Money in UtilityModels and examplesJean-Paul Lam1February 24, 20211These notes are the property of the author. They are intended for the Winter 2021 Econ 406class and can only be shared with students registered in the class.1 / 150Outline of lecture◮ MIU Model◮ Model set up◮ Steady-state◮ Neutrality and Superneutrality◮ Seignorage and fiscal considerations◮ Optimal rate of … Continue reading “Money in Utility Models and examples | My Assignment Tutor”
Money in UtilityModels and examplesJean-Paul Lam1February 24, 20211These notes are the property of the author. They are intended for the Winter 2021 Econ 406class and can only be shared with students registered in the class.1 / 150Outline of lecture◮ MIU Model◮ Model set up◮ Steady-state◮ Neutrality and Superneutrality◮ Seignorage and fiscal considerations◮ Optimal rate of inflation◮ Stochastic version of the model◮ Simulations of the MIU model◮ Application: Demonetization in India◮ Conclusion2 / 150Learning objectivesBy the end of the lecture, you will learn about:1. How money is introduced in macro models2. The meaning of neutrality and superneutrality of money3. The optimal rate of inflation4. The implications of money-financed deficits5. How to simulate the stochastic version of the model in Matlab3 / 150Introduction◮ Why do individuals want to hold money?◮ The above question is a fundamental issue in monetary and macroeconomics◮ It has been a challenge for macroeconomists to come up with answers to thisquestion◮ Many macroeconomic models assume that individuals want to hold moneybecause it facilitates transactions or it provides direct utility to the holder ofmoney◮ We are going to assume the second approach4 / 150Introduction◮ Typically economists have incorporated money in general equilibrium modelsby adopting these different approaches:1. Money directly in utility2. Cash-in-advance3. Shopping time and Real resource costs4. Search theory5. Imperfect information5 / 150Money in utility◮ The first approach, money in utility, assumes that money directly enters inthe utility function◮ This model highlights the role of money as a medium of exchange◮ Money is valued because it is needed for the exchange of goods and services◮ Under this approach, the demand for money does not arise endogenously butrather because agents receive utility from holding money◮ Putting money in the utility function is an easy way of giving money valuewithout explicitly having to model how the demand for money arises6 / 150Cash-in-advance approach◮ The cash-in advance (CIA) approach is very similar to the MIU approach◮ It captures the role of money as a medium of exchange◮ Under this approach, money is needed to purchase goods, and services andagents cannot transact without money◮ There demand for money in this model also does not arise endogenously; it isimposed in the model7 / 150Shopping time and real resource cost models◮ Under the shopping-time model, agents spend time on buying goods(shopping)◮ Since agents in the model devote their time endowment between shopping,leisure and work, there is a cost associated with shopping◮ Money in the model provides liquidity services and frees up time that agentswould otherwise spend on transactions◮ Money as a medium of exchange, therefore, allows agents to spend more timeon work or leisure as money facilitates transactiions◮ In this way, money not only provides liquidity services but it also reducestransaction costs8 / 150Shopping time and real resource cost models◮ The basic idea of the real resource cost model is very similar to theshopping-time model◮ Agents incur costs when transacting; where the costs are in terms of realresources (instead of time as in the shopping-time model)◮ Money does not provide utility directly or indirectly as in the MIU and CIAapproach; money lowers transaction costs and frees up real resources spenton transactions9 / 150Search models◮ In basic money search models, money is used as a medium of exchange andas in the OLG model, it solves the double coincidence of wants◮ Barter is possible, but barter entails much higher search costs because it isdifficult to satisfy the double coincidence of wants◮ Unlike the MIU, CIA and the other models above, in the search model, theneed for money arises endogenously because money solves the doublecoincidence of wants◮ Search models, therefore, offer an explicit reason or a micro-founded reasonfor the existence of money10 / 150Informational frictions◮ Related to the basic search theoretic models of money, some models assumethat money is important because it helps to overcome some informationfrictions◮ These models assume that agents have informational problems and cannotassess the value of goods that are used in exchange◮ Fiat money solves this problem since money can be used as a unit of accountand can therefore provide to agents the signal they need to value goods andservices11 / 150Introduction◮ This lecture focusses on the MIU models◮ The structure of the model that we employ is very similar to the neoclassicalgrowth model except that money enters directly in the utility function.