Notes on Cash-in-Advance PDF

Notes on Cash-in-Advance Steve Williamson University of Western Ontario December 2024 1 Introduction When making modeling choices in monetary economics, it is a good thing if we can be explicit about the role that m...

Notes on Cash-in-Advance Steve Williamson University of Western Ontario December 2024 1 Introduction When making modeling choices in monetary economics, it is a good thing if we can be explicit about the role that money and other assets play in the economy. But sometimes explicitness comes at a cost, and we need to make tradeoffs. For some purposes it is productive sometimes to take shortcuts, so that we can say something useful about economic policy, for example. In monetary economics, shortcuts are very common, particular in work on monetary policy. There are three commonly-used shortcuts, all of which involve assuming what money is at the outset of the modeling exercise. In the first approach, money is assumed to be an argument in the utility function – people hold money because they like it. In the second approach, money is assumed to enter the production technology, in that holding money frees up resources (time, for example), that can be used for other purposes. Third, in cash-in-advance models, it is assumed that money is necessary for executing particular transactions. Wallace (1998) makes a case that none of these approaches are desirable, but other economists disagree, as there is much published work using each of the three shortcuts. The cash-in-advance approach was first suggested by Clower (1967), and then developed by Lucas (1980), who used it extensively. In the model constructed here, it is assumed that consumption goods must be purchased with money ac- quired in advance of when purchases are made. The model is convenient for showing some standard results in monetary economics that are essentially ubiq- uitous – common to essentially all mainstream models of monetary economies. These results include the relationships between the real interest rate, the nomi- nal interest rate, and inflation. We can also show the effects of monetary policy on inflation, the costs of inflation, and a common prescription for monetary policy, the Friedman rule. 1 2 The Model Time is indexed by t = 0, 1, 2,..., and there is a single representative consumer who maximizes X∞ β t [u(ct ) − v(nst )] (1) t=0 In (1), β is the discount factor, where 0 < β < 1, ct is consumption in period t, and nst is labor supply in period t. The utility function u(·) is strictly increasing, strictly concave, and twice continuously differentiable, with u′ (0) = ∞. The function v(·) denotes the disutility of supplying labor for the consumer, where v(·) is assumed to be strictly increasing, strictly convex, and twice continuously differentiable, with v ′ (0) = 0 and v ′ (n̄) = ∞ for some n̄ > 0. The representative firm produces output yt using only labor as an input, according to yt = andt , (2) where a > 0 is productivity, and ndt is the firm’s labor input. The government can issue money through lump-sum transfers and, similarly, can retire money through lump-sum taxation. Letting τt denote the lump-sum transfer to the consumer in period t, with Mts denoting the nominal money supply, and Pt the price level, the government’s budget constraint can be written s Mts − Mt−1 = τt (3) Pt For convenience, we will use the consumption good in each period as the numeraire, and express all quantities in terms of consumption goods, with these real quantities denoted by lower case. Letting πt = Pt /Pt−1 − 1 denote the inflation rate, we can then rewrite equation (3) as mst−1 mst − = τt (4) 1 + πt 2.1 Timing Within the Period The assets in this economy are money, nominal bonds, and real bonds. Assuming a representative agent, as we have here, implies that any assets we add to this economy, other than money, will not matter for the determination of equilibrium quantities and prices. But, this feature will allow us to determine asset prices in a convenient way, which is a virtue of the representative agent paradigm. A nominal bond issued in period t is defined to be a promise to pay one unit of money at the beginning of period t + 1. A real bond – essentially an indexed bond – is a promise to pay one unit of consumption goods one period hence. md t−1 At the beginning of the period, the consumer has 1+πt units of money bt−1 1+πt nominal bonds, and ft−1 real bonds, all of which were acquired in the previous period. The consumer receives the transfer τt from the government, 2 and then takes money, the payoffs on bonds, and the transfer, and allocates this beginning-of-period wealth to purchases of newly issued nominal and real bonds, and to purchases of consumption goods. Consumption goods must be purchased with money. So, in the asset market that opens at the beginning of the period, the consumer must satisfy the asset market constraint mdt−1 bt−1 ct + qt bt + st ft ≤ + + ft−1 + τt (5) 1 + πt 1 + πt In (5) qt and st denote, respectively, the prices of nominal and real bonds. When the asset market closes, the consumer goes to work for the firm, supplying labor. But the consumer’s wages will not be paid until the end of the period (after the firm sells its output). So, the consumer must purchase consumption