk before the financial crisis of 2008/09. We illustrate this mechanism using a stylized model of an incentive problem between bank owners and bank managers and analyze the distributive implications.

Let us extend our benchmark model by assuming that bank owners have to hire a new set of agents called bank managers to conduct their business. Bank managers choose an unobservable level of risk-taking x in period 0. Bank owners are able to observe the realization of bank capital e in period 1 and to instruct managers to allocate any bank capital up to e* in financial intermediation, and managers carry any excess max{0, e - e*} in a storage technology. We can view financial intermediation versus storage as representative of lending to real projects versus financial investments, or commercial banking versus investment banking.

Suppose that bank managers do not have the ability to commit to exert effort in period 1 and can threaten to withdraw their monitoring effort for both bank loans and storage in period 1. If they do not monitor, the returns on intermediation and storage (real projects and financial investments) are diminished by a fraction ε and δε respectively, where δ > 1. In other words, the returns to financial investments are more sensitive to managerial effort than real investments. An alternative interpretation would be along the lines of Jensen (1986) that free cash provides managers with greater scope to abuse the resources under their control.

Assuming that managers have all the bargaining power and that we are in a symmetric equilibrium, the threat to withdraw their effort allows them to negotiate an incentive payment from bank owners of

The marginal benefit of bank equity for an individual manager is p₁ (e, e) = επ₁ (e, e) for e < e* and p₁ (e,e) = δεπ₁ (e,e) = δε for e ≥ e*. Since financial investments require a greater incentive payment, the payoff of managers is relatively more convex than the payoff of banks π (e, e) and managers benefit disproportionately from high realizations of bank capital. Comparing this extension to our benchmark setup, we can view Π (x) as the joint surplus of bank owners and managers, and the two functions Π (x) and W (x) remain unchanged compared to our earlier framework - the only thing that changes is the level of x that will be chosen by bank managers.

Managers internalize the asymmetric payoff profile when they choose the level of risk-taking in period 0. They maximize E[p (e_i, e)] where e_i = Ãx_i + 1 - x_i and their optimality condition is

It is then straightforward to obtain the following result:

Proposition 5 (Agency Problems and Risk-Taking) (i) The optimal choice of risk-taking of bank managers exceeds the optimal choice x^LF in our benchmark model if the payoff function of managers is asymmetric δ >1.

(ii) The expected welfare of workers is a declining function of δ.

Proof. For (i), observe that we can write

where 𝕀_eⁱ≥e* is an indicator variable that is equal to 1 when eⁱ ≥ e* and 0 otherwise. The preferred choice of x by managers, call it x^A, satisfies P₁(x) = E [(Ã-1)p₁(eⁱ,e) ≥ 0. We can write this as

where Π₁(x)= E [(Ã-1)π₁(eⁱ,e)] is the owner’s first-order condition. Now we argue that the second term is strictly positive. Note that we can write this term as

Since we have E[Ã - 1] > 0 by assumption, and since (Ã-1) is an increasing function of Ã, it follows that the integral over the upper half of the range must also be strictly positive. Therefore for all x, we have P₁(x) > Π₁(x), and in particular Π₁(x^LF > 0, and so x^A > x^LF.

To prove (ii), we begin by showing that x^A is strictly increasing in δ. Differentiating P₁(x) with respect to δ yields ϵE [(Ã-1)𝕀_eⁱ≥e*], which is strictly positive. At the old preferred level of x, we now have P₁(x) > 0, and so x^A will increase. Now we observe that increasing x for x > x^W will always make workers worse off. Given that x^W < x^LF < x^A, this implies that increasing δ will make workers worse off. ◾

5.2 Financial Institutions with Market Power

Assume that there is a finite number n of identical bankers in the economy who each have mass $\frac{1}{n}$. Banker i internalizes that his risk-taking decision xⁱ in period 0 affects aggregate bank capital $e=\frac{1}{n}{e}^i+\frac{n-1}{n}{e}^{-i}$, where e ^-i captures the capital of the other bankers. For a given e, we assume that bankers charge the competitive market interest rate R(e) in period 1.¹¹