◮ We start describing the basic structure of the model, and we begin with thedeterministic version of the model to explain and derive some key results◮ We then move to the stochastic version of the model which we employ toperform simulations and derive IRFs by performing a shock to money12 / 150MIU Model◮ We assume that there is a single agent (representative agent – all agents areidentical)◮ The agent derives (instantaneous) utility from two sources:1. real per capita consumption denoted by ctct =CtNtPt2. real per capita money balances denoted by mtmt =MtNtPt◮ Ct is the nominal consumption at time t,◮ Pt price level,◮ Nt population,◮ Mt nominal money balances13 / 150MIU Model◮ The utility function of a representative agent is given byU(ct, mt) =∞ X t=0βtu (ct, mt) , (1)◮ with β as the discount factor◮ The agent receives utility from consuming but also from holding real moneybalances◮ In this model, money is assumed to yield utility directly to consumers.◮ This may seem a little strange, but in fact, many models of money demandare written this way.◮ The intuitive reason is that money provides “liquidity” or “transactions”services which people value.14 / 150MIU Model◮ Money provides utility because it reduces the time needed to purchaseconsumption goods (liquidity services)◮ It also provides utility because agents can insure against income fluctuationsby holding money (precautionary motive)◮ Moreover, including money directly in the utility function, implies that thereis a demand for money◮ As we argued in the introduction, the demand for money in this model doesnot arise endogenously; instead, it is imposed15 / 150Population growth◮ We assume that population grows deterministically at the rate n such thatNt = (1 + φ)Nt-1◮ For simplicity, we are setting φ = 0, and hence we have no population growthin this baseline model◮ Nt = Nt-1 = …Nt-n = …◮ Since all agents are identical, we assume that there is a single person in thiseconomy and set Nt = 1◮ If this is the case then we can rewrite ct = PCtNt t ≡ C Pt t and mt = PMt Nt t ≡ MPtt◮ The same definition applies to all the other per-capita variables16 / 150The MIU model◮ Each period, the agent receives:1. real income, τt where Yt = F (Kt-1, Nt)2. a real lump-sum transfer, τt resulting from changes in the nominal moneysupply3. return from their investment or savings, it-14. The agent carries (unspent) money from the previous period, Mt-1; the realvalue at time t is MPt-t 1 ◮ Yt is aggregate level of real output, Kt-1 is the aggregate level of the realcapital stock at time (t – 1) and Nt is the amount of labour at time t◮ Note that each dollar the agents saves pays it-1 in interest at time t; theinvestor gets back (1 + it-1)BPt-t 1 at time t17 / 150Lump sum transfer◮ We assume that changes in the level of nominal money are distributed toagents in the form of lump-sum transfers◮ Denoting, nominal lump sum transfers as Zt, the real value of these transfers,τ = ZtPt is equal to: τt ==–= mt –(2)Mt – Mt-1MtMt-1mt-1PtPtPt1 + πt 18 / 150The MIU model◮ The production function satisfies the neoclassical properties:1. Constant return to scale2. Positive and diminishing marginal products, fk ≥ 0, f ′′(k) ≤ 0 where f (k) isthe production function in intensive form3. Inada conditions are satisfied, limk→0 f ′(k) = ∞,limk→∞ f ′(k) = 0, wheref ′(k) is the marginal product of capital◮ The Inada conditions imply that as the capital stock tends to zero (infinity),the marginal product fo capital tends to infinity (zero), implying that theaddition of a new unit of capital is very valuable (useless)◮ The Inada conditions guarantees that there is a unique and stable solution tothe model19 / 150The MIU model◮ We can write the production function in per-capita terms as:yt =YtNt = F KNt-t 1 , NNtt = F NKt t- -1 1 , 1 = f (kt-1)◮ The above is also known as the intensive form of the production function◮ We can replace Nt with Nt-1 since Nt = Nt-1 when n=020 / 150MIU Model◮ The agent must choose to allocate their resources in four ways at time t:1. real consumption (ct = CPtt ),2. real money holdings mt = MPtt ,3. real investment xt = kt – (1 – δ)kt-1, where δ is the depreciation rate of lastperiod capital (set δ = 0 for now)4. real bonds (savings), bt = BPtt .◮ Hence we can write each period’s budget constraint as: ct + xt + mt + bt=yt + τt ++(3)Mt-1(1 + it-1)Bt-1PtPt ct + (kt – kt-1) + mt + bt = f (kt-1) + τt + Mt-1Pt +(1 + it-1)Bt-1Pt◮ Since xt = (kt – kt-1) and yt = f (kt-1)21 / 150Optimization problem◮ Denote the rate of inflation (the rate of change in prices) as πt = PtP-tP-t1-1 ,◮ We can rewrite Mt-1Pt as:Mt-1Pt = MPt-t 1 PPtt–11 = MPtt–11 PPt-t 1 ◮ Since Mt-1Pt-1 = mt and PPt-t 1 = 1+1πt , we have rewrite the above in per-capitaterms Mt-1PtPt-11 +mtπt = MPtt–11 Pt = ◮ You can use the same trick to write Bt-1Pt in per capita terms22 / 150Optimization problem◮ We can therefore rewrite the budget constraints as:ct + kt + mt + bt = f (kt-1) + kt-1 + τt (4)+mt-1(1 + πt) +(1 + it-1)bt-1(1 + πt)◮ The consumer maximizes utility (equation 1) subject to their budgetconstraint (equation 4) by choosing a sequence paths for ct, kt, bt, mt andλt.◮ λ denotes the set of Lagrange multipliers for the time t budget constraint.