(Incomplete) summary of the course

(Incomplete) summary of the course Lecture 19, ECON 4310 Tord Krogh November 20, 2012 Tord Krogh () ECON 4310 November 20, 2012 1 / 68

Main topics This semester we have been through: Ramsey OLG RBC methodology for analyzing business cycles Search and matching models Applications of microfounded intertemporal models to discuss e.g. Optimal taxation (intertemporally not the optimal combination of taxes) Savings, risk and asset prices Debt crisis Tord Krogh () ECON 4310 November 20, 2012 2 / 68

Intention? 1 Give you the necessary background to understand modern macroeconomic theories which are all developed using the same microfounded intertemporal approach as the Ramsey and OLG models. 2 Make you capable of solving RBC models for the business cycle involving derivation of first-order conditions and lineariziation which you can use to understand the sources of economic fluctuations. 3 Understand important implications one can derive with regards to asset pricing and optimal taxation principles 4 Know central theories for explaining unemployment For most of you, this course is quite challenging, since the models are extremely different from the tradititional Keynesian models that you learn at the bachelor level. Tord Krogh () ECON 4310 November 20, 2012 3 / 68

This summary 90 minutes on 18 lectures 5 minutes per lecture. What I go through will be central, but I may also leave out important issues. Plan: 1 Quick review of Ramsey and OLG 2 Taxation and debt 3 Risk and asset pricing 4 RBC methodology 5 Unemployment Tord Krogh () ECON 4310 November 20, 2012 4 / 68

Ramsey and OLG Ramsey model The basic Ramsey model is also known as the neoclassical growth model. Think of it as an extended Solow model. We no longer assume a constant savings rate. Instead we microfound the savings-decision Important ingredient: Agents are assumed to have rational expectations If there is no uncertainty this implies perfect foresight Tord Krogh () ECON 4310 November 20, 2012 5 / 68

Ramsey and OLG Ramsey model II Solution method: Solve the social planner s problem Second Welfare Theorem: A social planner can achieve any Pareto optimal allocation With several agents, the wanted allocation is obtained by choosing the appropriate welfare weights With a representative agent, the Pareto optimum is unique Tord Krogh () ECON 4310 November 20, 2012 6 / 68

Ramsey and OLG Ramsey model III Consider the deterministic case: max {c s,k s+1 } s=t s.t. { } β s t u(c s) s=t c t + k t+1 = Akt α n1 α t + (1 δ)k t c t 0 k t+1 0 0 n t 1 with k t > 0 given. Can simplify by setting n t = 1 and ignore c t 0 and k t+1 0 under normal assumptions. Tord Krogh () ECON 4310 November 20, 2012 7 / 68

Ramsey and OLG Ramsey model III Lagrangian is: [ L = β s t u(c s) λ s (c s + k s+1 Aks α + (1 δ)k ] s) s=t and the first-order conditions with resepect to c s and k s+1 are: c s : β s t u (c s) λ s = 0 ( ) k s+1 : λ s + λ s+1 1 δ + αak α 1 s+1 = 0 Tord Krogh () ECON 4310 November 20, 2012 8 / 68

Ramsey and OLG Ramsey model IV Combine the first-order conditions to see that optimum is characterized by the Euler equation: u (c t) = β[1 δ + αak α 1 t+1 ]u (c t+1 ) the resource constriant: as well as a transversality condition: c t + k t+1 = Ak α t + (1 δ)k t lim T βt u (c T ) = 0 Tord Krogh () ECON 4310 November 20, 2012 9 / 68

Ramsey and OLG Ramsey model V In a steady state we have c t = c s = c and k t = k s = k (no technological or population growth). From the Euler equation we get ( ) αak α 1 = ρ + δ where ρ = 1 1. From the resource constraint we have β ( ) Ak α δk = c Note that k < k gr (the golden-rule level of capital) as long as ρ > 0. Tord Krogh () ECON 4310 November 20, 2012 10 / 68

Ramsey and OLG Ramsey model VI We can draw ( ) and ( ) in a k, c diagram Tord Krogh () ECON 4310 November 20, 2012 11 / 68

Ramsey and OLG Ramsey model VII Saddle path: By showing which way consumption and capital moves when we are off the steady state curves, we trace out the saddle path. Can use the phase diagram to analyze the effects of an exogenous shift in e.g. productivity (A). Tord Krogh () ECON 4310 November 20, 2012 12 / 68

