Microfoundations of DSGE Models: III Lecture

Microfoundations of DSGE Models: III Lecture Barbara Annicchiarico BBLM del Dipartimento del Tesoro 2 Giugno 2. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 / 65

Contents A New Keynesian model with capital accumulation A New Keynesian model with capital accumulation, habit persistence and adjustment costs on labour and investments A New Keynesian model with capital accumulation and rule-of-thumb households A complete model. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 2 / 65

Main Features Agents a continuum of households that consume, own capital and supply differentiated labour services; a continuum of trade unions each of which representing workers of a certain type; a continuum of intermediate goods producers that employ labour inputs, rent capital from households and produce differentiated intermediate goods (monopolistic competition); a continuum of final goods producers that use intermediate goods to produce a homogenous final good consumed by households (perfect competition); the government that set public spending; the central bank that implements monetary policy. Imperfect competition Prices rigidities Business cycles driven by nominal and real shocks Rational expectations and no asymmetric information. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 3 / 65

Final-Good Firms Each firm takes the other firms prices as given. The representative final-good firm uses Y t (j) units of each intermediate good j 2 [, ] purchased at a nominal price P t (j) to produce Y t units of the final good with a constant returns to scale technology: Z Y t = Y t (j) θ θ θ θ dj θ = the elasticity of substitution across intermediate goods, θ >. As θ! higher and higher degree of substitution! less market power of intermediate-goods producers.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 4 / 65

Final-Good Firms The problem of the representative firm is to max their profits wrt Y t (j) with j 2 [, ] (static problem) given the available technology. Z P t Y t P t (j)y t (j) dj Profit maximization yields the following set of demands for intermediate goods: Pt (j) θ Y t (j) = Y t P t Perfect competition and free entry drives the final good-producing firms profits to zero, so that from the zero-profit condition we obtain: Z P t = P t (j) θ dj θ. which defines the aggregate price index of the economy and is such that P t Y t = R P t (j) Y t (j) dj.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 5 / 65

Intermediate-Goods Firms Each intermediate good j is produced by a monopolist which has a production function of the form: Y t (j) = A t L t (j) α K t (j) α where < α < A t = Total factor productivity, A t = A exp(ε A,t ) and ε A,t = ρ A ε A,t + ξ A,t with ξ A,t iid.n(, σ 2 A ) L t (j) = CES aggregate of labor inputs supplied by unionized workers defined below (see below) K t (j) = physical capital. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 6 / 65

Price Rigidities à la Rotemberg According to Rotemberg (983) each monopolistic firm faces a quadratic cost of adjusting nominal prices, measured in terms of the final-good: ADJ_P t (j) = γ p 2 Pt (j) P t (j) 2 Y t where γ p =degree of nominal price rigidities. Firm j will not always choose to charge the optimal price (i.e. Pt (j) = θ θ MC t(j)) since it is assumed to face convex costs to changing its price.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 7 / 65

Price Rigidities à la Rotemberg Why are price changes costly? There is the cost of physically changing posted prices (probably a fixed cost per price change). A firm that changes its prices may face a cost which results from the negative reaction of its costumers (reputation loss). From this point of view, probably customers react more strongly to large price changes than to gradual variations.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 8 / 65

Price Rigidities à la Rotemberg What is the main difference between the Calvo price setting and the Rotemberg adjustment costs hypothesis? In the Rotemberg model there is no price dispersion. In each period t firm j can change its price price P t (j) s.t. the payment of the adjustment cost. All firms face the same problem and will set the same price and produce the same quantity of each differentiated good (symmetry). In the Calvo model there s price dispersion. Firms will not set the same price (asymmetry). Different types of inefficiency in the two models In the Rotemberg model the cost of nominal rigidities consists in a wedge between aggregate demand (C t + I t + G t ) and aggregate output (Y t ), since a fraction of the output goes in the price adjustment cost. In the Calvo model, nominal rigidities through price dispersion, makes aggregate output less efficient.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 9 / 65