First consider the optimal level of capital supplied by bankers who partially internalize their effect on interest rates and are not subject to a leverage constraint. Bankers solve

whose solution is $\frac{1}{n}{F}_{kk}{k}^i+F_k=1$. Assuming a symmetric solution, this will be satisfied at

which will be achieved at a level of equity $e^{*,n}=\left(1-\frac{\varphi }{1-\frac{1}{n}\left(1-\alpha \right)}\right)k^{*,n}$. If e <e^*w,ⁿ, the deposit constraint binds and bankers receive the same profits as before.

The marginal valuation of bank capital is now

This falls in between the marginal value of bank capital for the sector as a whole and for a competitive banker, i.e. ${\pi }^{\prime }<{\pi }_1^{i,n}<{\pi }_1^i$.

Since we have ${\pi }_1^{i,n}(e^i,e)={\pi }_1^i+\frac{1}{n}({\pi }^{\prime }-{\pi }_1^i)$, we can write the optimality condition for one of n large firms as

We immediately see that for n = 1, this reduces to ${\prod }^{\prime }=0$, which has solution x^B, and for $n\to \infty$ this reduces to Π₁ = 0, which has solution x^LF < x^B.

Now suppose that for a given n, we have $x^n\in \left(x^{LF},x^B\right)$. At xⁿ, we differentiate the optimality condition w.r.t. n and find

Since Π₁ and ${\pi }^{\prime }$ are both strictly decreasing in x, and since they are zero at x^LF and x^B > x^LF respectively, in the interval (x^LF,x^B)we have ${\text{Π}}_1<{\text{Π}}^{\prime }$. Therefore for higher n we have $\frac{d}{dn}{\text{Π}}_1^{i,n}<0$, and so xⁿ is decreasing in n. We summarize these results as follows:

Proposition 6 The optimal risk allocation xⁿ of bankers is a declining function of the number n of banks in the market. In particular, we observe that x¹ = x^B ≥ x^∞ = x^LF, with strict inequality unless they are corner solutions.

Intuitively, bankers with market power internalize that the benefits of additional equity when they are constrained accrue in part to the rest of the economy by relieving the credit crunch. This reduces their scarcity rents and therefore provides lower incentives for precautionary behavior. Our example illustrates that socially excessive risk-taking is an important dimension of non-competitive behavior by banks.

5.3 Financial Innovation

An important manifestation of financial innovation is to allow financial market players to access new investment opportunities, frequently projects that are characterized by both higher risk and higher expected returns. For example, financial innovation may enable bankers to invest in new activities, as made possible e.g. by the 1999 repeal of the 1933 Glass-Steagall Act, or to lend in new areas, to new sectors or to new borrowers, as e.g. during the subprime boom of the 2000s.

Formally we capture this type of financial innovation by expanding the set of risky assets to which bankers have access in period 0. For a simple example, assume an economy in which bankers can only access the safe investment projects in period 0 before financial innovation takes place, and that financial innovation expands the set of investable projects to include the risky project with stochastic return Ã. Furthermore, assume that e* < 1, i.e. the safe return in period 0 generates sufficient period 1 equity for bankers to intermediate the first-best level of capital. The pre-innovation equilibrium corresponds to x = 0 in our benchmark setup and this maximizes worker welfare.

Example 1 (Distributive Effects of Financial Innovation) In the described economy, expanding the set of investment projects to include à increases banker welfare but reduces worker welfare.

