Monetary Economics Concepts

• Focus on understanding the roles of money and assets in the economy when making modeling choices in monetary economics.
• There are trade-offs between explicitness and simplicity.
• Shortcuts are common in monetary economics, especially in monetary policy work.
• Three common shortcuts involve assumptions about money at the outset of modeling.
  • Money is an argument in the utility function because people like it.
  • Money enters the production technology, freeing up resources.
  • Money is necessary for executing transactions in cash-in-advance models.
• Cash-in-advance models assume consumption goods are purchased with previously acquired money.
• The model demonstrates relationships between real interest rates, nominal interest rates, and inflation.
• Also shows monetary policy's effects on inflation, inflation costs, and the Friedman rule.

Model Basics

• Time is indexed by t = 0, 1, 2, ...
• There is a single representative consumer who maximizes utility.

Consumer Preferences

• Consumer maximizes Σβ^t [u(ct) – v(n)], where β is the discount factor (0 < β < 1), ct is consumption, and nt is labor supply.
• The utility function u(·) is strictly increasing, strictly concave, and twice continuously differentiable, with u′(0) = ∞.
• The disutility function v(·) is strictly increasing, strictly convex, and twice continuously differentiable, with v'(0) = 0 and v'(n) = ∞ for some ñ > 0.

Production

• Representative firm produces output yt using labor: yt = an^d, where a > 0 is productivity, and n^d is labor input.

Government

• The government issues or retires money via lump-sum transfers (Tt).
• The government budget constraint is (Mt - Mt-1)/Pt = Tt, where Mt is the nominal money supply and Pt is the price level.
• The consumption good numeraire expresses quantities in terms of consumption goods (lower case denotes real quantities).
• Inflation rate is πt = Pt/Pt-1 - 1, rewriting the government budget constraint as (mt - mt-1)/(1 + πt) = Tt.

Period Timing

• Assets are money, nominal bonds, and real bonds.
• Representative agent implies assets other than money don't affect equilibrium quantities/prices, but determine asset prices.

Bonds

• Nominal bonds issued in period t promise one unit of money at the beginning of t + 1.
• Real bonds promise one unit of consumption goods one period hence.
• At the period's start, the consumer has (mt-1)/(1 + πt) money, bt-1 nominal bonds, and ft-1 real bonds from the last period.
• Consumers receive transfer Tt, allocate wealth to new bonds and consumption: ct + qtbt + stft ≤ (mt-1)/(1 + πt) + bt-1/(1 + πt) + ft-1 + Tt.
• qt and st are prices of nominal and real bonds, respectively.
• The asset market closes, and the consumer supplies labor to the firm.
• Wages are paid at the period's end (after the firm sells output).
• Consumers buy goods with assets on hand, firms accept only money.
• At the period's end, the firm pays wages, and the consumer carries assets into the next period.
• The consumer's budget constraint: ct + qtbt + stft + mt = (mt-1)/(1 + πt) + bt-1/(1 + πt) + ft-1 + wtnt + Tt.
• The difference between asset holdings (m) and wage earnings (wtnt) determines money holdings and wage earnings occur on opposite sides of the equations.

Optimization Problem

• The consumer solves a dynamic optimization problem knowing prices and inflation, (wt, qt, st, πt+1), for t = 0, 1, 2, ....
• Consumers observe w0, q0, and s0, and form expectations for t = 1, 2, 3, ..., where forecasts are correct in equilibrium.
• Without aggregate uncertainty, consumers' forecasts are always correct in a rational expectations equilibrium.
• The consumer chooses ct, nt, mt, bt, and ft, with consumption-savings and asset allocation decisions across three assets.
• Choices about assets in period t affect consumption-savings decisions in future periods.

Lagrangian Optimization

• The Lagrangian for the consumer's constrained optimization is infinite-dimensional.
• A discounted sum of period Lagrangians is given by Lt = u(ct) - v(nt) + λt[ (mt-1)/(1 + πt) + (bt-1)/(1 + πt) + ft-1 + Tt - ct - qtbt - stft] + μt[ mt + (bt-1)/(1 + πt) + (ft-1 + wtnt + Tt) - ct - qtbt - stft].
• The first-order conditions help maximize consumer utility (ct, nt, mt, bt, ft, λt, μt) for all periods.