◮ We ignore uncertainty, and there is no labour-leisure choice in the baselinemodel23 / 150Optimization problem◮ Formally, the problem of the consumer consists ofmaxct,mt,bt,kt∞ X t=0βtu (ct, mt) ,s.t. ct + kt + mt + bt = f (kt-1) + kt-1 + τt+mt-1(1 + πt) +(1 + it-1)bt-1(1 + πt)(5)◮ And some non-negativity conditions on ct, mt, bt and kt24 / 150Optimization problem◮ Setting up the Lagrangean, we have the followingL =∞ X t=0βtu(ct, mt) + λt[f (kt-1) + kt-1 + τt+mt-1(1 + πt) +(1 + it-1)bt-1(1 + πt) – ct – kt – mt – bt25 / 150First-order conditions◮ The first order conditions (foc) are:ct : βtuc (ct, mt) – λt = 0 (6)mt : βtum (ct, mt) – λt + λt+1(1 + πt+1) = 0 (7)kt : -λt + λt+1 [f ′(kt) + 1] = 0 (8)bt : -λt + λt+1 (1 + 1 +πtit+1) = 0 (9)λt : f (kt-1) + kt-1 + τt + mt-1(1 + πt) (10)+(1 + it-1)bt-1(1 + πt) – ct – kt – mt – bt◮ where uc (ct, mt) is the marginal utility of consumption, um (ct, mt) is themarginal utility of money26 / 150First-order conditions◮ We also have the three transversality conditions:limt→∞βtktλt = 0 (11)limt→∞βtmtλt = 0 (12)limt→∞βtbtλt = 0 (13)◮ These transversality (or terminal) conditions imply that in the limit, nocapital, money or bonds are left unused; there is no overaccumulation ofwealth27 / 150Intertemporal Condition◮ Using the first-order conditions, we can easily derive an intertemporalcondition (Euler equation) and an intratemporal condition relating money toconsumption.◮ First derive the Euler equation relating consumption at time t and t + 1,recall the foc for ct and kt are:βtuc (ct, mt) – λt = 0-λt + λt+1 [1 + f ′(kt)] = 0◮ Substituting the foc for c into foc for k for λ, we haveuc(ct, mt) = β(1 + f ′(kt))uc(ct+1, mt+1) (14)28 / 150Intertemporal Condition◮ This consumption Euler equation is very similar to the consumption Eulerequation in the Neoclassical growth model.◮ There is a crucial difference as money enters the utility function, which is notthe case in the standard neoclassical growth model◮ If the marginal utility of consumption and the marginal utility of real moneybalances are correlated, the consumption Euler equation will contain termsinvolving the real balances.29 / 150Intertemporal Conditionuc(ct, mt) = β(1 + f ′(kt))uc(ct+1, mt+1)◮ The Euler equation shows the dynamic optimal decision path for consumptionbetween time t and t + 1◮ This equation implies that the marginal utility of consuming an extra unit ofresources today (uc(ct, mt)) must be equal to the marginal utility of savingan extra unit of resources today (the savings today yield, when discounted byβ, (1 + f ′(kt))uc(ct+1mt+1) in utility tomorrow)30 / 150Intertemporal Condition◮ The consumer when faced with an extra unit of resource can either consumeit and obtain utility uc(ct, mt) or they can invest the extra unit of resourceand obtain a return of (1 + f ′(kt)) next period which they can then consumeand enjoy utility (1 + f ′(kt))uc(ct+1, mt+1).◮ The consumer is indifferent between these two streams of utility, if themarginal utility from consuming that extra unit of resources today is preciselyequal to the present discounted value of the utility from consuming that extraunit of resources tomorrow◮ The consumer cannot do better in terms of utility by deviating form the Eulercondition31 / 150Intratemporal Condition◮ To obtain the intratemporal equation relating money and consumption, wesubstitute the foc ct in the foc for mt, simplify and we obtain:um (ct, mt) + βuc (ct+1, mt+1)(1 + πt+1) = uc (ct, mt) (15)◮ The marginal benefit of an extra unit of money holdings has two components.1. Since money enters the utility function, it provides direct utility to theconsumer just like consumption goods 2. Real money balances at time t add(1+π1t+1) to real per capita resources a time t + 1.◮ A dollar carried forward, deflated by the rate of inflation prevailing in period t + 1, is worth1(1+πt+1) in real terms◮ In terms of utility, it is worth βuc (ct+1,mt+1)(1+πt+1) at time t when discounted by thediscount factor.32 / 150Intratemporal Condition um (ct, mt) +(1 + πt+1) = uc (ct, mt)(16) βuc (ct+1, mt+1)◮ This intratemporal optimal equation implies that is an agent should beindifferent between holding an additional unit of money (marginal benefit hastwo components) and consuming an additional unit of a consumption good.◮ The agent cannot do better in terms of utility if they deviate from theoptimal path33 / 150Fisher relationship◮ Another way to express this intratemporal condition is as follows:◮ Use the first order condition for mt and divide through by uc (ct, mt) on bothsides, we obtain: um (ct, mt)uc (ct, mt) = 1 – βucu(cc(tc+1 t,,mmtt)+1) (1 +1πt+1)(17) ◮ Using the consumption Euler equation that we derived earlier, we know that:11 + f ′(kt) =βuc(ct+1, mt+1)uc(ct, mt) (18)34 / 150Fisher relationship◮ Substituting the Euler equation (eq18) in the revised intratemporal condition(eq17) uc (ct, mt), we have the following:um (ct, mt)uc (ct, mt) = 1 – (1 + 1f ′(kt) (1 +1πt+1)= (1 + (1 +f ′(fk′(t))(1 + kt))(1 +πtπ+1 t+1 ) -) 1 (19)◮ Substituting the foc for kt in the foc for bt, we obtain:(1 + f ′(kt))(1 + πt+1) = 1 + it (20)◮ The above equation is the Fisher equation that relates the nominal interestrate to the real interest rate (equal to the marginal product of capital) andinflation35 / 150Opportunity cost of holding money◮ Substituting equation(20) into equation (19), we obtain um (ct, mt)ituc (ct, mt)1 + it=(21) ◮ This equation implies that the MRS between money and consumption or theprice of real money balances relative to consumption is equal 1+it it , which isapproximately equal to it, the nominal interest rate, for small values of it◮ Therefore, the opportunity cost of holding money is the interest rate forgo onthe bond payment.36 / 150Money Demand◮ Equation (21) also allows us to derive a money demand equation and theinterest rate elasticity of money◮ To derive an expression for money demand, let us assume a log-utilityfunction where money and consumption are separable:u(ct, mt) = ln(ct) + ln(mt) (22)◮ Using the intratemporal condition given by equation 21um (ct, mt)uc (ct, mt) =it1 + it◮ and the fact that uc(ct, mt) = c1t and um(ct, mt) = m1t , we have the followingexpression:ct = 1 +it it mt ⇒ mt = 1 +it it ct (23)37 / 150Money Demand◮ Taking logs, we obtain:log mtd = log(1 + it) – log(it) + log(ct) (24)◮ The superscript d is added so that it is clear we are taking about moneydemand◮ If we assume that it is small, then we can assume that log(1 + it) ≈ 0, themoney demand equation simplifies to:log mtd = – log(it) + log(ct) (25)◮ Equation (25) is a standard money demand equation relating real moneybalances to the nominal interest rate and expenditure (here, consumption).