After financial innovation introduces the risky project, bankers allocate a strictly positive fraction of their endowment x^LF > 1 - e* to the risky project and incur the risk of being financially constrained in low states of nature. This is their optimal choice because the expected return E[Ã] > 1 delivers a first-order benefit over the safe return, but bankers perceive the cost of being marginally constrained as second-order since π₁ (e^l,e) is continuous at e*. Worker welfare, on the other hand, unambiguously declines as a result of the increased risk-taking. Workers have nothing to gain from bank capital that exceeds e^* but they experience first-order losses if bank capital declines below e*, constraining capital investment and reducing wages.

This illustrates that financial innovation that increases the set of investable projects so as to include more high-risk-high-return options may redistribute from workers to bankers, akin to financial deregulation, even though total surplus may be increased. The problem in the described economy is that workers would be happy for bankers to increase risk-taking if they could participate in both the upside and the downside via complete insurance markets.

Restrictions on the risk-taking activities of banks, e.g. via regulations such as the Volcker rule, may benefit workers by acting as a second-best device to complete financial markets. In the example described above this would be the case.

5.4 Bailouts

Bailouts have perhaps raised more redistributive concerns than any other form of public financial intervention, presumably because they involve redistributions in the form of explicit transfers that are much more transparent than other implicit forms of redistributions.

However, the redistributive effects of bailouts are both more subtle and potentially more pernicious than this simple view suggests. Ex post, i.e. once bankers have suffered large losses and the economy experiences a credit crunch, bailouts may actually lead to a Pareto improvement and workers may be better off by providing a transfer. However, ex-ante, bailout expectations increase risk-taking. This redistributes surplus from workers to the financial sector in a less explicit and therefore more subtle way, as we have emphasized throughout this paper.

Model of Endogenous Bailouts Since bank capital is essential for the real economy, workers in our model find it optimal to coordinate to provide bailouts to bankers during episodes of severe capital shortages in order to mitigate the adverse effects of credit crunches on the real economy. Such bailouts typically come in two broad categories, emergency lending, frequently at a subsidized interest rate, and equity injections, frequently at a subsidized (or even zero) price. We show in Appendix A.3 that no matter how exactly a bailout is provided, what matters in our model is the total amount of resources (subsidies) given to bankers to relax their binding financial constraints. For the remainder of this section, we focus for simplicity on bailouts in the form of direct transfers. The appendix generalizes our results.

Lemma 7 (Optimal Bailout Policy) If aggregate bank capital in period 1 is below a threshold 0 < ê < e* workers find it collectively optimal to provide lump-sum transfer t = ê–e to bankers. The threshold ê is determined by the expression $w^{\prime }(\widehat{e})=1$ or

Proof. The welfare of workers who collectively provide a transfer t ≥ 0 to bankers is given by ω(e +1) – t. An interior optimum satisfies $w^{\prime }(e+t)=1$ We define the resulting equity level as ê = e+t which satisfies equation (6). Observe that $w^{\prime }(e)$ is strictly declining from $w^{\prime }(0)=1/\varphi >1\text{ to }{w}^{\prime }(e^{*})=\frac{1-\alpha }{1-\varphi \alpha }<1$ over the interval [0,e*]so that ê is uniquely defined. If aggregate bank capital is below this threshold e<ê, bankers find it collectively optimal to transfer the shortfall. If e is above this threshold, it does not pay off for workers to provide a transfer since $w^{\prime }<1$ and the optimal transfer is given by the minimum t = 0.

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The intuition stems from the pecuniary externalities of bank capital on wages: increasing bank capital via lump-sum transfers relaxes the financial constraint of bankers and enables them to intermediate more capital, which in turn expands output and increases wages. As long as e< ê, the cost of a transfer on workers is less than the collective benefit in the form of higher wages. If workers can coordinate, they will be collectively better off by providing a transfer that lifts bank capital to ê.

For the remainder of our analysis of bailouts, we make the following assumption:

Assumption 1 The parameters α, ø and A are such that ê< 1.

