SINTERIZACIÓN

Entrepreneurs

Suppose there is a single, representative competitive entrepreneur. At the end of period

t, the entrepreneur is endowed with a net worth,Ne,t≥0, from the household’s transfer. The entrepreneur obtains a loan,Le,t, from home banks, and combines with its own net worth,Ne,t, in order to purchase raw capital, ¯Kt+1, at a competitive price ofQ_K,t¯ . That

is,

(3.40) Le,t =Q_K,t¯ K¯t+1−Ne,t.

Given period t+ 1 aggregate rates of return and prices, the entrepreneur chooses the utilization rate, ut+1, which transforms its raw capital, ¯Kt+1, into the services of

effective capital according to

(3.41) Kt+1 =ut+1K¯t,

subject to the same utilization cost a(ut+1) as in the EFP version. The entrepreneur

also supplies effective capital services, Kt+1, for the competitive rental rateZk,t+1, and

sells undepreciated capital back to households at the priceQ_K,t¯ ₊₁. The optimal capital

utilization and the zero profit condition are as follows:

(3.42) Zk,t+1=Pi,t+1a0(ut+1), (3.43) Rk,t = Zk,t+1ut+1−Pi,t+1a(ut+1) + (1−δ)Q_K,t¯ ₊₁ ω_K¯Q_K,t¯ −(1−ωK¯)QK,t¯ +1 ω_K¯Q_K,t¯ ,

where ω_K¯ is the share of aggregate capital purchase financed by the aggregate en-

trepreneurial loan and Pi,ta(ut) is the unit utilization cost denominated in the price of investment goods. The entrepreneur breaks even every period, transfers all net worth back to its household and hence accumulates no net worth state by state.

Bankers

I assume the presence of a competitive banking sector that efficiently monitors various borrowers and enforces their obligations. Thus risk neutral banks frictionlessly lend available funds to borrowers. At the end of period t, the state of a bank is summarized

by its net worth, Nb,t(j)≥0. At this point, each bank raises a nominal deposit, Dt(j), from households at the risk-free rate,Rt, and then extends three types of risk-free loans. The first type is working capital loans, Lf,t(j), to intermediate goods producers who must finance part of their expenditures for capital and labor services:

(3.44) Lf,t(j) =vKZk,tKt(j) +vHWtht(j).

The second type is credit, Lx,t(j), to exporters who must finance a fraction of their homogeneous goods bill:

(3.45) Lx,t(j) =vXPtXt(j).

Both these types of intraperiod loans are due at the end of periodt. The third loan type is interperiod loans to entrepreneurs, Le,t(j), who need to purchase raw capital. This type of loan is due at the beginning of period t+ 1. The bank’s balance sheet simply states that

Lt(j)≡Lf,t(j) +Lx,t(j) +Le,t(j) =Nb,t(j) +Dt(j).

The three types of loans pay out the non-contingent norminal returns, Rf,t, Rx,t and

Rk,t, respectively.

Optimal credit allocation

I also assume that an exogenous random fraction,θb,t, of the banking earnings is retained to grow the business while the complementary fraction is paid out to shareholders as dividends at the end of period t. Given my assumption, it is clear that the larger the net worth of the bank, the greater the financial resources available to its shareholders. Thus it is in the interests of shareholders to request that their bankers maximize net worth. Also, I assume that shareholders value a particular portfolio of loans proposed by bankers according to the expected future discounted value of the owned funds. Formally, the banker solves the following problem:

max ˜ Nb,t+1(j) Et ( _∞ X s=0 (1−θb,t+1)θb,t+1+sβs+1Ξt,t+1+sNb,t+1+s(j) ) =Vt(j)

However, a moral hazard problem arises when bankers can discretionarily divert an exogenous time-varying fraction$tof assets on the balance sheet for their own interest.10 Shareholders would therefore only approve the project of loan allocations proposed by bankers if the discounted funds owned from eacht+ 1 period were no less than the assets diverted in the period t. Therefore the following cash constraint

(3.46) Vt(j)≥$t(Lf,t(j) +Lx,t(j) +Le,t(j)), 0< $t<1. 10_{In Gertler and Karadi’s (2011) framework,$}

t is traditionally referred to as a time-invariant steal-

ing fraction of assets and thus financial frictions on the liability side originally come from a fear that bankers would steal. I believe that an interpretation of asset diversion at the banker’s discretion is better suited for a time-varying parameter. Therefore, following this interpretation, I present a slightly different framework in which the scheme of incentive constraint works due to the skewed nature of banks’ compensation.