◮ In this example, the income elasticity is, therefore, +1 while the interestelasticity of money demand is simply -138 / 150Steady-State◮ We now turn to the steady-state conditions of the model◮ The steady-state is where the model economy converges to in the absence ofany shocks (similar to long-run equilibrium)◮ Since we assume in the baseline model that there is no population growth. insteady-state, the real variables of the model (consumption, output,investment, etc) will be constant◮ Before defining steady-state more formally, we assume the following aboutthe rate of growth of nominal money and the relationship between changes innominal money balances and transfers39 / 150Steady-State◮ To analyze the effects of money on this economy in the long-run, we firstderive the steady-state equilibrium of the model.◮ Denote steady-state values by a bar over the variables; in steady-state, wedrop the time index and we have xt = xt+1 = … = ¯ x.◮ Using the first-order conditions of the model, assuming that the nominalsupply of money is growing at the rate of θ, we can derive the steady-stateconditions of the model40 / 150Steady-Stateuc (¯ c, m¯) = λ¯ (26)um (¯ c, m¯) = λ¯ 1 – 1 +β θ (27)1 β= f ′(k¯) + 1 ⇒ f ′(k¯) = 1β – 1 (28)1 β=1 + i¯1 + θ ⇒ 1 + i¯= 1 +β θ (29)41 / 150Steady-State◮ Since in steady-state, the amount of bonds outstanding is zero, the budgetconstraint is given by: c¯ + k¯ + ¯ m = f (k¯) + k¯ +m¯+ ¯ τ(30)(1 + ¯ π) ◮ As τt = mt – 1+ mt-π1t , therefore in steady-state ¯ τ = ¯ m – 1+¯ m¯π = 1+ m¯θθ.◮ Therefore, the budget constraint in steady-state simplifies to:c¯ = f (k¯) (31)◮ From the production function, we know thaty¯ = f (k¯) (32)◮ Since investment, xt = kt – kt-1, thereforex¯ = k¯ – k¯ = 0 (33)42 / 150Money growth◮ So far, in our simple model we have not defined how nominal money growsover time◮ Assume nominal money grows at the rate θ: Mt – Mt-1Mt= θ ⇒ = 1 + θ(34) Mt-1 Mt-1 43 / 150Steady-State◮ ThereforeMt – Mt-1Mt-1 = θ ⇒ MMt-t 1 = 1 + θ ⇒ MMPt-tt 1Pt= 1 + θMtPtMt-1Pt 1 + θ ⇒= 1 + θ ⇒= 1 + θmt-11+πtmt-1mt(1 + πt)mt ◮ This implies that in steady-state, the inflation rate grows at the same rate asthe growth rate of nominal moneyπ¯ = θ44 / 150Money neutrality◮ The steady-state conditions reveal that nominal money balances, that is Mtdoes not appear in any of the steady-state equations◮ This implies that changes in the level of nominal money balances do notaffect any real variables in steady-state or in the long-run◮ In other words, money is neutral and the model obeys the classicaldichotomy◮ Classical dichotomy implies that nominal money balances affect only nominalvariables and do not affect any real variables in steady-state45 / 150Neutrality◮ To see this further, let us assume that the production function is given by thefollowing Cobb-Douglas production function:yt = f (kt) = ktα ⇒ f ′(k) = αkα-1 (35)◮ Using the steady-state condition that we derived for kt, that is equation 28,we can solve for the steady-state value of k, that is k¯f ′(k¯) = 1β – 1αk¯α-1 = 1β – 1k¯ = 1αβ – β 1α-146 / 150Neutrality◮ Given the steady-state value for k¯, we can find the steady-state values for yand c:y¯ = ¯ c = f (k¯) = k¯α = 1αβ – β αα-1(36)◮ The capital stock, output and consumption depends on the discount factorand the share of capital α and not on the level of nominal money balances◮ Since k, c or y are independent on the level of nominal balances, money isneutral in this model in the long-run.47 / 150Neutrality◮ Monetary neutrality is a result that one can find in many different models◮ Economists generally agree that money is neutral in the long-run; where thedisagreement lies is whether money is neutral or not in the short-run◮ Recall that the money market is in equilibrium when MPtts = MPttd = MPtt48 / 150Neutrality◮ Assume again that our utility function is given byu(ct, mt) = ln(ct) + ln(mt) (37)◮ Using the utility function given above, we have the following Euler conditionand equilibrium condition in the money market:1 ct= β(1 + f ′(kt)) ct1+1 MtPt = mt = 1 +it it ct49 / 150Neutrality◮ Consider two cases:1. Prices adjust instantaneously in the model◮ In this case, prices adjust immediately as the money supply changes◮ In this case changes in the level of Mt will lead to an equi-proportionate changein Pt, so that MtsPt =MdtPt =MtPt remain unchanged◮ A change in the nominal money supply will leave mt unchanged and hence anyreal variables unchanged◮ When prices adjust instantaneously, money is neutral in the short-run and in thelong-run50 / 150Neutrality2. Prices adjust fully only in the long-run but slowly in the short-run◮ In this case changes in the level of Mt will lead to changes in mt since pricesare adjusting slowly in the short-run◮ It is clear that as the demand for money increases, the nominal interest rate itfalls◮ Since prices do not adjust fully in the short-run, the fall in the nominal interestrate also causes a fall in the real rate of return on capital, that is f ′(kt)◮ From the Euler condition, as the price of consumption today relative totomorrow falls, ct increases in the short-run (the other real variables alsoincrease)◮ Money is not neutral in the short-run but as prices fully adjust in the long-run,money becomes neutral in the long-run51 / 150Superneutrality◮ Neutrality in the long-run implies that changes in the level of nominal moneybalances do not have any effect on the real variables of the model◮ Changes in the level of nominal money balances only affect nominal variablesin the long-run◮ If changes in the level of nominal money balances do not affect the realvariables of the model, what about the growth rate of money balances?