Households live forever and have standard intertemporal preferences over consumption (Ct) and hours worked (Ht). Investment (It), which is subject to adjustment costs, cannot be funded directly by households and instead must be financed by borrowing from a bank. Bank loans (Lt) are financed by the capital of bankers (Bt) and deposits (Dt) from households. Banks are owned and run by bankers. A proportion (1-γ) of bankers die at the end of each period and are replaced by a new generation. Bankers reinvest all their earnings in the capital of the bank whilst they live, and leave their net wealth to households when they die. In log-linear form (carets denote log-deviations from steady-state), the standard block of the model is described by: 1)  Market-clearing condition for output (Yt);  2)  Production function (Zt is TFP and Kt is capital);  3)  Labour market equilibrium (Xt is marginal costs);  4)  Price of physical capital (qt);  5)  Return on capital (rkt);  6)  Law of motion for physical capital;  7)  Consumption Euler equation (rnt is deposit rate);  8)  New Keynesian Phillips curve (πt is inflation rate). The special ingredient is a moral hazard friction in the banking sector.
Changes in bank profits, in turn, reflect the average spread between the loan rate and the deposit rate on changes on the deposit base, the average return on lending on changes in bank capital, positive shocks to the rate of return on current lending and negative shocks to the deposit rate. We assume that policy makers seek to minimise a loss function:   13)      L    j  t2 j   y yt  j  yt* j j 0     k 2  l  t j   k t* j   2  The addition of the arguments in kt is designed to reflect financial stability concerns. We assume equal weights of 1 on inflation and 0.01 on the output gap and capital gap. The starred variables are the levels of output and lending which would hold in a frictionless economy. Policymakers are set the nominal interest rate and the bank levy/subsidy. The model was calibrated with standard parameter values at a quarterly frequency (see, for example, Bernanke, Gertler and Gilchrist, 1999). Bank leverage was set at 10 in the steady state and the steady-state spread between the return on capital and the deposit rate is 1 percent.