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Chapter Two Banks

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Chapter Two Banks Chapter Two Banks 2.1 THE BASIC EQUATIONS In Table 2-l the important symbols used in this chapter are listed in alphabetic order. The first half of the table presents the notation for the non-condensed model, and the second half presents the notation for the condensedmodel. The notation for the condensed model pertains only to the discussion in Section 2.6. Each bank, say bank i, receives money from households in the form of savings deposits (SD&), on which interest is paid, and from firms, households, and &e bond dealer in the form of demand deposits @D&t), on which no interest is paid. Each bank lends money to tirms and households (L&) and buys gOVeINIXW bills (VBILLBi~) and bonds (BONDBit). Bank loans XC one-period loans, bills are one-period securities, and bonds are consols. Each bank holds reserves in the form of deposits with the government (B&). Each bank sets its own loan rate (R&).The three main decision variables of each bank are its loan rate, the value of bills and bonds to purchase (V~&), and the max- imum amount of money that it will lend in the period (LLWfA&). Banks are assumed not to compete for wings deposits, and the rate paid on all savings de- posits is assumed to be the bill rate (rr)I 79 20 A Model of Macroeconomic AC tivity Volume I: The Theoretical Model Table 2-I. Notation for Banks in Alphabetic Order Subscript i denotes variable far bank i. Subscript j denotes variable for bankj. Subscript f denotes variable for period f. An e superscript in the text denotes an expected value of the = number of bonds held, each bond yielding one dollar per period = actual reswves = required mserves = prclfit tax rate = penalty tax mte on the composition of banks’ portfolios = demand deposits = dividends paid = largest error the bank expects to make in overestimating its demand deposits for any period = latgest error the bank expects to make in overestimating its savings deposits for any period = amount that the bank knows it will have available to lend to households and firms and to buv bills and bonds even if it overestimates its demand and savings depositiby the maximum amounts = reserve xquirement ratio = no-tax proportion of banks’ portfolios held in bills and bonds = total value of loans of the bank sactor = value of loans = maximum value of loans that the bank will make = toial unconsuained demand for loans = bill rate = bond mte = loan rate (of bank i) = loan fate (of bankj) = average loan rate in the economy = savings deposits = taxes paid = length of decision horizon = value of bills and bonds that the bank chooses to purchase [ VBILLBit + 80NDBit/R ?] = value of bills held = before-ax profits Subscript I denotes variable for period f Only notation that differs from the notation for the non-condensed model is presented here. EMAXDD = largest enor the bank sector expects to make in overestimating its demand deposits for any periad EMAXSD = largest erra the bank sector expects to make in overestimating its savings deposits for any period FUNDS+ = amount the bank sector knows it will have available to lend to households and firms and to buy bills and bonds even if it overestimates its demand and savings deposits by the maximum amounts LBluAX* = maximum value of loans that the bank sector will make RLt = loan rate of the bank sector SD, = savings deposits of the bank sector TAXB, = taxes paid by the bank sector VBB, = value of bills and bonds that the bank S~CLOIchooses to purchase ( VBILLB, + BONDB</RJ VBILLB, = value of bills held by the bank sector w = before-tax profits of the bank sector Banks 21 The basic equations for bank i for period t are the following: VBB,, = VBILLB,, + BONDBJR,, [value of bills and bonds held] (2.1) II& =RBi,LBi, + I? VBILLB,, + BONDBit - r,SDBi, + (BONDBir/Rr+I - BONDBJR,), (before-tax profits] (2.2) TAXB, = d, Wit fdZ [VSS,, - gz(VBBi, + LBit)]‘, [taxes paid] (2.3) DIV& = llBi, - TAXBi,, [dividends paid] (2.4) BR,, =DDBit + SDB,, - LBi, - VBB,, - (BOND13JRr+l - BONDB<,/R,) =DDBi, + SDB?, - LB!, - VBILLB,, - BONDBi,/R,+I, (actual reserves] (2.5) BR *if =glDDBit, [required reserves] (2.6) BR,, >BR*!, [actual reserves must be greater than OI equal to required IeRKVeS] (2.7) Equation (2.1) merely defines the value of bills and bonds held. Since bonds are consols and since each bond is assumed to yield one dollar each period, the value of bonds held is merely the number held divided by the bond