goods on the goods market with assets he or she has on hand, and the firm will only accept money in exchange for the goods it sells. At the end of the period, the firm makes wage payments to the consumer, and the consumer carries its assets into the next period. The consumer’s budget constraint is mdt−1 bt−1 ct + qt bt + st ft + mdt = + + ft−1 + wt nst + τt (6) 1 + πt 1 + πt Note that the difference between (5) and (6) is that money holdings mdt appear on the left-hand side, and wage earnings wt nst appear on the right-hand side of (6), where wt is the real wage. 2.2 Optimization In this model, the representative consumer solves a dynamic optimization prob- lem. We will assume that the consumer knows the complete path for prices and the inflation rate, (wt , qt , st , πt+1 ), for t = 0, 1, 2,.... Equivalently, con- sumers observe w0 , q0 , and s0 in period 0, form expectations of (wt , qt , st , πt ) for t = 1, 2, 3,..., and in equilibrium those forecasts are correct. That is, we are looking for a rational expectations equilibrium for this economy. If there were random aggregate shocks in this economy, then in a rational expectations equi- librium consumers would use information efficiently to form their forecasts of future prices. In such a world, with aggregate uncertainty, consumers would not always forecast correctly, but in equilibrium they would not make systematic errors. Here, without aggregate uncertainty, consumers’ forecasts are always correct in a rational expectations equilibrium. To solve for a rational expectations equilibrium, we need to start by solving the representative consumer’s optimization problem. This optimization prob- lem looks formidable, as it is infinite-dimensional. In any period t, the consumer chooses consumption ct , labor supply nst , and assets (mdt , bt , ft ). This is essen- tially a consumption-savings decision, with the additional complication that the consumer chooses how to allocate savings among the three assets. Choices about which assets to hold in period t then have implications for future consumption- savings decisions in periods t + 1, t + 2,... One approach to solving this opti- mization problem is to formulate it as a dynamic programming problem, but it 3 will be more straightforward in this context to structure the problem the way we would a static constrained optimization problem. The Lagrangian for the consumer’s constrained optimization problem is infinite-dimensional, but we can write it as the disounted sum of period La- grangians, where the period Lagrangian for period t is given by Lt = u(ct ) − v(nst )  d  mt−1 bt−1 + λt + + ft−1 + τt − ct − qt bt − st ft 1 + πt 1 + πt  d  mt−1 bt−1 + µt + + ft−1 + wt nst + τt − ct − qt bt − st ft − mdt (7) 1 + πt 1 + πt In (7), note that the period Lagrangian consists of the period utility of the representative consumer, from (1), and includes the two constraints (5) and (6), just as we would write this in a standard problem. Here, λt is the multiplier as- sociated with the asset market constraint (5), and µt is the multiplier associated with the budget constraint (6). Then, we can write the consumer’s optimization problem as choosing (ct ,nst ,mdt , bt , ft , λt , µt ) for t = 0, 1, 2, 3,..., to maximize ∞ X β t Lt (8) t=0 The first order conditions for a maximum are u′ (ct ) − (λt + µt ) = 0, (9) −v ′ (nst ) + µt wt = 0, (10)   λt+1 + µt+1 −µt + β = 0, (11) 1 + πt+1   λt+1 + µt+1 −qt (λt + µt ) + β = 0, (12) 1 + πt+1 −st (λt + µt ) + β (λt+1 + µt+1 ) = 0, (13) mdt−1 bt−1 + + ft−1 + τt − ct − qt bt − st ft = 0, (14) 1 + πt 1 + πt mdt−1 bt−1 + + ft−1 + wt nst + τt − ct − qt bt − st ft − mdt = 0. (15) 1 + πt 1 + πt Next simplify, by substituting in (11)-(13) using (9) and (10), to get wt u′ (ct+1 ) v ′ (nst ) = β , (16) 1 + πt+1 u′ (ct+1 ) qt u′ (ct ) = β , (17) 1 + πt+1 4  st u′ (ct ) = βu′ (ct+1 ). (18) Equation (16) states that, when the consumer is optimizing, the marginal disu- tility of supplying labor must be equal to the marginal payoff for the consumer. When the consumer works, wages go into money balances that are held over and spent in the next period. Therefore, the effective real wage for the consumer is (wt Pt )/Pt+1 = wt /(1 + πt+1 ). At the margin then, wages earned are valued according to the discounted marginal utility of future consumption βu′ (ct+1 ). In equation (17), when the consumer buys one nominal bond in period t, what is foregone, in units of consumption goods, is qt /Pt , which is valued at the marginal utility of consumption u′ (ct ). The payoff for the consumer comes in period t + 1, when the bond pays off one unit of money, or 1/Pt+1 units of consumption goods, valued at the marginal utility of future consumption u′ (ct+1 ). When the consumer is optimizing, marginal utility foregone in the present must equal marginal utility gained in the future (discounted), which gives (17). Similarly, in (18), when the consumer acquires a real bond in period t, foregone marginal utility is st u′ (ct ), and the discounted marginal future benefit (discounted) is βu′ (ct+1 ). From (11) and (12), we obtain µt qt = , (19) λt + µt So, if qt = 1, then λt = 0. That is, the nominal interest rate is Rt = q1t − 1,so if the nominal interest rate is zero the asset market constraint does not bind. So, if