Price Rigidities à la Rotemberg Despite the economic difference between these two models of price rigidities, up to a first order approximation and around a zero inflation steady state, they imply the same reduced form of the New Keynesian Phillips curve. Otherwise, the Rotemberg model seems to be more robust to non-linearities (implying more robust results). The implications of of having trend inflation in the two pricing models. On these issues: see Ascari and Merkl (29); Ascari and Ropele (27); Ascari and Rossi (29, 2).. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 / 65

Intermediate-Goods Firms Given the wage index W t and the rental rate of capital rt k, the problem for firm j is to choose fl t (j), K t (j), P t (j)gt = in order to maximize the sum of expected discounted real profits ( ) E β t λ R Pt (j) W t t= λ R P t Y t (j) t P t L t (j) rt k K t (j)+, ADJ_P t (j) given Y t (j) = ADJ_P t (j) = γ p 2 Pt (j) P t θ Yt and where Pt (j) P t (j) 2 Yt.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 / 65

Intermediate-Goods Firms To solve firm s j problem, 8 2 consider the Lagrangian function 3 P t (j) W >< P L = E t= β t 4 t Y t (j) t P t L t (j) rt k K t (j) λr t γ p Pt (j) 2 5 + λ R 2 P t Yt (j) >: MC t (j) Y t (j) A t L t (j) α K t (j) α where MC t (j) =real marginal cost. 9 >= >;. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 2 / 65

Intermediate-Goods Firms Using Y t (j) = Pt (j) P t FOC wrt P t (j) ( θ) + MC t (j) θ P t θ Yt we have: P t (j) Y t (j) = ADJ_P t(j) + P t (j) +βe t λ R t+ λ R t ADJ_P t+ (j) P t (j) Remark: for γ p = under symmetry: MC t = θ θ, that is P t = θ θ MC N t.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 3 / 65

Intermediate-Goods Firms FOC wrt K t (j) FOC wrt L t (j) r k t = ( α) MC t (j) A t L t (j) α K t (j) α W t P t = αmc t (j)a t L t (j) α K t (j) α. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 4 / 65

Households and Preferences Consider a continuum of households index by i 2 [, ]. Household i is characterized by the following lifetime utility function: E β t log(c i,t ) + ω t= v ( L i,t) v C i,t consumption; L i,t specific labour inputs; β is the time discount factor; v < The period-by-period budget constraint is P t C i,t + B i,t + P t I i,t = W i,t L i,t + ( + r t )B i,t + +P t r K t K i,t + D i,t P t TAX i,t where I i,t = K i,t+ ( δ) K i,t ; D i,t = dividends; W i,t =nominal wage; TAX t =lump-sum taxes; r = nominal interest rate; B i,t =nominal bonds issued by the government.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 5 / 65

Households and Preferences The representative household will choose fc i,t, B i,t, K i,t+ gt= so as to max the lifetime utility function given the sequence of budget constraint. To solve household s j problem, consider the Lagrangian function 8 L i >< = E β t t= >: λ i,t " log (C i,t ) + ω v ( L i,t) v + Wi,t P t L i,t + ( + r t ) B i,t P t + rt K K i,t + D i,t P t + B TAX i,t C i,t i,t P t K i,t+ + ( δ) K i,t # 9 >= >; FOC wrt C i,t : C i,t = λ i,t FOC wrt B i,t : λ i,t λ P t = βe i,t+ t P t+ ( + r t ) FOC wrt K i,t+ : λ i,t = βe t λ i,t+ r K t+ + ( δ). Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 6 / 65

Households and Preferences Combining the previous conditions we derive the Euler equation governing the time path of consumption C i,t = βe t C i,t+ + r t + π t+ and the asset equation according to which at the optimum a household is indifferent between the two assets (capital and risk-free public debt bonds) since the expected benefit in terms of utility is the same: + r h i t E t λ i,t+ = E t λ i,t+ rt+ K + ( δ) + π t+. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 7 / 65

Wage Setting There is a continuum of unions each of which represents workers of a certain type. Effective labour input hired by the intermediate-good firm j is a CES function of the quantities of the different labour types employed:! σ R L t (j) = L L i,t (i) σ L σ L σ L di where σ L > elasticity of substitution across different types of labour inputs. At the optimum (and under symmetry among firms) the demand for each variety of labour input i is σl Wi,t L i,t = L t h R where W t = W σ L i,t i di W t σ L R such that W t L t = W i,t L i,t di.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 8 / 65

