Economics 502 Nominal Rigidities Geoffrey Dunbar UBC, Fall 2012 November 22, 2012 Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 1 / 68
Money Our models thusfar have been real models. We have not considered whether introducing money would change our conclusions. Q: How to introduce money? If money doesn t change real relative prices, then it will have no effect (Cooley and Hansen (1995)). Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 2 / 68
Money As an object in dynamic models, money is hard to motivate. Why would agents hold savings in terms of paper money? It s rate of return is the negative of the inflation rate. Other savings instruments are better. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 3 / 68
Inflation Milton Friedman is famous for saying Inflation is everywhere and anywhere a monetary phenomenon. By implication, at least in the long-run, we think that inflation is caused by changes in the supply of money. Governments have reasons to print money to pay for spending. So little reason to believe that the inflation rate will be negative (or perhaps even predictable). Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 4 / 68
Introducing Money Theoretically, money is introduced into models by an assumption. 1 Money in the utility function - preferences are defined over money holdings. 2 Cash-in-advance constraints - by assumption at least some consumption must be paid for with cash. 3 Search money - the most modern approach. Two flavours. One is essentially a cash-in-advance constraint with a matching problem. The second assumes that there are some goods (nudge, nudge, wink, wink) for which you don t want anyone to observe your consumption. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 5 / 68
The Role Money The price of money in macro models is a relative price and typically affects other relative prices. So restrictions on how the price of money changes over time may affect how other shocks affect the economy. These nominal shocks are usually particularly relevant for labour supply because price shocks change the intratemporal price of labour effort. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 6 / 68
Aggregate Supply The main focus in nominal adjustment models is to model aggregate supply in the short-run. Most macroeconomists would typical expect that there is some long-run level of aggregate supply that only depends on real variables. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 7 / 68
Expectations Augmented Phillips Curve A modern Keynesian model might have an equation for aggregate supply that looks like: π t = π t + λ[lny t ln Ȳ t ] + ɛ S t where π t is actual inflation, Y t is actual output, Ȳ t is potential output, πt is core inflation that might be expected if Y t = Ȳ t and ɛ S t is a shock. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 8 / 68
What is Expected Inflation? This equation is basically a mechanistic rule linking price growth and output. But how is π t determined? Friedman argued that it was expected long-run inflation. This implies that there is no room for monetary policy. Keynesians disagree but the proper specification is open to debate. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 9 / 68
Three Possibilities Early models tended to assume that πt was a weighted sum of past inflation πt = k i β i π t i. This implies no forward looking behaviour. You could also impose a learning approach where agents forecast and update their beliefs (or model). A compromise is: πt = φe[π t ] + (1 φ)π t 1. This implies agents have room to make errors in perpetuity. Adjustment is sluggish. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 10 / 68
Nominal Rigidities and Market Imperfections Lucas(1972, 1973) suggested a microfoundation for sluggish price adjustment. He imagines a world with differentiated producers who each sell a different good. The key idea is that producers cannot tell what is causing prices for their products to change. Is is a general price change or a relative price change. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 11 / 68
A Perfect Information Benchmark Suppose that producers produce output according to: Q i = L i, where Q is output and L is the labour input. The producers real income is P i Q i /P, where P i is the price of good i and P is the general price level. P = 1/n n i P i Utility is given by U i = C i L γ i /γ Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 12 / 68
Optimizing Producers Taking aggregate prices as given, an optimizing producer sets C i = P i Q i assuming no savings. Thus, Maximixing with respect to L i yields: Taking logarithms yields U i = P i L i L γ i /γ L i = (P i /P) 1/(γ 1) l i = 1 γ 1 (p i p) where lowercase indicates logged values. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 13 / 68