money and bonds bear the same rate of return, the consumer is willing to hold money from this period to the next without spending it. We will assume that the asset market constraint (5) holds with equality, which amounts to assuming that when the consumer is indifferent between money and bonds, he or she only holds bonds until the next period. Equation (19) also tells us that a positive nominal interest rate is associated with a binding asset market constraint, i.e. λt > 0. Also, from (9), (10), and (19), we get wt v ′ (ns ) = ′ t (20) 1 + Rt u (ct ) The quantity on the right-hand side is the rate at which the consumer is willing to substitute labor for consumption which, if there were efficiency, would be equated to the real wage wt. Thus, a positive nominal interest rate reflects an inefficiency – consumers work and consume too little relative to what is efficient. The firm maximizes profits each period, solving max ndt (a − wt )   (21) nd t That is, the firm’s optimization problem is much simpler than for the consumer, as it’s static. For the firm, the solution is ndt = 0 if a < wt , ndt = ∞ if a > wt , and any ndt ≥ 0 is optimal if a = wt. 5 3 Equilibrium A competitive equilibrium satisfies three properties: 1. Consumers and firms optimize. 2. The government satisfies its budget constraint (3) in each period. 3. Markets in consumption goods, labor, money, nominal bonds, and real bonds, clear in each period. Consumer optimization is summarized by the first-order conditions from the consumer’s optimization problem, and our solution to the firm’s optimization problem, which tells us the firm’s demand for labor as a function of the market real wage. The market-clearing conditions are ct = yt , (22) ndt = nst , (23) mdt = mst , (24) bt = 0, (25) ft = 0. (26) Note that, for bonds, there has to be zero net supply in equilibrium. In the model, the bond market is a market in private bonds. That is, the consumer sees himself or herself as being able to issue bonds or purchase bonds at will, i.e. to borrow or lend on the bond market. But, in equilibrium, prices have to be such that it is not in the representative consumer’s interest to either issue bonds or to purchase them. We will assume that the government sets transfers over time so that the nominal money supply grows at a constant rate i, that is Mt+1 = (1 + i)Mt , (27) for t = 0, 1, 2,.... In the next steps, we’ll characterize the equilibrium. First, from (4), (14), and the equilibrium conditions (22)-26), and letting mt = mst = mdt , and nt = nst = ndt , we get mt = ant (28) or M t = Pt y t , (29) which is just the quantity theory of money, usually stated as M V = P y, where M is the money supply, V is the velocity of money, and P y is aggregate nominal income. In our model, money turns over once very period, so the velocity of money is 1. 6  Given the firm’s demand for labor, the labor market can clear only when wt = a, so this is the equilibrium real wage rate. Then, from (16), (28), and (22)-24), and (27), we get ant+1 u′ (ant+1 ) v ′ (nt )nt = β , (30) 1+i Equation (30) is then the key to solving the model. We want to find a sequence n0 , n1 , n2 ,..., that solves this equation (a first-order nonlinear difference equa- tion). In principle, there could be many solutions, but we’ll focus on just one, which is nt = n, a constant, for all t. Then, from (30), we get au′ (an) v ′ (n) = β (31) 1+i Equation (31) then solves for n, and then we can work backward, and solve for all the other variables of interest. From (2) and market-clearing, c = y = an, and from (29), Pt = M an , and the inflation rate is π = i. In one sense money is t neutral here, as its level has no bearing on determining n. That is, the money supply in period t is Mt = M0 (1 + i)t , (32) but M0 appears in no equations, except that it matters for Pt , i.e. M0 (1 + i)t Pt =. (33) an So, increasing M0 just increases all nominal variables in proportion, and has no real effects. But, money is non-neutral, in the sense that its growth rate matters. From equation (31), an increase in i causes n to fall, so that output and consump- tion fall as well. That is, higher money growth causes a one-for-one increase in inflation, and this reduces labor supply, because income today is spent tomor- row, when its real value has depreciated due to inflation. That is, inflation is a distortion in this economy – it works like a tax. For interest rates, from (17) and (18), and the solution above, we get β q= (34) 1+i s=β (35) So, the equilibrium nominal interest rate, and real interest rate, respectively, are 1+i R= −1 (36) β 1 r = −1 (37) β Let 1 β= , (38) 1+ρ 7 where ρ is the consumer’s subjective discount rate. Then r = ρ, a constant. That is, the real interest rate is invariant to monetary policy, and inflation. Further, as an approximation, R ≈ i + ρ, (39) or, R = i + r. (40) This is sometimes called the “Fisher equation,” as it illustrates the Fisher ef- fect, which is that anticipated inflation is reflected in nominal interest rates. Here, the Fisher effect is one-for-one – a permanent increase of 1% in the money growth rate, which implies a 1% permanent increase in the inflation rate, re- sults in a 1% permanent increase in the nominal interest rate. This might seem counterintuitive, as we might think that higher money growth is a more “accom- modative” monetary policy, which should make nominal interest rates go down (and maybe inflation should go down as well). But that is intuition for possi- ble short-run effects, which we have not modeled. For now, it is important to recognize that this is a model of the long-run effects from perfectly anticipated changes in money growth. 