Wage Setting The union representing worker of type i will set W i,t in order to max σl the expected utility of household i given L i,t = Wi,t W Lt t. The relevant part of the Lagrangian is ω E β t v ( L i,t) v W i,t + λ i,t L i,t At the optimum: W i,t σ = L ω ( P t σ L {z } wage markup λ i,t P t L i,t ) v. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 9 / 65

The Government The government budget constraint is B t = ( + r t )B t + P t G t P t TAX t where G t = G exp(ε g,t ) and ε g,t = ρ g ε g,t + ξ g,t with ξ g,t iid.n(, σ 2 g ), while P t TAX t = P t TAX + τp t B t where τ is set in order to rule out any explosive path of the public debt ("passive" rule as meant by Leeper 99).. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 2 / 65

The Central Bank The monetary authority sets the short-term nominal interest rate in accordance with an interest rate rule R t R = Rt R ιr ιπ ιy Πt Yt Π Y ιr u R t. where R t = + r t, Π t = + π t, ut R with ξ u,t iid.n(, σ 2 u). = exp(ε u,t ) and ε u,t = ρ u ε u,t + ξ u,t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 2 / 65

Equilibrium Combining the above conditions, imposing symmetry between firms, households and unions the equilibrium of the economy is described by the following equations. The Euler equation + r t λ t = βe t λ t+ + π t+ The capital asset equation i λ t = βe t λ t+ hr t+ K + ( δ) The Lagrange multiplier λ t = C t The wage equation W t = σ L ω ( P t σ L λ t L t ) v. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 22 / 65

Equilibrium The aggregate production function Y t = A t L α t K t α The demand of capital The demand of labour The inflation equation γ p (Π t ) Π t + βγ p E t λ R t+ λ R t r k t = ( α) MC t A t L α t K α t W t = αmc t A t Lt α Kt α P t (Π t+ ) Y t+ Y t Π t+ = ( MC t ) θ. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 23 / 65

Equilibrium The budget constraint of the government B t = ( + r t )B t + P t G t P t TAX t The tax rule The interest rate rule P t TAX t = P t TAX + τp t B t R t R = Rt R ιr ιπ ιy Πt Yt Π Y ιr u R t.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 24 / 65

Equilibrium The capital accumulation equation K t+ = ( δ)k t + I t The resource constraint of the economy Y t = C t + I t + G t + γ p 2 (Π t ) 2 Y t The exogenous processes governing A t, G t and u R t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 25 / 65

A remark on the inflation equation Given the inflation equation γ p (Π t ) Π t + βγ p E t λ R t+ λ R t (Π t+ ) Y t+ Y t Π t+ = ( MC t ) θ in a zero-inflation steady state MC = θ θ. With trend inflation (and no indexation): MC = θ ( β)(π )Π θ + γ p The markup will be: θ markup = θ ( β) (Π ) Π + γ p θ θ. The markup is decreasing in the level of trend inflation. As a result: output (and so employment) is higher the higher Π. However, a higher fraction of output is eaten up by the adjustment costs.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 26 / 65

Calibration α = 2/3 labour share β =.99 discount factor δ =./4 depreciation rate v =.8 preference parameter γ p = 58.25 degree of price rigidities θ =.25 θ = 6 elasticity of subst. between goods σ L = 5 elasticity of subst. between labour inputs ρ A =.9 persistence of tech shock ρ G =.9 persistence of the public spending shock ρ R =.9 persistence of monetary policy shock. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 27 / 65

Calibration Π = L =.3 Y = ι π =.5 ι R = ι y =.5/4 inflation employment output monetary policy parameter monetary policy parameter monetary policy parameter. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 28 / 65

% % Effects of a Technology Shock.6 Consumption, c.4 Output, Y.6 Labour, L 6 Investments, I.5.2.4 5 4.4.3.8.6.2 3 2.2.4.2. 2 4 2 4.2 2 4 2 4 Inflation.8 Real Wage, W/P.5 Real Interest Rate.25 Public Debt, B/P.5.7.6.4.3.2.5..5.2..5.4..5.2.3.2..5.25 2 4. 2 4.2 2 4. 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 29 / 65