◮ What is the effect of a change in θ on real variables in the long-run?52 / 150Superneutrality◮ With a Cobb-Douglas production function, we have the following insteady-state:k¯ = 1αβ – β 1α-1y¯ = ¯ c = 1αβ – β αα-1x¯ = 0◮ We also found that:π¯ = θ53 / 150Superneutrality◮ The steady-state values of capital (k¯), output (¯ y), and consumption (¯ c) arenot only independent of the level of money but also of the growth rate ofmoney (θ)◮ The baseline model, therefore, implies that money is neutral but alsosuperneutral since all the real variables of the model are independent of therate of inflation or the money growth rate in steady-state◮ Note the difference between neutrality and superneutrality of money◮ Neutrality of money implies that changes in the level of money do not haveany effect on the real variables in the long-run◮ Superneutrality of money implies that changes in the growth rate of moneydo not have any effect on the real variables in the long-run54 / 150Superneutrality◮ An intuition for the superneutrality result can be gained by looking at theintertemporal condition given by equation (15)◮ Rearranging this equation we have: uc(ct+1, mt+1)1/βuc(ct, mt)1 + f ′(kt)=(38) ◮ In steady-state, the left-hand side of the equation is always one, which meansthat the right-hand side must also be one55 / 150Superneutrality◮ Therefore in steady-state we have1 =1/β1 + f ′(k¯)◮ Assume that we are not in steady-state and we start with an initial capitalstock that is below the steady-state level of capital, that is kt f ′(k¯) because of diminishingreturns◮ The right-hand side of the equation is, therefore, less than one when kt k¯.◮ In this case, the capital stock would have to decline over time until it reachesits steady-state level◮ The fall in the capital stock also leads to a fall in consumption over time◮ In both cases, whether the economy starts with a capital stock that isdifferent from its steady-state level, the level and the growth rate of moneydoes not affect how the capital stock evolves over time◮ The capital stock is influenced by real factors only57 / 150Superneutrality◮ While neutrality of money is a pervasive result in many macro models,superneutrality is more model dependent◮ For example, we can easily modify our model such that inflation has realeffects in the long-run (the superneutrality result no longer holds)◮ By introducing money in the production function, the steady-state level ofcapital is no longer independent of the rate at which money grows◮ The basic intuition is that real money balances affect the marginal product ofcapital, thus the steady-state level of the capital stock58 / 150Superneutrality◮ For example, if the production function is given by yt = f (kt, mt) and∂MPK/∂m > 0 (so that money, and capital are complements), higherinflation, by lowering real money balances, leads to a lower steady-state levelof capital stock (opposite of the Tobin effect)◮ In this case, it is clear that neutrality will not hold◮ When we introduce a stochastic version of the model with a labour-leisurechoice, superneutrality will also not hold as inflation will affect the laboursupply decision of workers in the model◮ As the amount of labour supplied enters the production function, inflation inthis modified model will have real effects59 / 150Evidence on neutrality and superneutrality◮ We present some evidence on the neutrality and superneutrality of money◮ We have not witnessed many episodes of hyperinflation; but when we did, itwas almost always preceded by rapid increases in the money supply◮ The evidence in favour of neutrality is robust while the evidence onsuperneutrality is not60 / 150German Hyperinflation1Source: The Economist61 / 150Zimbabwe Hyperinflation2Source: Financial Times62 / 150Venezuela Hyperinflation3Source: The Economist63 / 150Long-run correlation: M, Y , π◮ Several other cross-country studies have shown a strong relationship betweenmoney and prices and money growth and inflation in the long-run.