Demand Assume that demand for good i is given by: q i = y + z i ξ(p i p) where y is logged aggregate income, z i is an idiosyncratic demand shock for good i (a relative demand shock). Assume: z i N(0, σ 2 ) Aggregating the individual demands over the i goods yields: q = y = m p where the last equality assumes a vertical LM curve. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 14 / 68
Equilibrium Market clearing for each good implies: l i = q i Thus, Which yields: 1 γ 1 (p i p) = y + z i ξ(p i p) Averaging over all i gives: p i = γ 1 1 + ξγ ξ (y + z i) + p p = γ 1 1 + ξγ ξ (y) + p Thus, y = 1 and so Y = 1. Consequently, m = p. Money is neutral. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 15 / 68
Imperfect Information Of course, there is nothing surprising that money has no effect in a world with perfect information. Mispricing here is costly, and so no producer will willingly choose to set prices such that q i l i. This serves as a benchmark to a case with imperfect information. With imperfect information, producers may not correctly observe z i and may misprice their goods in real terms. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 16 / 68
Imperfect Information Define r i = p i p as the relative price of good i. Assume that a producer observes: p i = p + r i but cannot tell whether a change in the equilibrium price for his good is because of an aggregate price component or a relative price movement. Lucas (1972) makes a few simplifying assumptions. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 17 / 68
Simplifying Assumptions 1 producers determine E[r i p i ] and then act as if this expectation is certain. (Note this implies they do not maximize expected utility). 2 producers find E[r i p i ] rationally. Thus the true expected value of r i given p i is implied by the joint distribution. 3 Assume that m N(E[m], σ 2 m). Since the model is linear this implies that r i and p i are also normally distributed with mean E[r], E[p] and variances V r and V p respectively. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 18 / 68
Rational Expectations What the producers want to know is E[r i p i ] because: l i = 1 γ 1 (p i p) 1 γ 1 E[r i p i ] By assumption E[r i p E[p]] = 0. So, We can express E[p p i ] as: E[p p i ] = E[p] + E[r i p i ] = E[p i p p i ] = p i E[p p i ] V r V r + V p (p i E[p]) = ˆβp i + (1 ˆβ)E[p] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 19 / 68
The Producers Problem Returning to the producers problem: which reduces to: l i = 1 γ 1 E[r i p i ] = 1 γ 1 (p i ˆβp i + (1 ˆβ)E[p]) Averaging over all producers gives: l i = 1 γ 1 (1 ˆβ)(p i E[p]) y = 1 ˆβ (p E[p]) γ 1 Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 20 / 68
A Microfounded Phillips Curve y = b(p E[p]); b = 1 ˆβ γ 1 Is sometimes referred to as the Lucas supply function. Casting this as a Phillips curve yields, after simple rearranging: where λ = 1/b. π t = π e t + λ[ln Y t ln Ȳ ] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 21 / 68
Policy Implications Recall that y = m p. So, This yields: m p = b(p E[p]) and, p = 1 1 + b m + b 1 + b E[p] y = b 1 + b m b 1 + b E[p] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 22 / 68
Policy Implications Applying the law of iterated expectations, Which implies, E[p] = 1 1 + b E[m] + b 1 + b E[p] E[p] = E[m] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 23 / 68
Output Thus, and y = b (m E[m]) 1 + b p = E[m] + 1 (m E[m])) 1 + b This model implies a positive relationship between inflation and output. But it cannot be exploited by a monetary authority if producers update their expectations rationally. The Lucas Critique: If policymakers attempt to take advantage of a statistical relationship, effects operating through expectations may case the relationship to breakdown. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 24 / 68
Some Objections There are some objections to this island model of Lucas. First, there is evidence that monetary policy can influence the economy. Ball (1991) finds that output growth is on average below trend following policy announcements of tighter monetary policy. Second, in the model, output fluctuations arise from labour supply changes. Thus, to match output volatility one needs significant labour supply elasticity. This seems to be counter to the empirical evidence. Third, high quality pricing information is available quickly so the source of confusion seems unrealistic. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 25 / 68
Imperfect Competition and Staggered Pricing An alternative model is to assume both imperfect competition and staggered pricing. Staggered pricing means that producers may be unable to adjust pricing every period. There are a few flavours of pricing rigidities. Three of the main pricing rules are: staggered, size-dependent and state-dependent. The mechanisms that generate these rigidities are usually assumed. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 26 / 68
Imperfect Competition and Staggered Pricing Assume that demand for good i is given by: In logs this is simply, Q i = Y ( P i P ) ϕ q i = y ϕ(p i p) Assuming ϕ > 1 gives each producer a monopoly power over pricing. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 27 / 68