Effects of a Technology Shock: Propagation Higher Productivity! lower marginal costs! lower inflation As in the basic RBC model a transitory productivity shock, which temporarily raises the real wage rate, increases employment (maybe we have a too low degree of price rigidity) Productivity "!MPK "!rental rate of capital " Substitution effect increases savings (prevails) Consumption increases gradually (consumption smoothing) Investments increases on impact (the volatile component) As a result y increases more than proportionally.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 3 / 65

% % Effects of a Technology Shock with a higher degree of price rigidities.7 Consumption, c.4 Output, Y.6 Labour, L 6 Inv estments, I.6.2.4 5.5 4.4.8.2 3.3.6 2.2.4..2.2 2 4.4 2 4 2 4 2 4 Inflation.7 R eal W age, W /P R eal Interes t R ate.5.2 Public Debt, B/P.5.6.5.4.3...5.4.3.2..2.2....25 2 4 2 4.2 2 4.2 2 4 Reponses under the baseline calibration are plotted in red.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 3 / 65

% % Effects of a Government Spending Shock. Consumption, c.8 Output, Y.2 Labour, L. Investments, I.2.3.4.5.6.7.6.4.2..8.6.4.2..2.3.8.2 2 4 2 4 2 4.4 2 4 6 x 3 Inflation.5 Real Wage, W/P Real 3 x Interest Rate 3 Public Debt, B/P 5..5 2.5.8 4.2 2.6 3.25.5.4 2.3.35.2 2 4.4 2 4.5 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 32 / 65

Effects of a Government Spending Shock: Propagation Government spending "!taxes increase "!net wage income# income effect (agents need to work more)!employment increases then gradually returns to normal consumption falls, but the rise in government supply is temporary, hence agents respond by decreasing their capital holdings (consumption smoothing) firms will revise their prices upward (as real marginal costs are higher)! the Central Bank will react to inflation by increasing the nominal interest rate more than proportionally! as a result the real interest rate will increase (lean against the wind policy). As a result y will increase less than proportionally. Remark: no comovement between c and g.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 33 / 65

% % Effects of a Government Spending Shock with a weak response to deviations from targets ι π =.; ι y = Consumption, c. Output, Y.4 Labour, L. Inv estments, I.2.4.6.8.6.4.2.2..8.6.4.2..2.3.8.2 2 4 2 4 2 4.4 2 4.3 Inflation R eal W age, W /P R 3.5 x eal 3 Interes t R ate Public Debt, B/P.25.2.5..5..2.3 3 2.5 2.5.5.8.6.4.2 2 4.4 2 4 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 34 / 65

% % Effects of a Monetary Policy Shock Consumption, c Output, Y.5 Labour, L 2 Investments, I..2.3.4.5.6.5.5.5.5 2 2.5 2 4 6 8.7 2 4 2 2 4 3 2 4 2 4.5 Inflation Real Wage, W/P Real Interest Rate.45 Public Debt, B/P.4.2.4.4.35.2.5.6.8.3.25.8.6.2.4.2.5.2.5 2 4.4 2 4. 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 35 / 65

Effects of a Monetary Policy Shock: Propagation The presence of price rigidities is a source of nontrivial real effects of monetary policy shocks. Firms cannot immediatly adjust the price of their good when they receive new information about costs or demand conditions. The shock generates an increase in the real rate, a decrease in inflation, output and employment.. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 36 / 65

% % Effects of a Monetary Policy Shock with a higher degree of price rigidities Consumption, c Output, Y 2 Labour, L 5 Investments, I.5.5 2 3 4 5 2 4 6 5 5 2 2 2 4 6 2 4 8 2 4 25 2 4.5 Inflation Real Wage, W/P Real Interest Rate.5 Public Debt, B/P.5.5.5.5 2 2.5..5.5..5.5.5 2 4 3 2 4.2 2 4.5 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 37 / 65