◮ The correlation between inflation and the growth rate of the money supply inmost of these studies is almost 1, and this result is robust over time andacross countries◮ This correlation is taken to support the main tenet of the Quantity Theory ofMoney: a given change in the rate of money growth induces an equal changein the inflation rate (the growth rate of prices)64 / 150Long-run monetary facts◮ One of the frequently cited studies is from Mc Candless and Weber (1995)◮ They analyze a sample of 110 countries for the period 1960-1990 andcompute a 30-year (1960-1990) average of1. rate of growth of real GDP2. consumer price3. M0, M1 and M2◮ They also compute the correlation between money growth and inflation,inflation and GDP growth and money growth and GDP65 / 150Correlation – money growth and inflation◮ The correlation betweenthe rate of growth ofthe money supply andthe rate at which pricesis increasing (inflation)is almost unity for allcountries andsub-samples◮ This correlation isrobust for variousdefinitions of money(M0, M1 and M2)Table: Correlation coefficients for money growth andinflation SampleAll 110 countriesM00.925M10.958M20.950 Subsamples21 OECD countries 0.894 0.940 0.95814 Latin American 0.973 0.992 0.99366 / 150Correlation – money growth and output growth◮ When the whole sampleis considered, thecorrelation between thegrowth rates of moneyand real output isalmost zero◮ This result is, however,not robust whensub-samples areconsidered; thecorrelation is low butnegative for LatinAmerican economiesand high but positivefor OECD countriesTable: Correlation coefficients for money growth andoutput growth SampleAll 110 countriesM0-0.027M1-0.05M2-0.014 Subsamples21 OECD countries 0.707 0.511 0.51814 Latin American -0.171 -0.239 -0.24367 / 150Correlation – inflation and output growth◮ The correlation betweeninflation and realoutput growth is lowwhen the whole sampleis considered◮ The correlation is highand positive when 21OECD countries areconsidered and high butnegative when onlyLatin Americancountries are in thesampleTable: Correlation coefficients for output growth andinflation Sampleoutlierincluded-0.243outlierexcluded–0.101All 110 countries Subsamples21 OECD countries 0.390 0.39014 Latin American — -0.34268 / 150M2+ and inflation in Canada – 1968-2020-48 4 012162024-2 0 2 4 6 8 10 12 14CPI (y/y, percent)Growth rate of M2+ (y/y, percent)69 / 150M2+ and inflation in Canada – 1980-2020-48 4 01216208 6 4 2 010121980 1988 1996 2004 2012 2020CPI (right scale) M2+ growth (left scale)M2+ y/y growth rateCPI y/y growth rateR-Squared = 0.72-Year Averages8 6 4 2 010121416188 6 4 2 010121416181980 1988 1996 2004 2012 2020CPI (right scale) M2+ growth (left scale)M2+ y/y growth rateCPI y/y growth rateR-Squared = 0.784-Year Averages8 6 4 21012141618209 8 7 6 5 4 3 2 1101980 1988 1996 2004 2012 2020CPI (right scale) M2+ growth (left scale)M2+ y/y growth rateCPI y/y growth rateR-Squared = 0.878-Year Averages70 / 150Seignorage revenue◮ In lecture 2, seignorage, the revenue the government obtains from printingmoney is a non-negligible source of revenue for many governments◮ In many countries, seignorage can represent a significant share of GDP –sometimes over 10% of GDP◮ We also explained how seignorage revenue is created from printing money71 / 150Seignorage revenue◮ We explore seignorage revenue in the context of the model and draw someconclusion on the optimal seignorage revenue◮ As the government prints and issues more money, this leads eventually toinflation, thereby reducing the real value of newly issued money.◮ In other words, there is a tax on holding money, when ceteris paribus, themoney supply increases (inflation acts as a tax on holding money)72 / 150Seignorage revenue◮ Recall that the transfer form the government in the model was given by:τt =Mt – Mt-1Pt (39)◮ The money supply evolves according to:Mt = (1 + θ)Mt-1◮ Therefore,Mt-1 = 1 +1 θ Mt73 / 150Seignorage revenue◮ Substituting for Mt-1 in equation 39, we obtain an expression for theseignorage revenue for the government.τt =MtPt –Mt-1Pt (40)=MtPt – 1 +1 θ MPtt= 1 +θ θ MPtt◮ It is clear that if money growth is zero (that is θ = 0) and hence inflation iszero in the long-run, the revenue the government raises from printing moneyis also zero.74 / 150Optimal seignorage revenue◮ Since seignorage revenue and real money balances in the model are functionsof the growth rate of the money supply, θ, we can rewrite the seignoragerevenue as:I(θ) = 1 +θ θ µ(θ) (41)◮ We have dropped the time subscripts for simplicity, denoted seignoragerevenue as a function of θ, that is I(θ) and assumed that real money balancesare given by µ(θ)◮ Since the demand for real money balances falls as inflation increases,therefore, µ′(θ) 0, there is an optimal θ∗ that maximizes I ′(θ) and(43) I ′(θ) θ∗◮ One can show that the shape of the revenue from seignorage is hump-shaped,and there is a level of inflation that would maximize this seignorage revenue.76 / 150Optimal seignorage revenueθI (θ)I (θ∗)θ∗◮ The shape of the seignorage revenue is hump-shaped, and there is a level ofinflation (θ∗) maximizes this seignorage revenue.77 / 150Optimal Rate of Inflation◮ In the model, the rate of inflation is directly connected to the rate of moneygrowth◮ Since money holdings yield direct utility and higher inflation reduces realmoney balances; inflation therefore generates a welfare loss◮ In the model, inflation represents a tax for holding money◮ This raises two questions1. Is there an optimal rate of inflation that maximizes the welfare of households?◮ The model provides a very clear answer; the optimal rate of inflation is thenegative value of the real interest rate2. How high is the welfare cost of inflation?