Imperfect Competition and Staggered Pricing Producers have utility given by: U i = C i (L i) γ γ Production is again given by: Q i = L i One important assumption is that individuals sell their labour and hire labour from others in a competitive market. They do not produce their own good. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 28 / 68
Imperfect Competition and Staggered Pricing With this assumption over labour, utility for a producer can be written: U i = (P i w)q i + wl i P (L i) γ γ Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 29 / 68
Producer s Problem without Staggering Given the expression for Q i, the producer s problem becomes: U i = (P i w)y ( P i P ) ϕ + wl i P (L i) γ γ The producer has two choices, P i and L i. The FOC s are: U i = Y ( Pi P ) ϕ (P i w)ϕy ( P i P ) ϕ P 1 i P i P = 0 U i L i = w P Lγ 1 i = 0 Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 30 / 68
Rearranging Using the FOC for P i yields: ϕ(p i w)/p i = 1 This implies, or equivalently, Using the FOC for L i gives, P i = P i P = ϕ ϕ 1 w ϕ w ϕ 1 P L i = ( w P )1/(γ 1) Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 31 / 68
Symmetric Equilibrium without Staggering In a symmetric equilibrium, L i = L for all i; Q i = Q for all i and; L = Q = Y. Thus, Y = ( w P )1/(γ 1) Substituting this into the pricing equation yields: P i P = ϕ ϕ 1 Y γ 1 = 1 where the last equality follows from P i = P for all i. Thus, Y = ( ϕ 1 ϕ )1/(γ 1) < 1 Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 32 / 68
Aggregates If we maintain the assumption that aggregate real income is equal to aggregate real expenditures then Y = M/P. So, A little rearranging yields: M/P = ( ϕ 1 ϕ )1/(γ 1) ϕ P = ( ϕ 1 )1/(γ 1) M In this model without price staggering, money is still neutral. It affects P but not Y. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 33 / 68
Price Staggering Return to the expression for relative prices, P i P = ϕ ϕ 1 Y γ 1 Using the fact that Y = M/P, rewrite this as: P i P = ϕ ϕ 1 γ 1 M P Taking logarithms and rearranging yields (lowercases are logged values): p i = ln ϕ + (γ 1)m γp ϕ 1 This is the log pricing rule. Usually write as: p i = (θ)m + (1 θ)p Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 34 / 68
Staggered Pricing To investigate the effect of staggered pricing, we need to specify a rule for how prices are set. Assume that firms set prices for two periods. Assume that half the firms set prices in one period and the remainder set prices in the second period. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 35 / 68
Prices Define p 1 t as the price for period t set in period t 1. Similarly, define p 2 t as the price for period t set in period t 2. In the aggregate: p t = 1 2 (p1 t + p 2 t ) Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 36 / 68
Prices Assume that firms set prices equal to the expected profit maximizing price. Then: and p 1 t = E t 1 [p i,t] p 2 t = E t 2 [p i,t] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 37 / 68
Prices Since we have an expression for the optimal price, then: pt 1 = E t 1 [(θ)m t + (1 θ)p t ] and pt 2 = E t 2 [(θ)m t + (1 θ)p t ] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 38 / 68
Iterative Expectations In period t 1 then: so, Solving for p 1 t, p 1 t = E t 1 [(θ)m t + p 1 t = (θ)e t 1 [m t ] + (1 θ) (pt 1 + pt 2 )] 2 (1 θ) (pt 1 + pt 2 ) 2 p 1 t = 2θ 1 + θ E t 1[m t ] + 1 θ 1 + θ p2 t Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 39 / 68
Iterative Expectations Applying iterative expectations, in period t 2 we get: E t 2 [p 1 t ] = 2θ 1 + θ E t 2[m t ] + 1 θ 1 + θ p2 t Now using the previous expression for p 2 t and the staggered pricing rule, we get. p 2 t = E t 2 [(θ)m t ] + (1 θ)(e t 2 [p 1 t ] + p 2 t )] But notice that we have an expression for E t 2 [p 1 t ] in the equation above. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 40 / 68
Iterative Expectations Substituting for E t 2 [p 1 t ] yields, p 2 t = θe t 2 [m t ] + 1 θ 2 [ 2θ 1 + θ E t 2[m t ] + 1 θ 1 + θ p2 t + p 2 t ] Solving for p 2 t yields: p 2 t = E t 2 [m t ] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 41 / 68
The Other Price... Now we can substitute back into the expression for p 1 t to get, p 1 t = 2θ 1 + θ E t 1[m t ] + 1 θ 1 + θ E t 2[m t ] Finally, we can solve for the price level using the fact that aggregate prices are a linear combination, p t = θ 1 + θ E t 1[m t ] + 1 1 + θ E t 2[m t ] So prices depend on the expectations of monetary policy. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 42 / 68
Innovations to Monetary Policy Matter To see how prices evolve over time because of monetary policy, notice that we can rewrite p 1 t as: Substituting in for p t and p 2 t yields: p 1 t = E t 2 [m t ] + p 1 t = 2p t p 2 t 2θ 1 + θ (E t 1[m t ] E t 2 [m t ]) Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 43 / 68