rate, BONDBi,/R,. Equation (2.2) defines before-tax profits. The first three terms on the right-hand side of the equation are the interest revenue received on loans, bills, and bonds, respective1y.a The fourth term is the interest paid on savings deposits. llw last term is the capital gain or loss made on bonds held in period t. Taxes are defined in Equation (2.3), where dl is the profit tax rate. With respect to the second term on the right-hand-side of the equation, the government is assumed through its taxing policy to try to induce banks to hold a certain proportion, g2, of their assets in bills and bonds. In practice, commercial banks and other financial intermediaries are under certain pressures to hold particular kinds of securities, and here these pressures are assumed to take the form of government taxing policy. If, in the model, banks were not induced in some way to hold bills and bonds, they would never want to hold bills and bonds as long as their loan rates were higher than the bill rate. The introduction of government taxing policy is a simple way of explaining why banks hold nvxe than one kind of asset. In Equation (2.3), bank i is assumed to be taxed at rate dz on the square of the difference between the value of bills and bonds held and &!z times the value of loans issued plus bills and bonds held. Since capital gains and losses are, included in the definition of profits, Equation(2.3) also reflects the assumption that capital gains and losses are taxed a$ regular income. Bank i is 22 A Model of Macroeconomic Activity Volume I: The Theoretical Model assumed not to retain any earnings, so that the level of dividends, as defined in Equation (2.4), is merely the difference between before-tax profits and taxes. Bank reserves are defined in Equation (2.5). Slice bank i pays out in the form of taxes and dividends any capital gains made in the period (and conversely for capital losses), and yet does not receive any actual cash flow from the capital gains, capital gains take away from (and conversely capital losses add to) bank reserves, as specified in (2.5). Required reserves are defined in Equation (2.6). For simplicity, no rewve requirements ax placed on savings deposits. Actual reserves must be greater than or equal to required reserves, as indicated in (2.7). 2.2 THE FORMATION OF EXPECTATIONS Let T+I be the length of the decision horizon. In order for the bank to solve its control problem at the beginning of period t, it must form expectations of a number of variables for periods r through t+T Bank i is assumed to form the following expectations.b fqt ,z [expected loan rate of‘ bank q-1 j for period t] W) , [expected loan rate of bank j for period r+k (k=l,Z,...,T)] (2.9) q+k =(R&r+k * qr+,):, [expected average loan rate for period t+k (k=O,I,. .,7)] (2.10) a3 > 0, [expected aggregate unconstrained demand for loans for period f] (2.11) LUIY;;, =Luq+k_, , [expected aggregate unconstrained demand for loans for period r+k (k = I,&. .,7’J] (2.12) L;+k = Luiv !+k, [expected aggregate constrained demand for loans for period wk (k=O,l,. .,TJ] (2.13) , a4 < 0, [expected market share of loans for period t] (2.14) Banks 23 [expected market share of loans for period t+k(k=I,Z,. ..7)] (2.15) DD%+k=DDB,f_I, [expected level of demand deposits for period t+k (k=O,l,. .,Tf] (2.16) sD%+k= SDB,t_l, [expected level of savings deposits for period f+k (k=O,I,. ..T)] (2.17) I& =rr, [expected bill rate for period t+k (k=I,Z,. .)] (2.18) I 1 I 1 +... R& = %) + (1+&k) (l+<+k+l) + (I+‘?+k) (~‘%+I) (l+<+k+z) = 1. [equation determining expected bond rate for period t+k rt (k = OJ,. .,T+I)] (2.19) The first term on the right-hand side of Equation (2.8) reflects the fact that bank i expects its rate setting behavior in period t-2 to have an effect on bank j’s rate setting behavior in period t The second term is designed to represent the effect of general market conditions on bank i’s expectation of bank j’s rate. The bond dealer sets the bill rate for period tat the end of period t-l, and bank i knows the bill rate for period f at the time that it makes its decisions for period t. If the bill rate for period i has changed, then bank i is assumed to expect that this change will have an effect in the same direction on the rate that bank j sets in period t Bank i must also form expectations of bank j’s rate for periods WI and beyond. These expectations are specified in Equation (2.9), which is the same as Equation (2.8) without the final term.
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