First Order Conditions for a Maximum

• Key equations from utility maximization:
• u'(ct) – (λt + μτ) = 0
• -v'(n) + µtwt = 0
• -qt (λt + μτ) + β [λt+1 + μt+1 / (1 + πτ+1) ]= 0
• -St (At + μt) + β (At+1 + μt+1) = 0

Simplified Equations

• By substituting, we can get the equations: -ν' (η) = β [wtu'(ct+1) / 1 + πτ+1 ] -qtu' (ct) = β [u'(ct+1) / 1 + πτ+1 ]

Consumer Behavior Insights

• When optimizing, marginal disutility of supplying labor equals the marginal payoff.
• The effective real wage is (wtPt)/Pt+1 = wt/(1 + πt+1).
• Wages are valued according to the discounted marginal utility of future consumption, ßu' (ct+1). Foregone consumption equals marginal utility gained when buying a nominal bond.
• Pt/Pt valued at marginal utility u'(ct) yields a payoff of 1/Pt+1 units of consumption
• This can be valued at future consumption u'(ct+1) where u’ct = ßπ+1 u’ct+1.
• Marginal utility foregone must equal the discounted marginal utility gained in the future. Similarly when acquiring a read bond, the foregone marginal utility is stu’(Ct)

Interest Rates

• If qt = 1, then At = 0.
• The nominal interest rate is R = 1/β, and the asset market constraint does not bind here.
• Positive nominal rates are associated with a binding asset market constraint.
• Also, (wt / 1 + Rt) = (v'(nt) / u'(ct)).
• The quantity of the right-hand side is the consumer’s willingness to substitute labor for consumption.
• A positive nominal interest rate reflects inefficiency.

Firm Optimization

• Static optimization: max [nđ(a - wt)].
• Firm solution: nɖ= 0 if a < wt, nđ = ∞ if a > wt, and any n^d ≥ 0 is optimal if a = wt.

Equilibrium Conditions

Competitive Equilibrium

• Consumers and firms optimize; the government satisfies its budget constraint. Markets clear for goods, labor, money, nominal bonds, and real bonds. Consumer optimization is based on first-order conditions and firm optimization is linked to market real wages.

Market Clearing Conditions

• ct = yt
• n^d = nt
• m^d = mt
• bt = 0
• ft = 0 For bonds, there must be zero net supply in equilibrium. Also, the consumer borrows or lends at will on the bond market.

Government Policy

• The government sets transfers over time so that the nominal money supply grows at a constant rate: Mt+1 = (1 + i)Mt. From equilibrium conditions, m = an.

Key Equations

mt = ant Mt = Pt * yt

Quantity Theory of Money

• Quantity theory of money: MV = Py, where money turns over once per period. The labor market clears only when wt = a, and the equilibrium real wage rate is achieved . Then use equations 16, 28 and 22-24 to get v’(nt)nt = ß * (ant+1(u’ant+1)) / 1 + i Equation 30 solves for the amount and output consumption n = nt. So, if nt is a constant from 30, you get v’n = ß * au’(an / 1 + I. Solves the variables of interest

Monetary Neutrality

• Pt = M/an, with the inflation rate (i) of 1%, the money is supply is Mo * (1+i). Money is neutral, meaning that the money supply growth rate has an affect on the real variables. The quantity (i) causes (n) to fall, output falls.

Interest Rates and Other Insights

ß / 1 + I

• Interest rates are the ratio of money over bonds q = β =s
• Real interest rate has not effect on rate or inflation R ≈ i + p which is the Fisher Equation: an increase 1% growth equates to increase in nominals rate.

Optimal Monetary Policy

• The way the model is setup, the model is for the setting of (i) which money growth rate.

Benchmark For Monetary Policy

• Setting (i) should result in maximum welfare for the people. The social planner uses the model to choose no, n1 which maximizes the production and consumption to plan economic outputs.

Optimal labor Supply

Σβt [u(ant) – v(nt)] the consumer would have a optimal labor supply of nt = n* au' (an*) – v' (n*) = 0 with consumers working a set amount that would cause maximum economic equilibrium. Achieving the optimal money growth rate = optimal outcome forever.