Output Recall the aggregate output relationship y t = m t p t, then: Rearranging yields, y t = m t θ 1 + θ E t 1[m t ] 1 1 + θ E t 2[m t ] y t = m t E t 1 [m t ] + 1 1 + θ (E t 1[m t ] E t 2 [m t ]) Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 44 / 68
Conclusion As in Lucas, unanticipated aggregate demand shifts have real effects, including monetary policy changes. A big difference is that anticipated demand shifts affect output as well because prices are not flexible in the short run. A proportion 1/(1 + θ) of a change in m t that becomes anticipated between t 2 and t 1 is passed into output because some prices can t adjust. The size of the effect depends on θ - the responsiveness of firm prices to aggregate demand. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 45 / 68
Microfoundations As we have noted in class, these models rely on pricing imperfections that could easily be overcome. Let s consider the sort of costs a firm must face in order to not set prices more flexibly. As a shorthand, let s imagine that firms face a menu cost for adjusting prices and calculate how large this menu cost would have to be for firms to not wish to set prices more flexibly. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 46 / 68
Recall Recall the model: Q i = Y ( P i P ) ϕ = L i Firm real profits were given by: π i = ( P i P w P )Y (P i And the labour market equilibrium yielded: P ) ϕ w P = Y γ 1 = Y 1/v where by redefining we have v = 1/(γ 1) which is the elasticity of labour supply to income. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 47 / 68
Profits Rewriting the profit equation gives: π i = ( P i P Y 1/v )Y ( P i P ) ϕ = ( P i P )1 ϕ Y Y (1+v)/v ( P i P ) ϕ Imposing the equilibrium condition Y = M/P gives: π i = ( P i P )1 ϕ M P (M P )(1+v)/v ( P i P ) ϕ Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 48 / 68
The Optimal Price Solving for the optimal price yields: This reduces to: π i P i = (1 ϕ)( P i P )1 ϕ P 1 i P i P = M P + ϕ(m P )(1+v)/v ( P i P ) ϕ P 1 i ϕ ϕ 1 (M P )1/v Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 49 / 68
Profit Under Flexibility Profit under flexibility given that other prices are fixed is: ˆπ i = 1 ϕ 1 ( ϕ ϕ 1 ) ϕ ( M P )(1+v ϕ)/v This is the profit a firm could earn by having flexible prices if all other firms do not. In contrast, if the firm has a fixed price then P i = P for all i. In this case, π i = M P (M P )(1+v)/v Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 50 / 68
Some Parameters Let s follow conventional wisdom and set v = 0.1 as the elasticity of labour supply, ϕ = 5 which corresponds to a mark-up of 25 per cent. These parameters imply, Y = ( ϕ 1 ϕ )v = 0.978 Consider an unanticipated 3 per cent fall in M and assume P is fixed. Then M new /P = 0.97M old /P. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 51 / 68
Menu Costs Plugging these numbers into the profit equation yields: Thus, ˆπ i = 0.605 π i = 0.381 ˆπ i π i ˆπ i = 0.224 0.601 1 3 These costs would have to be unreasonably high. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 52 / 68
Menu Costs The key problem is that labour supply is pretty inelastic. Thus a fall in aggregate output leads to a large fall in the real wage. This implies that the producers costs fall and so they have a strong profit incentive to cut prices and increase production. By implication, there has to be a large cost to induce them to keep prices unchanged. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 53 / 68
Dynamic New Keynesian Models So far we have examined nominal price setting frictions in static models. But we motivated some of the discussion by noting that nominal frictions may arise because of dynamic contracts. And we spent a fair bit of time on dynamic models a few weeks ago. So we will now try to combine the two. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 54 / 68
A Simple NK Model Assume there is a fixed number of infinitely-lived households. Helpful to imagine a continuum of finite measure. Assume households can be represented by a single household with preferences: U = β t [U(C t ) V (L t )], β (0, 1) t=0 Assume that the subutilities have the forms: U(C t ) = C t 1 σ 1 σ, σ > 0 V (L t ) = BLγ t γ, B, γ > 0 Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 55 / 68
Optimizing Households Assume that W is the nominal wage and that P is the average price level. The real wage is therefore W /P. We will also assume that households consume their period income. Any incremental increase in the labour supply of the household will therefore yield Wt P t dl t extra income. Thus, V (L t ) = U (C t ) W t P t Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 56 / 68
Market Clearing Assume that, like in the Lucas nominal rigidity models, that Y i = L i for all firms. For the representative firm then Y = L. Since the household consumes its period income then Y = L = C and so by substitution, V (Y t ) = U (Y t ) W t P t Using the functional forms for V and U we should find: This is the real wage. W t P t = BY σ+γ 1 t Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 57 / 68