This guarantees that the banking sector will not require a bailout if the period 0 endowment is invested in the safe project. It also implies that bailouts are not desirable in states of nature in which the risky project yields higher returns than the safe project. This is a mild assumption as we typically expect bailouts to occur in bad states of nature.

Period 1 Equilibrium We analyze the optimal bailout transfer policy of the worker sector in period 1 as a function of the aggregate bank capital position e. (We continue to assume that workers’ period 1 wealth m is large so that it does not limit the size of the desired bailout.)

For e ≥ ê, the collective welfare of workers w (e) and bankers π (e) are unchanged from the expressions in the benchmark model since no bailouts are given. For e< ê, the possibility of bailouts modifies the expressions for welfare as follows:

We illustrate our findings in Figure 5. The threshold ê below which bankers receive bailouts is indicated by the left dotted vertical line. Panel 1 shows bailouts t (e) and welfare of workers and bankers as a function of bank capital. Bailouts are positive but decrease to zero over the interval [0, ê]. Within this interval, they stabilize the profits of bankers at the level π (ê). The welfare of workers is increasing at slope 1 since each additional dollar of bank capital implies that the bailout is reduced by one dollar. Bailouts therefore make the payoff functions of all agents less concave and, in the case of bankers, locally convex.

Welfare and marginal value of bank capital under bailouts

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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Welfare and marginal value of bank capital under bailouts

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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Welfare and marginal value of bank capital under bailouts

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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Marginal Value of Bank Capital From the above expressions, we derive the marginal value of bank capital in the bailout region, i.e. for e < ê. We note that $1+t^{\prime }(e)=0$ in this region and find that the welfare effects of a marginal increase in bank capital are

Panel 2 of Figure 5 depicts the marginal welfare effects of bank equity under bailouts. The marginal benefit for workers $w^{BL}’(e)$ is 1 within the bailout region e< ê, since each additional dollar of bank equity reduces the size of the required bailout that they inject into bankers by a dollar. (In fact, we can determine the level of ê by equating the marginal benefit of bank capital to workers in the absence of bailouts, corresponding to the downward-sloping dotted line $w^{BL}’(e)=(1-\alpha )F_k$ in the figure, to the marginal cost which is unity. This point is marked by a circle in the figure.)

Bailouts constitute straight transfers from workers to bankers, but in our model, they nonetheless generate a Pareto improvement for a given e< ê in period 1 because they mitigate the market incompleteness that is created by the financial constraint (1) and that prevents bankers from raising deposit finance and intermediating capital to the productive sector. At the margin, each additional unit of bailout generates a surplus F_k (e, 1) - 1, of which $w^{\prime }(e)-1-$ arises to workers and π’ (e) to bankers. For the last marginal unit of the bailout, the benefit to workers is ${\omega }^{\prime }\left(\hat{e}\right)-1=0-$ they are indifferent between providing the last unit or not. However, the marginal benefit to bankers for the last unit is strictly positive ${\pi }^{\prime }(\widehat{e})=\frac{\alpha }{1-\alpha }$. One interpretation of this is that bankers are able to extract “bailout rents” from workers because of their bottleneck role in financial intermediation.

Period 0 Risk-Taking Optimal discretionary bailouts impose a ceiling on the market interest rate $R^{BL}(e)\le R(\widehat{e})=\frac{1}{1-\alpha }$ since they ensure that aggregate capital investment is greater than the threshold k ≥ ê at all times. This mitigates the precautionary incentives of bankers and increases their optimal risk-taking, corresponding to a “wealth effect” of bailouts. This effect exists even if bailouts are conditional on aggregate bank capital e and are provided in the form of lump-sum transfers.

In addition, the adverse incentive effects of bailouts are aggravated if they are conditional on individual bank capital eⁱ, which provide bankers with incentives to increase risk-taking so as to increase the expected bailout rents received, corresponding to a “substitution effect” of bailouts.