Ramsey and OLG OLG What about overlapping generations? Gives us models where The first welfare theorem no longer holds Ricardian equivalence is broken Good framework for studying rational bubbles Tord Krogh () ECON 4310 November 20, 2012 13 / 68

Ramsey and OLG OLG II How to solve such models? First solve the problem for each single agent Then see what it implies for the aggregate Tord Krogh () ECON 4310 November 20, 2012 14 / 68

Ramsey and OLG OLG III Important results: Dynamic inefficiency is possible Government may improve welfare A bubble might also improve welfare Leave it here, since I don t want to distort what Kjetil managed to teach you. Tord Krogh () ECON 4310 November 20, 2012 15 / 68

Taxation and debt Taxation and debt The Ramsey and OLG framework you have learned is useful to discuss a number of issues. Among them: What is optimal the best policy for financing government expenditure using taxes or debt? Tord Krogh () ECON 4310 November 20, 2012 16 / 68

Taxation and debt Taxation and debt II 1 In basic frictionless Ramsey model: Irrelevant question (Ricardian equivalence) 2 With distortinary taxes: Tax smoothing optimal. Level of debt follows residually. Debt grows in periods it is optimal to have relatively low taxes 3 In OLG model: Not irrelevant but answer depends on social planner s preference for intergenerational equity 4 With default risk: Migth be optimal to limit the level of debt to reduce the likelihood of self-fullfilling crisis Let us go through these four cases. Tord Krogh () ECON 4310 November 20, 2012 17 / 68

Taxation and debt Taxation and debt, Case 1 Under Ricardian equivalence, the timing of taxes and government debt is irrelevant. Implication: You might as well finance increases in government expenditure using taxes instead of issuing debt. If true, you should doubt the potential for expansionary fiscal policy. Tord Krogh () ECON 4310 November 20, 2012 18 / 68

Taxation and debt Taxation and debt, Case 1: II How to see that Ricardian equivalence holds? First look at the intertemporal BC of the government Then see how the agent makes optimal choices Derive the intertemp. BC of the agent Tord Krogh () ECON 4310 November 20, 2012 19 / 68

Taxation and debt Taxation and debt, Case 1: III Consider an infinitely lived government with expenditure g t, tax revenue tax t and government assets b t+1. Period t budget constraint for a constant interest rate r: g t + b t+1 = tax t + (1 + r)b t Intertemporal budget constraint, starting in period 0 (assuming no Ponzi-game condition holds): (1 + r)b 0 = t=0 g t tax t (1 + r) t Assuming b 0 = 0, this implies t=0 g t (1 + r) t = tax t (1 + r) t=0 t Tord Krogh () ECON 4310 November 20, 2012 20 / 68

Taxation and debt Taxation and debt, Case 1: IV What about the agent? Assume that the agent maximizes [ ] β t log c t φ h1+θ t 1 + θ t=0 subject to c t + a t+1 = wh t tax t + (1 + r)a t Assume also β(1 + r) = 1. In the frictionless model lump-sum taxes are possible. So tax t is taken as given by the agent. The first-order conditions for optimum are: c t = c t+1 φh θ t = w c t This implies a constant consumption level ( c) and a constant labor supply, h = (w/φ c) 1/θ. Tord Krogh () ECON 4310 November 20, 2012 21 / 68

Taxation and debt Taxation and debt, Case 1: IV The intertemporal budget constraint (imposing no Ponzi-game), starting from period 0 is: (1 + r)a 0 = t=0 c t wh t + tax t (1 + r) t Then we can insert for optimal c t and h t, and also use the government budget constraint to write ( ) 1/θ c w w φ c + gt (1 + r)a 0 = (1 + r) t=0 t Tord Krogh () ECON 4310 November 20, 2012 22 / 68

Taxation and debt Taxation and debt, Case 1: V Hence with lump-sum taxes The optimal choice of consumption and labor supply is independent of how taxes are timed Whether you choose to set tax t = g t every period, or tax t = constant and let deficits vary with the business cycle, it doesn t matter. Why not? Since the level of a lump-sum tax will not affect the consumption trade-off in the Euler equation or the consumption-leisure trade-off in the intratemporal optimality condition. The presence of a government will still have an impact, since it reduces c. But it is only the NPV (i.e. total costs) that matters The economy features Ricardian equivalence Tord Krogh () ECON 4310 November 20, 2012 23 / 68