Model #2: Model #+ Habit+Adjust. Costs Extend the previous model to account for: external habit (see Lecture I) adjustment costs on investments (see Lecture I) adjustment costs on investments on labour (see Lecture I). Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 38 / 65

Model #2: Model #+ Habit+Adjust. Costs Households and Preferences The typical household will solve the following problem Household i is characterized by the following lifetime utility function: E β t log(c i,t h e C t ) + ω t= v ( L i,t) v C average aggregate consumption; h e measure of habit intensity The period-by-period budget constraint is P t C i,t + B i,t + P t I i,t = W i,t L i,t + ( + r t )B i,t + P t r K t K i,t +D i,t P t TAX t P t ADJ(I t, K t ) where K i,t+ = ( δ) K i,t + I i,t ADJ(I i,t, K i,t ) = γ 2 I Ii,t δ K i,t 2 K i,t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 39 / 65

Model #2: Model #+ Habit+Adjust. Costs Households At the optimum we now have (dropping index i) C t h e C t = λ t λ t P t = βe t λ t+ P t+ ( + r t ) γ I It K t δ K = q t q t λ {z} t = βe t λ t+ r t+ + β ( δ) E t q t+ λ {z t+ } ξ t where q t is the Tobin s marginal q. βe t λ t+ ADJ(I t+, K t+ ) K t+. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 4 / 65 ξ t+

Model #2: Model #+ Habit+Adjust. Costs Intermediate-goods firms the adjustment costs on labour We now assume that hiring and firing unionized workers is costly, in particular we have: ADJ_L t (j) = γ L 2 Lt (j) L t (j) 2 Y t As a result the optimal demand of labor is W t = α L MC t (J) Y t (j) P t L t (j) +βγ L E t λ R t+ λ R t Lt (j) γ L L t (j) Lt+ (j) L t (j) 2 Y t+ Lt+ (j) L t (j) Yt L t (j) +. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 4 / 65

Model #2: Model #+ Habit+Adjust. Costs The resource constrain of the economy The resource constraint of the economy is now Y t = C t + I t + G t + + γ p 2 (Π t ) 2 Y t + + γ L Lt (j) 2 Y t + 2 L t (j) + γ 2 I It δ K t 2 K t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 42 / 65

% % Model #2: Model #+ Habit+Adjust. Costs Effects of a Technology Shock.8 Consumption, c.4 Output, Y.6 Labour, L 6 Investments, I.6.2.4 5 4.4.8.6.2 3 2.2.4.2 2 4 2 4.2 2 4 2 4 Inflation Real Wage, W/P Real Interest Rate..2 Public Debt, B/P..8.5..2.6.4..2.3.2.5.3.4.4 2 4 2 4. 2 4.5 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 43 / 65

% % Model #2: Model #+ Habit+Adjust. Costs Effects of a Government Spending Shock Consumption, c.8 Output, Y.2 Labour, L.2 Investments, I.2.4.6.8..2.6.4.2..8.6.4.2...2.3.4.4.2 2 4 2 4 2 4.5 2 4.6 Inflation Real Wage, W/P Real Interest Rate.3 Public Debt, B/P.5.4.3.2..5..5.25.2.5..5.8.6.4.2 2 4.2 2 4 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 44 / 65

% % Model #2: Model #+ Habit+Adjust. Costs Effects of a Monetary Policy Shock Consumption, c Output, Y.5 Labour, L 2 Investments, I..2.3.4.5.6.5.5.5.5 2 2.5 2 4 6 8.7 2 4 2 2 4 3 2 4 2 4.5 Inflation Real Wage, W/P Real Interest Rate. Public Debt, B/P 2.2.4.8.5.5.6.8.6.4.5.2.2.5 2 2 4.4 2 4 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 45 / 65

Model #3: Model #+RoT Extend the Model # to account for: Consider rule of thumb households as in Galí, López-Salido and Vallés (27).. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 46 / 65

Model #3: Model #+RoT Households There is a continuum of households. Population is constant and normalized to. A fraction s NR of households do not borrow and save, and just consume their current labor income (hand-to-mouth households)! extreme form of non-ricardian behavior Motivation: an extensive empirical literature provides evidence of excessive dependence of consumption on current income; deviations from the permanent income hypothesis. As a result now we have two types of households: Non-Ricardian households (population share s NR ) Ricardian households (population share s NR ). Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 47 / 65