must be satisfied in each period t. The linearity of the banker’s optimization problem implies (3.47) Vt(j) =τf,tLf,t(j) +τx,tLx,t(j) +τe,tLe,t(j) +γtNb,t(j), with τf,t=βEt Ξt,t+1 (1−θb,t+1)(Rf,t−Rt) +θb,t+1 Lf,t+1(j) Lf,t(j) τf,t+1 (3.48) τx,t =βEt Ξt,t+1 (1−θb,t+1)(Rx,t−Rt) +θb,t+1 Lx,t+1(j) Lx,t(j) τx,t+1 (3.49) τe,t=βEt Ξt,t+1 (1−θb,t+1)(Rk,t−Rt) +θb,t+1 Le,t+1(j) Le,t(j) τe,t+1 (3.50) γt=βEt Ξt,t+1 (1−θb,t+1)Rt+θb,t+1 Nb,t+1(j) Nb,t(j) γt+1 (3.51)

whereτf,t,τx,tandτe,tare the expected discounted marginal gains of various loans, while

γtis the expected discounted marginal value of net worth.

Observe that the left term in (3.46) cannot be strictly greater than the term on the right in each time period t. Otherwise, banks would make positive profits, which is incompatible with the competitive credit market. Thus the perfect competitiveness and the cash constraint in (3.46) jointly imply that (3.46) must hold as a strict equality in every period. Combining this fact with the linear value function (3.47), I obtain the aggregate incentive compatibility constraint for the banking sector,11

(3.52) Lt=φb,tNb,t, φb,t>1, (3.53) τf,t=τx,t =τe,t ≡τt

whereφb,t= _$tγ−tτt is the banking leverage multiple. The time-varying divertible fraction

of assets,$t, directly affects the bank’s balance sheet and its capability to lend, which I refer to below as the bank risk shock.12 Thus the moral hazard problem imposes an en- dogenous incentive constraint on the bank’s balance sheet. Note that in equilibrium the incentive constraint is symmetric across all types of loans and thus banks are indifferent to lending an additional unit among various borrowings.13 The (Lf,t(j), Lx,t(j), Le,t(j)) combinations that satisfy the (3.53) arbitrage condition define an optimal loan portfolio approved by the representative household’s shareholders and allocated byjth banker to various borrowers.

11

This aggregation is possible due to the constant number of bankers in the economy as well as their risk neutrality and perfect competitiveness. Also, there exists one aggregate loan demand curve that is identical for all bankers. Therefore thejindex has been dropped.

12_{Shocks of this kind occur when bankers pursue goals which have a high profile. Bankers can, for}

example, relax lending criteria for undesirable big projects or increase discretionary expenditure on perks. In this way, they give insights into prestige, job security and power.

13

The arbitrage condition, however, does not imply that the expected returns from various loans are identical (not even up to first order) as the marginal gain for each type of loan depends on the growth rate of each loan type.

Net worth evolution and modified resource constraint

Aggregating across all banks yields the following expression for end-of-periodtbanking profit:

(3.54) Vb,t= (Rf,t−Rt)Lf,t−1+ (Rx,t−Rt)Lx,t−1+ (Rk,t−Rt)Le,t−1+RtNb,t−1.

After bankers have received returns on the optimal portfolio of loans and settled their obligations to depositors, the complementary fraction, 1−θb,t, is paid out as dividends and consumed within the period. Therefore the per-capita consumption of shareholders is

(3.55) Pc,tCb,t= (1−θb,t)Vb,t.

Banks also raise exogenous additional capital, which corresponds to a fraction χb of the balanced-growth-path aggregate net worth, nbzt. Thus, after capital raising, aggregate net worth at the end of periodtis

(3.56) Nb,t=θb,tVb,t+χnbzt.

Disturbances to the retention ratio θb,t would cause sudden volatilities in the value of assets on the bank balance sheet, which in turn would impact on the lending capacity of the bank.14

The weighted average lending-deposit spread is given by

(3.57) sprb,t=

Rf,tLf,t+Rx,tLx,t+Rk,tLe,t

RtLt

.

The resource constraint from the Pure model is modified as follows:

(3.58) 1 gt Yt= ˜Cd,t+Id,t+Xt, with ˜ Cd,t= (1−ωc) Pt Pc,t −ηc (Ct+Cb,t) and C˜m,t =ωc Pcm,t Pc,t −ηc (Ct+Cb,t).

3.3.3 Foreign economy, exogenous disturbances and comparable per-