Taxation and debt Taxation and debt, Case 2 Then assume that lump-sum taxes are no longer an alternative. Instead the government must use an income tax τ t such that tax t = τ twh t Agent takes this into account when choosing h t. New optimality conditions: φh θ t c t = c t+1 = w(1 τt) c t Tord Krogh () ECON 4310 November 20, 2012 24 / 68

Taxation and debt Taxation and debt, Case 2: III Insert for constant consumption in the intratemporal optimality condition to ( ) w(1 τt) 1/θ h t = φ c Take this into the intertemporal BC: (1 + r)a 0 = c w t=0 ( ) w(1 τt ) 1/θ φ c + gt (1 + r) t Shows that, even though the government intertemp. BC still holds (NPV of taxes = NPV of expenditure), the timing of taxes will matter since it affects labor supply. Ricardian equivalence does not hold! Tord Krogh () ECON 4310 November 20, 2012 25 / 68

Taxation and debt Taxation and debt, Case 2: IV In this particular case, we could use this condition to solve for c. The optimal tax rate will maximize c subject to the government BC. Other applications we have looked at with a similar set-up: Tax smoothing model from Romer (Lecture #7) and the seminar problem on optimal use of oil money. In the Romer-model taxes are set to minimize a loss function (which summarizes the welfare loss from taxation). Robust result:tax rates should be smooth since that limits the extent of intertemporal distortions Tord Krogh () ECON 4310 November 20, 2012 26 / 68

Taxation and debt Taxation and debt, Case 2: V What about debt? By finding the opitmal path of taxes, this also determines the deficit. The explanation for deficits/surpluses becomes: Surpluses are due to periods of high output, such that the optimal tax revenue exceeds expenditure Deficits are due to periods of low output, such that the optimal tax revenue is lower than expenditure What are the implications? It implies that taxes should be set to smooth the tax burden over time. Rather than adjusting taxes when expenditure is fluctuating, one should let debt play that role. This resembles automatic stabilizers, but motivation is not to stabilize business cycles. Here the motivation is to minimize welfare losses associated with taxation. Tord Krogh () ECON 4310 November 20, 2012 27 / 68

Taxation and debt Taxation and debt, Case 3 Then we turn to an OLG model. For simplicity, let us use 2 generations. Only work when young. A working agent earns wh t. Young will pay tax t in taxes and recieve p t if they are old. Consumption is c y t when young and ct+1 o when old. Savings are a t+1. Budget constraints: c y t + a t+1 = wh t tax t c o t+1 = pt + (1 + r)a t+1 or, if we combine the two: c y t + co t+1 pt = wht taxt + 1 + r 1 + r Tord Krogh () ECON 4310 November 20, 2012 28 / 68

Taxation and debt Taxation and debt, Case 3: II The government is still infinitely lived. Hence the intertemporal BC it faces (assuming no assets initially) will be p t (1 + r) t=0 t = tax t (1 + r) t=0 t Even though the NPV of taxes equals NPV of expenditure, this is of little comfort for an agent that only lives for two periods (he might benefit from this, of course). Tord Krogh () ECON 4310 November 20, 2012 29 / 68

Taxation and debt Taxation and debt, Case 3: III Ricardian equivalence will now fail to hold: Even when lump-sum taxes are available, the government may transfer income from one generation to the other. Timing of taxes will have effects. If taxes are distortionary as well, this is an extra reason for why it fails Tord Krogh () ECON 4310 November 20, 2012 30 / 68

Taxation and debt Taxation and debt, Case 4 Finally, we should also remember one last model we have looked at that has some implications for optimal taxation and debt. That model focuses at the possibility of debt crisis. Tord Krogh () ECON 4310 November 20, 2012 31 / 68

Taxation and debt Taxation and debt, Case 4 II Graphical illustration (see Lecture #7): Tord Krogh () ECON 4310 November 20, 2012 32 / 68