◮ The answer is model dependent but studies that employ a similar frameworkfind that the welfare cost of inflation is small◮ The welfare cost of inflation in these models are underestimated; inflation ismore costly than what this model would predict78 / 150Optimal Rate of Inflation◮ Friedman argued that the social marginal cost of producing money is nearlyzero while the private marginal cost of holding money (the opportunity cost)depends on the nominal interest rate◮ According to Friedman, households can maximize their utility if the wedgebetween private and social marginal cost disappears◮ According to Friedman, the optimal rate of inflation is obtained when thenominal interest rate is zero (Friedman rule)79 / 150Optimal Rate of Inflation◮ To see this result recall that the intratemporal condition can be written as:um(c, m)uc(c, m) = 1 +i i = 1 +r +r +π π (44)◮ The marginal utility of money becomes zero when i = 0 or if the inflationrate equals the negative real interest rate or π = -r.◮ Hence the optimal rate of inflation is a rate of deflation approximately equalto the real return on capital.◮ According to Friedman, the optimal inflation rate is obtained when thenominal interest rate goes to zero.80 / 150Optimal Rate of Inflation◮ A significant criticism of Friedman’s optimal rate of inflation is that moneygenerates revenue for the government through the inflation tax.◮ Imposing the Friedman rule implies that the government has to forgo thatsource of revenue and has to replace it with another source, for exampleother distortionary taxes.◮ Thus reducing nominal interest rates to zero may simply introduce moreimportant sources of inefficiencies in the economy if the government has toreplace its lost revenue with other distortionary taxes.81 / 150Welfare cost of inflation◮ In the OLG and MIU models we have seen, inflation is similar to a tax onholding money◮ The welfare cost of inflation is obtained by measuring the area under thedemand curve for money◮ This is a similar methodology for calculating consumer surplus◮ As we argued earlier, the demand curve for money is dependent on the utilityfunction one assumes in the model82 / 150Welfare cost of inflation◮ Most studies using this methodology have found that the welfare cost ofinflation is small◮ For example, many papers have found that an increase in inflation from 0%to 10% creates a deadweight loss of around 0.5% of GDP◮ Most economists argue that these estimates are likely too low.◮ Other approaches, such as using search-theoretic frameworks, have foundmore significant costs associated with inflation83 / 150Fiscal considerations◮ So far, we have completely ignored the government as a sector◮ We introduce the government to analyze the implications of the modelregarding the deficit and inflation rate◮ In particular, we analyze what happens to inflation if the governmentmonetizes the deficit◮ We also investigate whether it is possible to generate low inflation whileletting the debt runs out of control84 / 150Fiscal considerations◮ The government (consolidated) budget constraint is given by:dt = gt – Tt = bt+11 + rt+1– bt + Mt – Mt-1Pt (45)◮ where gt denotes government spending, Tt taxes, b, public debt anddt = gt – Tt is the primary deficit.◮ The primary deficit of the government that is the portion of expenditure notcovered by taxes must be either financed by issuing more debt (borrowing) ormoney-financed (printing money).85 / 150Fiscal considerations◮ The debt level, in this case, is the accumulated amount of borrowing that thegovernment does over time◮ The debt to GDP ratio is the level of debt divided by the level off GDP◮ Let us assume that the government never borrows and set bt = 0, ∀t.◮ In this case, the primary deficit is always financed by printing money andnever debt-financed.86 / 150Fiscal considerations◮ Thus we have: dt =Ptµ(θ)(46) Mt – Mt-1= 1 +θ θ ◮ It is clear that the monetization of the primary deficit creates inflation.◮ Thus a government that finances its expenditure by printing money willcreate inflation◮ A government that monetizes large deficits by only printing money willgenerate very high inflation (see the hyperinflation examples earlier)87 / 150Fiscal considerations◮ A second fiscal implication of the model is about the importance of crediblefiscal policy to achieve low inflation◮ Fiscal adjustments and discipline are necessary to achieve low inflation◮ Assume that the economy starts with a deficit level given by dt = d, nooutstanding bonds, such that b0 = 0◮ Assume also that there is an upper bound on government debt where thedebt to GDP ratio is 100%◮ Once the debt to GDP ratio exceeds that threshold, the government cannotborrow more88 / 150Fiscal considerations◮ Suppose that the government promises zero inflation, that is, it promises toset θ to 0, how credible is that promise?◮ In this case, the deficit is financed by borrowing (debt-financed) and not byprinting money since the government has promised zero inflation.◮ Thus the debt every period grows according to:bt+1 1 + rt+1– bt(47) ◮ It is clear that sooner or later if the government continues to accumulatedeficits and finances them by borrowing, the government will hit the ceiling ithas imposed on debt, that is 100% of GDP89 / 150Fiscal considerations◮ If such is the case, the government will have no choice but to resort toprinting money to finance the deficit since it will not be able to borrow oncethe debt to GDP ratio is above 100%◮ The inflation tax or seignorage revenue will have to cover the primary deficitand the interest rate paid on the debt, that is:1 +θ θ µ(θ) = ry + d (48)◮ Where ry is the interest rate paid on the debt90 / 150Fiscal considerations◮ In other words, the promise the government made at the beginning periodzero (zero inflation) is not credible if the government continues toaccumulate deficits and continues to finance