Model #3: Model #+RoT The Ricardian Households The typical Ricardian household will solve the following problem Household i is characterized by the following lifetime utility function: E β t log(ci,t R h e C t ) + ω L R i,t t= v C average aggregate consumption; h e measure of habit intensity The period-by-period budget constraint is P t C R i,t + B r i,t + P t I r i,t = W i,t L R i,t + ( + r t )B R i,t + +P t r K t K R i,t + D R i,t v P t TAX R t where K R i,t+ = ( δ) K R i,t + I R i,t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 48 / 65

Model #3: Model #+RoT The Ricardian Households At the optimum we now have (dropping index i) we have the standard optimality conditions: C R t = λ R t λ R t λ R t+ = βe t ( + r t ) P t P t+ i λ i,t = βe t λ i,t+ hr t+ K + ( δ). Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 49 / 65

Model #3: Model #+RoT The Non-Ricardian Households Non-Ricardian households are assumed to behave in a hand-to-mouth fashion: they fully consume their current labor income (no consumption smoothing). The representative household of this category derives utility from consumption and leisure: log C NR i,t ω L NR i,t v given a flow budget constraint of the form: vna P t C NR i,t = W i,t L NR i,t P t TAX NR i,t Consumption function is then: C NR i,t = W i,t L NR i,t P t TAX NR i,t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 5 / 65

Model #3: Model #+RoT Aggregate variables Aggregate consumption is now defined as C t s NR C NR t + ( s NR )C R t while investments and capital aggregates of the economy are give by I t ( K t ( s NR )I R t s NR )K R t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 5 / 65

Model #3: Model #+RoT Wage Setting New assumption: The fraction of Non-Ricardian and Ricardian households is uniformly distributed across workers types and hence across unions. Each period a typical union representing worker i sets the wage for its workers in order to maximize the objective function of the form ω s NR v ( L i,t) v + λ NR W i,t i,t L i,t + P t +( s NR ) s.t. to the demand schedule: L i,t = wage equation is σ L σ L ω ( L t) v = ω v ( L i,t) v + λ R i,t W i,t L i,t P t Wi,t W t σl Lt. At the optimum the h i s NR λ NR t + ( s NR )λ R Wt t P t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 52 / 65

Model #3: Model #+RoT The government The government budget constraint is B t = ( + r t )B t + P t G t P t TAX t where G t = G exp(ε g,t ) and ε g,t = ρ g ε g,t + ξ g,t with ξ g,t iid.n(, σ 2 g ), while TAX t s NR TAXt NR + ( s NR )TAXt R where we assume that TAXt NR The fiscal rule is as before: Set s NR =.2. = TAX R t = TAX t P t TAX t = P t TAX + τp t B t. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 53 / 65

% % Model #3: Model #+ RoT Effects of a Technology Shock.6 Consumption, c.4 Output, Y.6 Labour, L 6 Investments, I.5.2.4 5 4.4.3.8.6.2 3 2.2.4.2. 2 4 2 4.2 2 4 2 4 Inflation.8 Real Wage, W/P.5 Real Interest Rate.25 Public Debt, B/P.5.6.4.3.2.5..5.4.2...5.2.2..5.25 2 4 2 4.2 2 4. 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 54 / 65

Model #3: Model #+ RoT Effects of a Technology Shock: Consumption paths.8.6 C CNR CR.4.2.8.6.4.2 5 5 2 25 3 35 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 55 / 65

% % Model #3: Model #+ RoT Effects of a Government Spending Shock Consumption, c.8 Output, Y.2 Labour, L. Investments, I.2.4.6.6.4.2..8.6.4.2..2.3.4.5.8 2 4.2 2 4 2 4.6 2 4 6 x 3 Inflation.5 Real Wage, W/P Real 3 x Interest Rate 3 Public Debt, B/P 5..5 2.5.8 4.2 2.6 3.25.5.4 2.3.35.2 2 4.4 2 4.5 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 56 / 65