Taxation and debt Taxation and debt, Case 4 III Implications? Useful model to think about the dynamics of debt crisis Tells us that there might be additional costs from a higher debt level that models without default risk ignore Also tells us that not all debt-crisis are inevitable can have self-fulfilling prophesies. More broadly, this is the only model we look at in the course with multiple equilibria. Maybe we could get interesting business cycle models using the same mechanism? Tord Krogh () ECON 4310 November 20, 2012 33 / 68

Risk and asset pricing Saving and uncertainty Another useful extension of the basic model is to the case with stochasticity. In class you have been through How to work with decisions under uncertainty Asset-pricing implications of neoclassical model Consumption CAPM Equity-premium puzzle Tord Krogh () ECON 4310 November 20, 2012 34 / 68

RBC methodology RBC methodology Ambition: To start out with the simplest possible neoclassical growth model, calibrate it to match important long-run moments, and then see how well it does in accounting for business cycle facts. Tord Krogh () ECON 4310 November 20, 2012 35 / 68

RBC methodology RBC methodology II What facts? From Kydland and Prescott (1990): Hours worked are about as volatile as output, highly procyclical and is contemporaneously correlated with the cycle (or slightly lagging) Most of the volatility of hours is due to the extensive, not the intensive margin (i.e. employment, not hours per worker) Consumption is overall a procyclical variable, contemporaneously correlated with the cycle, almost leading. Durables most volatile Investment expenditures is another strongly procyclical variable contemporaneously correlated with the cycle. The most volatile variable Government purchases seem to be completely uncorrelated with the cycle The price level is countercyclical and leading the cycle. The CPI is almost as volatile as output. Tord Krogh () ECON 4310 November 20, 2012 36 / 68

RBC methodology The basic steps for an RBC model Write down the optimization problem Find first-order conditions Characterize steady state Calibrate the model Linearize optimality conditions around steady state Solve the set of linearized equations Plot impulse-response functions, simulate, calculate moments etc. Tord Krogh () ECON 4310 November 20, 2012 37 / 68

RBC methodology Labor supply response Individuals are often assumed to have period t utility functions such as log c t φ n1+θ 1 + θ With reasonable values for θ, it is hard to make the model generate sufficient labor supply response Relevant concept: Frisch elasticity of labor supply. But if we use the labor lotteries setup, we only change labor at the intensive margin. Allows us to work with a model where the representative agent has utility function u(c t) Dn t even though each single individual has utility given by u(c t) v(n). The micro Frisch elasticity is 1/θ. Under labor lotteries, the macro elasticity is infinite! Today: Use the labor lotteries model. Tord Krogh () ECON 4310 November 20, 2012 38 / 68

RBC methodology RBC optimization problem The basic RBC model will often take the following form: max {c t,n t,k t+1 } t=0 s.t. E 0 β t [u(c t) Dn t] t=0 c t + k t+1 = A tkt α n1 α t + (1 δ)k t A t = Ae zt z t = ρz t 1 + ε t c t 0 k t+1 0 0 n t 1 with k 0 > 0 given. Can as before simplify by ignoring the conditions on n, c and k under normal assumptions. Tord Krogh () ECON 4310 November 20, 2012 39 / 68

RBC methodology RBC optimization problem II To solve the problem, insert for c t in the utility function using the resource constraint. The first-order conditions with respect to k 1 and n 0 are now: [ ] u (A 0 k0 α n1 α 0 + (1 δ)k 0 k 1 ) = βe 0 (1 + r 1 )u (A 1 k1 α n1 α 1 + (1 δ)k 1 k 2 ) w 0 u (A 0 k α 0 n1 α 0 + (1 δ)k 0 k 1 ) = D where I have defined w t = (1 α)a tkt α nt α r t+1 = αa tkt α 1 nt 1 α δ Tord Krogh () ECON 4310 November 20, 2012 40 / 68

RBC methodology RBC optimization problem III Simplify the first-order condition for k t+1 by re-introducing c t as defined by the resource constraint. That gives the familiar expressions for the Euler equation and the intratemporal condition: u (c 0 ) = βe 0 [ (1 + r1 )u (c 1 ) ] w 0 u (c 0 ) = D These two conditions, together with the resource constraint and the definition of w t and r t+1 : is what we need to describe optimum. c t + k t+1 = A tkt α n1 α t + (1 δ)k t w t = (1 α)a tkt α n α t r t+1 = αa tkt α 1 nt 1 α δ Tord Krogh () ECON 4310 November 20, 2012 41 / 68