At a general level, we assume that the bailout received by an individual banker i for a given level of individual and aggregate bank equity (eⁱ,e) is allocated according to the function

where γϵ [0, 1] captures the extent to which the bailout depends on individual bank equity. This specification nests bailouts that are entirely conditional on aggregate bank capital (for γ = 0) as well as those conditional solely on individual bank capital (for γ = 1). Alternatively, if banks are non-atomistic and bailouts are conditional on aggregate bank capital e, we can interpret the parameter γ as the market share of individual banks, since each bank will internalize that its bank equity makes up a fraction γ of aggregate bank equity.

We denote the amount of their endowment that bankers allocate to the risky project in period 0 by x^BL (γ), and we find that bailouts have the following effects:

Proposition 8 (Risk-Taking Effects of Bailouts) (i) Introducing bailout transfers increases period 0 risk-taking x^BL (γ) > x^LF for any γ = 0.

(ii) Risk-taking x^BL (γ) is an increasing function of γ

Proof. See appendix A.1. ◾

Intuitively, point (i) reflects that bailouts reduce the tightness of constraints and therefore the returns on capital π₁ in low states of nature. This lowers the precautionary incentives of bankers and induces them to take on more risk, even if the bailouts are provided in a lump-sum fashion. Observe that this effect is similar to the effects of any countercyclical policy or any improvement in risk-sharing via markets. The effect is also visible in Panel 1 of Figure 5, in which the payoff function of bankers under bailouts is more convex than in the absence of bailouts, inducing them to increase their risk-taking.

For γ > 0, point (ii) captures that the risk-taking incentives of bankers rise further because they internalize that one more dollar in losses will increase their bailout by γ dollars. This captures the standard notion of moral hazard, i.e. that bailouts targeted at individual losses increase risk-taking.

Redistributive Effects The welfare effects of introducing bailouts on bankers and workers can be decomposed into two parts, the change in expected welfare from introducing bailouts for a given level of risk-taking x^BL, corresponding to the market completion effect of bailouts, and the change in the level of risk-taking, corresponding to the incentive effects of bailouts,:

Corollary 9 (Distributive Effects of Bailouts) (i) Bankers always benefit from introducing bailouts and their expected profits Π^BL (γ) are an increasing function of γ.

(ii) Workers benefit from the market completion effect of bailouts but are hurt by the incentive effects of bailouts if e* < 1. The absolute magnitude of both effects is an increasing function of γ.

Proof. The first term is positive for both sets of agents because, for given x, bailouts generate a Pareto improvement.¹² The second term is always positive for bankers because ${\Pi }^{\prime }(x)>0$ and is always negative for workers for e* < 1 because ${\text{w}}^{\prime }(x)>0$ Higher γ increases risk-taking and therefore leads to more bailouts, increasing the absolute magnitude of all four terms. ◾

Although the market completion effect is positive for both sets of agents, the increase in risk-taking benefits bankers at the expense of workers. Bailouts increase banker welfare both directly because of the transfers received from workers and indirectly as a result of the higher risk-taking.

We illustrate our findings in Figure 6. The figure shows how the Pareto frontier depicted in Figure 4 is affected by the introduction of bailouts under the assumption that γ = 0, i.e. bailouts are distributed lump sum. The new Pareto frontier (solid line) is shifted out compared to the old frontier (dotted line) at its left end, i.e. Π^BL (x) > Π(x) and W^BL (x) > W (x) for all x> x^W but is unchanged at x = x^W as long as e* < 1 (which holds in our parameterization). The shift in the frontier is thus biased towards bankers and the introduction of bailouts constitutes banker-biased technological change. In our figure, risk-taking increases significantly even though γ = 0. Banker welfare rises by ΔΠ whereas worker welfare falls by ΔW.

Pareto frontier under bailouts

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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Pareto frontier under bailouts

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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Pareto frontier under bailouts

Citation: IMF Working Papers 2013, 247; 10.5089/9781475546088.001.A001

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