them by borrowing◮ Unless the government adopts some fiscal adjustment, it cannot go onforever to accumulate deficits while credibly promising zero inflation◮ Put it differently, the intention of the government to fight inflation withoutfiscal adjustment to reduce the deficit is not credible.91 / 150Main results of the MIU model◮ Before we move one to the stochastic model, introduce uncertainty and alabour-leisure choice, let us summarize our findings from our baseline model1. In the baseline MIU, money is neutral and superneutral in the long-run2. The Friedman rule is optimal in this model – optimal level of the nominalinterest rate is zero which implies a rate of inflation that is equal to thenegative of the real rate3. A central bank promising zero inflation is not credible if the governmentmonetizes the deficit92 / 150Stochastic MIU model◮ The analysis of our baseline non-stochastic MIU model has mostly focussedon the long-run properties of the model.◮ We have not paid much attention to the short-run dynamics of the model◮ To analyze the short-run dynamics of the model, we need to modify thesimple MIU model, and we need to simulate the model as we will not be ableto solve the model using pen and paper anymore93 / 150Stochastic MIU model◮ Modify the basic model by introducing:1. Uncertainty2. A stochastic disturbance in the production function, a technology shock3. Stochastic money shocks4. Labour-leisure choice◮ Our objective is to specify, calibrate and simulate monetary policy shock◮ We want to understand the dynamics of the model and see if it canreproduce some of the dynamics of a VAR which is intended to replicate theactual economy94 / 150Stochastic MIU model◮ The stochastic model is very similar to the baseline model we described earlier◮ We start by describing the optimization problem of the consumer/worker◮ The utility function is very similar to the baseline model◮ In addition to real consumption and money balances, the agent values leisurein this model also◮ We denote leisure by ℓt, and we assume that the individual is endowed withone unit of time that they can devote to leisure or work (Nt)95 / 150Stochastic MIU model◮ The representative agent maximizes maxc,m,N,k,bEtβtu (ct, mt, Nt) ∞ X t=0subject toct + kt + mt + bt = wtNt + (1 + rt-1)kt-1 + τt (50)+mt-1(1 + πt) +(1 + it-1)bt-1(1 + πt) + Πtℓt + Nt = 1 (51)◮ The budget constraint is very similar to the baseline model except that in thismodel. each consumer receives wtNt when they supply labour to the firm,and since consumers own the firms, they also receive the profits (Π, whichwill be zero) of the firm96 / 150Stochastic MIU model◮ The utility function takes the following form:u(ct, mt, Nt) =act1-b + (1 – a)mt1-b1-Φ1-b1 – Φ –N1+ηt1 + η(52)◮ The utility function is non-separable in consumption and money, but it isseparable in labour◮ Φ represents the relative risk aversion of agents (the inverse of theintertemporal elasticity of substitution)◮ η is the inverse of the elasticity of work with respect to the real wage◮ b is the elasticity of substitution between consumption and real moneybalances◮ a represents the utility of real money balances relative to consumption;removal of cash (demonetization) from the system would increase the valueof money relative to consumption and hence would lead to a fall in a in ourmodel97 / 150Stochastic MIU modelu(ct, mt, Nt) =act1-b + (1 – a)mt1-b1-Φ1-b1 – Φ –N1+ηt1 + η◮ The non-separability of money and consumption (unless when b = Φ) impliesthat the marginal utility of consumption depends directly on changes in realmoney balances◮ The values that b and Φ take determines whether money and consumptionare complements or substitutes1. b = Φ, money does not affect the marginal utility of consumption2. b > Φ, money and consumption are complements3. b Φ and b Φ = 2.5, money and consumption are complements2. b = 2 Φ, m and c arecomplements10 20 30 40-10-50 10-3 c10 20 30 40-4-20m10 20 30 403 2 1 0pie10 20 30 40-2-10 10-4 n10 20 30 40-10-5010-4 y10 20 30 4000.020.04i10 20 30 4000.010.020.03k10 20 30 4000.51theta10 20 30 406 4 2 010-4 w127 / 150IRF – Money growth shock – b Φ, m and c are complements, as a result of the shock to moneygrowth, we have the following response from the model:1. Inflation and expected inflation increases and this leads to an increase in thenominal interest rate (the Fisher effect operates)2. Real money balances, consumption, output, labour supply falls3. The capital stock and real wages increase129 / 150IRF m and c are substitutes◮ When b Φ (b Φ◮ Output falls, and this leads to a fall in demand for labour and to a fall in realwages◮ Since the amount of labour falls, the marginal product of labour increasesexplaining why initially real wages increase before moving back to steady-state◮ Over time as the effect of the shock dissipates, all the variables recover andmove back to their steady-state values132 / 150IRF m and c are substitutes◮ When consumption and money balances are substitutes, the positive moneyshock leads to an increase in consumption◮ Output increases, in this case, leading to an increase in the demand for labour◮ The increase in the demand for labour leads to a fall in the marginal productof labour, explaining why real wages initially fall when b Φ), the model predictsthat:1. Real quantity of money falls2. Nominal interest rate and inflation increase3. y, N and c fall4. k and real wages increase◮ When money and consumption are substitutes (b