RBC methodology Steady state The non-stochastic steady state will have constant consumption and capital (since there is no secular growth). Requires from the Euler equation and r = 1 β 1 w = Dc from the intratemporal optimality condition (if we assume log utility). Remaining conditions: c = A k α n 1 α + δk w = (1 α)a k α n α r = αa k α 1 n 1 α δ Tord Krogh () ECON 4310 November 20, 2012 42 / 68

RBC methodology Calibration Ignoring productivity, the model has four structural parameters: Discount factor β Deprecitation rate δ Cobb-Douglas parameter α Disutility of labor supply D To calibrate the model we must find four moments (usually averages) we want our model to match. By this we mean that the steady state properties of the model should match the data. Tord Krogh () ECON 4310 November 20, 2012 43 / 68

RBC methodology Calibration II A standard set of moments to match are: Average capital share of income Average investment to capital ratio Average long-term real interest rate Average share of available hours spent on work Tord Krogh () ECON 4310 November 20, 2012 44 / 68

RBC methodology Calibration III If we want to ensure a capital share equal to 1/3, an investment to capital ratio of 2.5%, a real interest rate of 1% and n = 1/3 in our model we just choose: α = 1/3 δ = 0.025 β = 1/1.01 D = 8/3 You should manage to use the steady state equations to show why we choose this particular calibration. Tord Krogh () ECON 4310 November 20, 2012 45 / 68

RBC methodology Calibration IV There are two other parameters we need to find values for as well. Remember, technology depends on z t, which is an AR(1) process z t = ρz t 1 + ε where ε is N(0, σɛ 2). Must pin down ρ and σ2 ɛ. How to choose? Find productivity as the Solow residual: Z t = log A t = log Y t α log K t (1 α) log N t (possibly also controlling for a linear trend) Estimate ρ from regressing Z t = ρz t 1 + e t Estimate σ 2 ɛ based on the residual sum of squares from the regression Krueger chooses ρ = 0.95 and σ ɛ = 0.007. Tord Krogh () ECON 4310 November 20, 2012 46 / 68

RBC methodology Linearization Next step is to linearize the optimality condition. We only solve an approximation of the model (around the steady state). How to do? Let ˆx t be a variable s percentage deviation from steady state: xt x ˆx t = x What we do next is to take first-order Taylor approximations of all the conditions Tord Krogh () ECON 4310 November 20, 2012 47 / 68

RBC methodology Linearization II Example: Marginal utility. u (c t) u (c ) + u (c )(c t c ) = u (c ) + u (c )c ĉ t = u (c ) [1 u (c )c ] u (c ĉ t ) = u (c ) [1 σĉ t] where σ is the steady-state value of the coefficient of relative risk aversion. If u(c) = log c, we know that σ = 1. Tord Krogh () ECON 4310 November 20, 2012 48 / 68

RBC methodology Linearization III Example II: Expected product of marginal utility and the interest rate. βe t[(1 + r t+1 )u (c t+1 )] β(1 + r )u (c ) + βu (c )E t(r t+1 r ) + β(1 + r )u (c )E t(c t+1 c ) Use β(1 + r ) = 1 to get βe t[(1 + r t+1 )u (c t+1 )] u (c ) (1 + βr E tˆr t+1 σe tĉ t+1 )] Tord Krogh () ECON 4310 November 20, 2012 49 / 68

RBC methodology Linearization IV Combine these two approximations to find the linearized Euler equation: ĉ t = E tĉ t+1 βr σ Etˆr t+1 Tord Krogh () ECON 4310 November 20, 2012 50 / 68

RBC methodology Linearization V The full set of linearized equilibrium conditions is: together with the process for z t: ĉ t = E tĉ t+1 βr σĉ t = ŵ t σ Etˆr t+1 ŷ t = c c ĉt + (1 y y )î t ˆk t+1 = (1 δ)ˆk t + δî t ŷ t = z t + αˆk t + (1 α)ˆn t ŵ t = z t + α(ˆk t ˆn t) ˆr t = r + δ ( ) r z t (1 α)(ˆk t ˆn t) z t = ρz t 1 + ε t Tord Krogh () ECON 4310 November 20, 2012 51 / 68

RBC methodology Using the model Can simulate the model to understand the dynamics, and to compare its predictions with what we observe in the data. Tord Krogh () ECON 4310 November 20, 2012 52 / 68

