Imperfect Information and Market Segmentation Walsh Chapter 5

Imperfect Information and Market Segmentation Walsh Chapter 5

1 Why Does Money Have Real Effects? Add market imperfections to eliminate short-run neutrality of money Imperfect information keeps price from fully reacting to money Limited participation in markets implies that only some agents are affected by a change in money Monetary policy can have real effects in the presence of market imperfections

2 Imperfect Information 2.1 Lucas Monetary Surprises Assumptions Each individual is a consumer and a producer. Each individual consumes many different goods and produces only a single good. Each individual is a price-taker in all markets Each individual has imperfect information

All variables are logs Information Agents know their own price P t (z everything dated t 1 or earlier including E t 1 P t = P t and the model for price Price in market z has an aggregate component (P t and an idiosyncratic component (z t P t (z = P t + z t where variance of P t is σ 2, variance of z t is τ 2, and covariance is zero

Each individual wants to work harder and produce and sell more of his good in periods when the price of his good is high relative to the price of all other goods he consumes. Since agents do not know the prices of other goods, must form an expectation y t (z = y + γ [P t (z E (P t I t (z] Expectations E ( P t P (z, P t = (1 θ Pt (z + θ P t where θ = τ 2 τ 2 + σ 2

Prove that this is a rational expectation: Choose a, b to min E [ ] 2 P t a (P t + z t b P t = min E [ (1 a ( ] 2 P t P t az (b 1 + a P t = min [ (1 a 2 σ 2 + a 2 τ 2 + (b 1 + a 2 P t 2 Substitute for expectations y t (z = y+γ [ ] [ ] P t (z (1 θ P t (z θ P t = y+γθ Pt (z P t ] Aggregate over markets by integrating over z y t = y + γθ [ P t P t ]

output is increasing in price relative to its expectation conditioned on information one period ago effect of an increase in price on output is greater the larger the fraction of idiosyncratic variance relative to total variance large fraction: an increase in price fools agents into thinking that an increase in aggregate price is an increase in relative price small fraction: agents know that if their price has risen, it is probably due to aggregrate price and they are not fooled countries with volatile monetary policy will have steep supply curves relative to countries with stable monetary policy equivalently, if use monetary policy often, will not work to raise output

Aggregate demand is the cash-in-advance constraint, determined by money y t + P t = M t M t = M t 1 + δ + ɛ t where ɛ t is iid with mean zero Equilibrium sets aggregate supply equal to aggregate demand y t = y + γθ [ P t P t ] = Mt P t Solution Remember that P t = E t 1 P t

Undetermined coeffi cients conjecture a solution for price P t = π 0 + π 1 M t + π 2 M t 1 take the expectation conditional on t 1 information E t 1 P t = π 0 + π 1 (M t 1 + δ + π 2 M t 1 substitute both into equilibrium equation and equate coeffi cients on variables to zero y + γθπ 1 (M t M t 1 δ M t + π 0 + π 1 M t + π 2 M t 1 = 0 π 1 = 1 1 + γθ ; π 2 = γθ 1 + γθ ; π 0 = γθδ 1 + γθ y

Price surprises become P t = γθ (δ + M t 1 1 + γθ y + M t 1 + γθ P t E t 1 P t = ɛ t 1 + γθ Money affects output only if it is unexpected y t = y + γθɛ t 1 + γθ Magnitude of effect depends on θ, which reflects extent to which shocks to P (z are usually idiosyncratic

2.2 Sticky Information (Mankiw and Reis Assumptions Firms can adjust price every period Optimal price (all variables in logs p t (j = p t + αx t p t is aggregate price level, implying that firms care about relative price x t is output gap, where higher output gap inplies increasing marginal cost

all firms are identical implying that the optimal aggregate price p t = p t + αx t price is equal to optimal if output gap is zero Each period a fraction λ of firms are randomly chosen and allowed to update information a firm which updated i periods ago sets price at p i t = E t i p t

Sticky Information Aggregate Price λ of firms update information at time t, setting their price at p 0 t = p t do not update continuously due to cost of information processing of the 1 λ firms which do not update at time t, λ of them updated at time t 1, setting their price in period t at p 1 t = E t 1 p t therefore the aggregate price in period t is given by an infinite backward recursion

p t = λp t + (1 λ λe t 1 p t + (1 λ 2 λe t 2 p t +... (1 = λ = λ i=0 i=0 (1 λ i E t i p t (1 λ i E t i (p t + αx t if λ is close to one, then firms update often and prices are based largely on recent information multi-period surprises, compared with one-period surprises in Lucas

Sticky Information Phillips Curve define optimal price z t = p t + αx t p t = λz t + λ i=1 (1 λ i E t i z t = λz t + λ (1 λ E t 1 z t + λ (1 λ 2 E t 2 z t +... p t 1 = λe t 1 z t 1 + λ i=1 (1 λ i E t 1 i z t 1 = λe t 1 z t 1 + λ (1 λ E t 2 z t 1 + λ (1 λ 2 E t 3 z t 1 +...