Keynesian IS Curve You might recall the IS curve from undergraduate classes. It describes equilibrium outcomes in the goods market investment and savings. Assume that any financial instrument is in zero net supply. In this model, since all households are identical, any financial instrument introduced to share consumption between households will be in zero supply. But we can still price this instrument it s price will be the price at which there is zero demand and zero supply! Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 58 / 68
Keynesian IS Curve Suppose that there is a financial instrument. Assume that this is a riskless one period bond with interest rate r t. Then from household optimization, we will get an Euler equation: Taking logarithms yields: C σ t = β(1 + r t )C σ t+1 ln C t = ln C t+1 1 σ ln[β(1 + r t)] Imposing C t = Y t and ignoring ln β we get: This is the Keynesian IS curve. ln Y t = ln Y t+1 1 σ r t Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 59 / 68
Firms Firms produce according to Y = L. Assume that demand for firm i s output in period t is given by: Y i,t = Y t ( P i,t P t ) η which is the same demand function we assumed previously. Real profits are then: π i,t = Y t [( P i,t P t ) 1 η ( W t P t )( P i,t P t ) η ] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 60 / 68
Firm s Problem A maximizing firm that faces some pricing rigidity faces a complicated problem. Define q t as the probability that the price set by the firm in period 0 is the price in period t. Define λ t = β t U (C t ) U (C 0 ) This is the discounted marginal utility of consumption in t relative to that in the price setting period. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 61 / 68
Firm s Problem The firms problem then is to maximize (note subsuming the 0 subscript for P i ): q t λ t Y t [( P i,t ) 1 η ( W t )( P i,t ) η ] P t P t P t t=0 Note that q t can be a deterministic sequence such as we studied above. That is, it can be zero for any sequence of t. Note as well that this problem requires choosing P i. There is no general solution. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 62 / 68
But... If we imagine that inflation is low, then there is really no change in P t. Thus, a price that maximizes in period t is likely to (almost) maximize in adjacent periods. Second, assume that the economy is always close to P i,t = P t. Finally, assume β 1. Then, the firm s problem approximates to: t=0 q t λ t Y t Pt η 1 [P 1 η i W t P η I ] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 63 / 68
Firm s Problem Now since the firm s production function implies constant marginal cost, W t, and the elasticity of demand is constant, then like in the Lucas model above, the optimal price is a mark-up on marginal cost W t. Equivalently, this means that W t is a constant times the optimal price P t. Using lower case as logs, then: t=0 q t λ t Y t P η 1 t F (p t, p i ) Now by the assumptions above, λ t Y t Pt η 1 firms problem depends on q t and F (pt, p i ) is basically a constant. So the Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 64 / 68
Firm s Problem Assume that a 2nd order Taylor expansion around F (p t, p i ) is a good approximation. Approximate around p t = p i. Note that since this is the profit maximizing price by definition then df /dp i = 0. Also note that d 2 F /dpi 2 < 0. So, the firm s problem is: min p i q t (p i pt ) 2 t=0 (note that the minimization is because d 2 F /dp 2 i < 0. The solution to this is: p i = t=0 In a world with uncertainty this becomes: p i = t=0 q t τ=0 q τ p t q t τ=0 q τ mathbbe 0 [p t ] Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 65 / 68
Phillips Curve Recall from the Lucas model above that the profit maximizing real price is: And we saw above that : P i P = η W η 1 P W t P t = BY σ+γ 1 t So in logarithms, we can write: p t = p + ln η + ln B + (σ + γ 1)y η 1 Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 66 / 68
NK Phillips Curve If φ = σ + γ 1 and y = m p then we get: or p i = t=0 p t = φm + (1 φ)p t q t τ=0 q τ mathbbe 0 [φm + (1 φ)p t ] This is basically the same expression we used in the price staggering model above. As one can see, the obvious question is how to model the supply of money. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 67 / 68
Stochastic New Keynesian Models Basically add shocks to the Phillips curve and IS curve we derived above. Can imagine the DGP for these shocks is autoregressive like for technology shocks. This generates hump-shaped responses in aggregate output growth. The main equation we have not studied is how the central bank determines the path of the real interest rate. There are several flavours of this rule but a common approach is a Taylor Rule: r t = r t + a(π t Π t ) + c(y t ȳ t ) With this type of rule, we have a system of equations that can simulate a path for y, r and p. Geoffrey Dunbar (UBC, Fall 2012) Economics 502 November 22, 2012 68 / 68