RBC methodology Impulse-responses What is the effect of a one percentage point shock to technology? Tord Krogh () ECON 4310 November 20, 2012 53 / 68

RBC methodology Impulse-responses II Investment is very volatile Employment goes up to take advantage of the temporarily higher wages Output increases by twice as much (in percentage points) as technology. The shock is propagated by the endogenous responses (labor supply and investment) Tord Krogh () ECON 4310 November 20, 2012 54 / 68

RBC methodology Impulse-responses III As we did in Lecture 13, we can also simulate the model assuming divisible labor instead of labor lotteries. Clear that this kills a lot of the propagation. Tord Krogh () ECON 4310 November 20, 2012 55 / 68

Summary RBC methodology: Summary RBC models describe business fluctuations as Driven by technology shocks in a perfectly competitive environment Since the representative agent maximizes utility and there are no externalities, there are no efficiency losses Business cycles are therefore the product of an economy s optimal response to temporary technology shocks Any stabilization policy will either be (at best) ineffective or harmful Tord Krogh () ECON 4310 November 20, 2012 56 / 68

Summary RBC methodology: Summary II The RBC methodology involves: Macroeconomics should always involve microfounded general equilibrium models with rational expectations. Family of such models: DSGE. One should take the quantitative implications of a model seriously. Tord Krogh () ECON 4310 November 20, 2012 57 / 68

Unemployment Unemployment How to think about involuntary unemployment in microfounded equilibrium models? Completely missing from RBC models. Two prominent theories that can help us understand unemployment better: Efficiency wages (Shapiro-Stiglitz) Search and matching models In addition, the paper by Blanchard is a great presentation of how theories for unemployment have developed since the 1970s. Tord Krogh () ECON 4310 November 20, 2012 58 / 68

Unemployment Unemployment II The Shapiro-Stiglitz model features unemployment because Workers must exert effort to produce anything Firms have limited monitoring abilities Will pay a higher wage than the market-clearing wage in order to induce sufficient effort Result is efficiency wages Tord Krogh () ECON 4310 November 20, 2012 59 / 68

Unemployment Unemployment III Search and matching models focus at a different friction: Search frictions put a limit to how may firms and (unemployed) workers that meet every period Result is an equilibrium where agents are willing to work at the market wage, but search frictions prevent them from meeting firms that want to hire them Tord Krogh () ECON 4310 November 20, 2012 60 / 68

Unemployment Unemployment IV We have looked at: A one-sided model with random wage offers A two-sided model where both workers and firms search Tord Krogh () ECON 4310 November 20, 2012 61 / 68

Unemployment Unemployment V Central for the one-sided model: To derive the reservation wage w. And then look at flows in and out of unemployment to find out how w affects the equilibrium unemployment rate. Tord Krogh () ECON 4310 November 20, 2012 62 / 68

Unemployment Unemployment VI Illustration: Tord Krogh () ECON 4310 November 20, 2012 63 / 68

Unemployment Unemployment VII For the two-sided model: Use the matching function to determine the number of mathces Nash-bargaining gives the wage rate Then look flows in and out of unemployment gives the unemployment rate as a function of labor market tightness. Tord Krogh () ECON 4310 November 20, 2012 64 / 68

Unemployment Unemployment VIII Illustration: Tord Krogh () ECON 4310 November 20, 2012 65 / 68

Unemployment Unemployment IX Illustration: Tord Krogh () ECON 4310 November 20, 2012 66 / 68

Unemployment Unemployment X Common for both the Shapiro-Stiglitz and the search and matching models we have looked at, is the use of dynamic programming. We use Bellman equations to define value functions. You should understand the intuition behind e.g. V e(w) = β [w + δv u + (1 δ)v e(w)] which is the value function for an employed worker with a wage w in the search and matching model. Tord Krogh () ECON 4310 November 20, 2012 67 / 68

Unemployment Unemployment XI Blanchard s focus (for explaining the transmission of shocks in the 70s) is at: Real wage rigidities Nominal wage rigidities Real wage rigidities can explain why shifts in productiviy give a temporary increase in the NAIRU. Reason? Real wages adjust too slowly down, so unemployment rate must go up. Tord Krogh () ECON 4310 November 20, 2012 68 / 68