Model #3: Model #+ RoT Effects of a Government Spending Shock: Consumption paths.6.4 C CNR CR.2.2.4.6.8. 5 5 2 25 3 35 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 57 / 65

% % Model #3: Model #+ RoT Effects of a Monetary Policy Shock. Consumption, c Output, Y.5 Labour, L 2 Investments, I.2.3.4.5.6.7.5.5.5.5 2 2.5 2 4 6 8.8 2 4 2 2 4 3 2 4 2 4.5 Inflation Real Wage, W/P Real Interest Rate. Public Debt, B/P 2.5.8.5.5.6.4.5.5.2.5 2 2 4 2 2 4 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 58 / 65

Model #3: Model #+ RoT Effects of a Monetary Policy Shock: Consumption Paths.5.5 2 2.5 3 3.5 4 4.5 C CNR CR 5 5 5 2 25 3 35 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 59 / 65

% % Model #4: Habit+Adjust. Costs+RoT Effects of a Technology Shock.7 Consumption, c.4 Output, Y.6 Labour, L 6 Investments, I.6.5.2.4 5 4.4.3.8.6.2 3 2.2..4.2 2 4 2 4.2 2 4 2 4 Inflation.4 Real Wage, W/P.6 Real Interest Rate.2 Public Debt, B/P.5..2.4..5.8.2.2.6..25.3.4.2.2.2.35 2 4.4 2 4 2 4.3 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 6 / 65

% % Model #4: Habit+Adjust. Costs+RoT Effects of a Government Spending Shock.5 Consumption, c.8 Output, Y.2 Labour, L. Investments, I.6..5..4.2.8.6.4..2.5.2.3.2 2 4.2 2 4 2 4.4 2 4.3 Inflation Real Wage, W/P Real Interest Rate.2 Public Debt, B/P.25.2.5..5.5..5.2.25.3.5..5.8.6.4.2 2 4.35 2 4 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 6 / 65

% % Model #4: Habit+Adjust. Costs+RoT Effects of a Monetary Policy Shock Consumption, c.5 Output, Y.5 Labour, L 2 Investments, I.2.4.6.8.2.5.5.5.5 2 2.5 2 4 6 8.4 2 4 2 2 4 3 2 4 2 4.5 Inflation Real Wage, W/P Real Interest Rate.25 Public Debt, B/P 2.5.5.2 2.5.5 2.5..5.5 2.5.5.5 2 2 4 3 2 4 2 4 2 4. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 62 / 65

Discussion What is still missing to have a more complete model? Indexation Wage rigidities Variable capacity utilization International trade and international capital markets. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 63 / 65

Discussion Challenges of DSGE modelling Labour migration and remittances Foreign direct investments Heterogenous workers (atypical, self-employed etc...) Informal sector Role of relative price movements Non-market sector (public goods) Financial market frictions Portfolio choice Term structure of interest rates Currency risk premia Endogenous growth Time varying parameters and structural breaks Estimation problems. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 64 / 65

References Galí, J., López-Salido, J.D., Vallés, J., (27), Understanding the Effects of Government Spending on Consumption, Journal of the European Economic Association, 5(). Rotemberg, J., (983), Aggregate Consequences of Fixed Costs of Price Adjustment, American Economic Review, 73(3). Ascari, G., Merkl, C., (29), Real Wage Rigidities and the Cost of Disinflations, Journal of Money, Credit and Banking, 4(2-3). Ascari, G., Ropele, T., (27), Optimal monetary policy under low trend inflation, Journal of Monetary Economics, 54(8). Ascari, G., Rossi, L. (29), Real Wage Rigidities and Disinflation Dynamics: Calvo vs. Rotemberg Pricing, University of Pavia. Ascari, G., Rossi, L. (2), Trend Inflation, Non-linearities and Firms Price-Setting: Rotemberg vs. Calvo, University of Pavia. Leeper, E. M., (99), Equilibria under active and passive monetary and fiscal policies, Journal of Monetary Economics 27().. Annicchiarico (Università di Tor Vergata) (Institute) Microfoundations of DSGE Models 2 Giugno 2 65 / 65