subtracting yields an expression for inflation π t = p t p t 1 (2 = λz t + [ λe t 1 (z t z t 1 λ 2 ] E t 1 z t + [ λ (1 λ E t 2 (z t z t 1 λ 2 ] (1 λ E t 2 z t +... = λz t + λ i=0 from equation (1 above p t = λ i=0 (1 λ i E t 1 i z t λ 2 (1 λ i E t i (p t + αx t = λ (p t + αx t + λ i=1 i=0 (1 λ i E t 1 i z t (1 λ i E t i (p t + αx t

solve for p t by sutbracting p t from both sides and dividing by 1 λ p t = = λ 1 λ αx t + λ 1 λ λ 1 λ αx t + λ i=0 i=1 (1 λ i E t i (p t + αx t (1 λ i E t 1 i z t this implies λ 2 i=0 (1 λ i E t 1 i z t = λp t λ2 1 λ αx t substituting for the last term in equation (2 yields π t = λz t + λ i=1 (1 λ i E t i z t λp t + λ2 1 λ αx t

substituting for z t yields π t = λ 1 λ αx t + λ i=1 (1 λ i E t i (π t + α x t inflation is increasing in the output gap, by a larger amount, the more frequently firms update (the larger is λ since firms update infrequently, a shock which occurs at a point in time in the past, slowly raises inflation expectations are not forward-looking because agents can adjust future prices to expectations of future inflation Add an equation for output (gap m t + v t = p t + x t

first differences m t m t 1 + v t = π t + x t money shock does not affect second term in Phillips curve in current period therefore, money up must raise both inflation and output simulations show that if money growth is positively correlated, inflation and output continue to rise before eventually falling hump-shaped responses we see in data

3 Limited Participation Model in which increase in money growth lowers nominal interest rate initially Assumptions Each household has a shopper, a firm manager, a worker, a bank Household allocates money (M t to bank deposit (D t and to balances to use for purchases P t C t M t D t

Firms must pay nominal wages (P t ω t to workers before they sell output, implying that they must use bank loans (L t Firm s nominal profits P t ω t N d t L t Π f t = P ty ( N d t Pt ω t N d t R L t L t Banks accept deposits from households and pay R D. Banks make loans to firms. Central bank makes transfers (H to banks. Bank profits are L t = D t + H t Π b t = RL t L t + H t R D t D t = ( R L t RD t Dt + ( 1 + R L t Ht

Competition and profit max assure interest rates are equal R L t = R D t = R t Divide by P t, writing everything in real terms m t d t C t λ 1 is the marginal value of money in consumption l t ω t Nt d λ 3 is the marginal value of money in loans

Household Problem Budget constraint P t ω t N s t + M t D t + (1 + R t D t + Π b t + Π f t P tc t = M t+1 P t ω t Nt s + M t D t + (1 + R t D t + ( 1 + Rt L P t ω t Nt d Rt L L t P t C t = M t+1 Simplifying and dividing by price Ht + P t Y ( N d t ω t N s t + m t + R t d t + ( 1 + R L t = m t+1 ( Pt+1 P t ht + Y ( N d t ωt N d t R L t l t C t

Household maximizes V (m t = max d E max C t,nt s,n t D,l,m t+1 [u (C t v (N s t + βv (m t+1 ] where d t is chosen before observing h t subject to budget constraint (λ 2, cash-in-advance constraint (λ 1, and loans-in-advance constraint (λ 3 First order conditions d t is determined before observing h t, so add E h d E h [ λ1t + R D λ 2t ] = 0 (3 C u (C t = λ 1t + λ 2t (4

marginal utility of income differs from marginal utility of consumption when cia constraint binds N s t v (N s t + ω t λ 2t = 0 (5 N d t λ 2t [ Y ( N d t ωt ] λ3t ω t = 0 (6 l t R L t λ 2t + λ 3t = 0 (7 where λ 3t is the value of liquidity in the loan market m t+1 Envelope condition λ 2t ( Pt+1 P t + βv m (m t+1 = 0 V m (m t = E h [λ 1t + λ 2t ]

Interpretations Marginal value of liquidity in goods and loan markets Subtracting (4 from (7 yields marginal value of money for loans less its marginal value for consumption λ 3t λ 1t = ( 1 + R L t λ2t u (C t Solving first order condition on money for λ 2t = β = β = β ( Pt P ( t+1 Pt P ( t+1 Pt P t+1 V m (m t+1 E h [λ 1t+1 + λ 2t+1 ] E h u (C t+1

Solving these two equations for marginal utility of consumption u (C t = ( 1 + Rt L λ2t λ 3t + λ 1t = (1 + R t β ( Pt P t+1 E h u (C t+1 λ 3t + λ 1t when the marginal value of money in consumption and loans differs, there is a wedge in the normal Euler equation Combining equations (3 and (7 E h [ λ 1t + λ 3t ] = 0 when a household makes its portfolio choice, the value of sending cash to goods market must equal value of sending it to loan market by depositing it in a bank ex poste, the values can differ because households cannot reallocate after central bank makes decision on h

Labor market Together, equations (6 and (7 imply that the interest rate drives a wedge between the marginal product of labor and the real wage Y ( N d t = (1 + R ωt Equation (5 implies v (N s t = ω t λ 2t Combining the two implies that the interest rate drives a wedge between the marginal rate of substitution of labor and its marginal product ( N d t v (Nt s = Y λ 2t 1 + R t

Unexpected increase in supply of money Received by banks and so increases loans because D t is pre-determined by households L t = D t + H t To get increase in loan demand, R t must fall Fall in R t increases labor demand and the real wage Increase in real wage causes households to increase labor supply works because households cannot respond to the reduction in the interest rate on deposits by withdrawing them and having more money for consumption, driving up price