4 Optimal Monetary Policy The way we have set this model up, the choice for the government is the setting for i, the money growth rate. If the government’s goal is to maximize the welfare of people living in this fictitious world, what should it do? As a benchmark, we can first ask what could be achieved by a fictitious social planner who can tell the representative firm what to produce, and can tell the representative consumer how much he or she should work. The social planner does not throw anything away, so ct = ant , and the problem the planner solves is to choose n0 , n1 , n2 ,..., to maximize X∞ β t [u(ant ) − v(nt )] , (41) t=0 so optimal labor supply is nt = n∗ for all t, where n∗ solves au′ (an∗ ) − v ′ (n∗ ) = 0. (42) If there is a money growth rate i that achieves nt = n∗ as an equilibrium outcome forever, then that money growth rate is optimal. From (31) and (41), the optimal money growth rate is i = β − 1. Therefore, at the optimum the nominal interest rate is R = 0, so the inflation rate is π = β − 1. As a result, money, real bonds, and nominal bonds all have the same real rate of return, which is β1 − 1 = ρ, the subjective rate of time preference. In general, a positive nominal interest rate reflects the existence of a distortion – a suboptimality – which can be corrected with the right monetary policy. But note that the optimal policy involves a constantly contracting money supply, which induces 8 deflation, i.e. a negative inflation rate. This optimal monetary policy is a Friedman rule (Friedman 1969). The Friedman rule is an optimal monetary policy rule in a wide class of monetary models. Generally, it can be defined as a monetary policy which implies that the nominal interest rate on safe assets is zero forever. What results from this model should we take seriously? Certainly, high money growth is associated with high inflation in practice, but the model leaves out many details of banking and financial markets that are important for the relationship between money growth and inflation. In the model, money is best interpreted as currency, and so we have left out means of payment that compete with currency – transactions accounts at banks and credit cards, for example. In the model there is a tight relationship between money growth and infla- tion because the velocity of money is fixed, but velocity is highly variable in practice. Indeed, central banks attempted to control inflation in the 1970s and 1980s by controlling money growth, but this proved unreliable, principally be- cause changes in technology, changes in regulation, and short-run shocks in the financial sector cause long run changes in velocity, and cause it to be volatile in the short run. In other words, money supply targeting by central banks, which was advocated by Milton Friedman and other “monetarists,” requires that the demand for money be stable over time, but it is not. The model tells us that monetary expansions lead to higher nominal interest rates and higher inflation. But, conventional central banking wisdom says that monetary expansions reduce nominal interest rates and increase inflation. Is the model missing something, or is conventional wisdom wrong? In defense of the model, we could say it is intended for studying the long run effects of money growth, and in the long run, it’s clear that higher money growth produces higher inflation, which is reflected in higher nominal interest rates. As well, central bankers have sometimes been surprised by low inflation in periods when nominal interest rates have been low for a considerable time. But our model would say they should not have been surprised. Finally, the model tells us that a Friedman rule, which generates zero nom- inal interest rates and persistent deflation, is optimal. But central bankers sometimes view zero nominal interest rates as an emergency monetary policy, and there is a common view that deflation is dangerous. But, some central banks in the world have persisted with interest rate policy close to zero, and in some cases have implemented negative nominal interest rates (in the Euro area, Sweden, Switzerland, Denmark, and Japan, for example). While in some cases this has produced deflation, it would be hard to characterize these periods with zero or subzero nominal interest rates as disasters. References Clower, R. 1967. “A Reconsideration of the Microfoundations of Monetary Theory,” Economic Inquiry 6, 1-8. 9  Friedman, M. 1969. “The Optimum Quantity of Money,” in The Optimum Quantity of Money and Other Essays, Aldine Publishing Company, New York. Lucas, R. 1980. “Equilibrium in a Pure Currency Economy,” in Kareken, J. and N. Wallace, eds., Models of Monetary Economies, pages 131-145, Federal Reserve Bank of Minneapolis, Minneapolis, MN. Wallace, N. 1998. “A Dictum for Monetary Theory,” Federal Reserve Bank of Minneapolis Quarterly Review, Winter, pages 20-26. 10