Basic New Keynesian Model
Andrés
3/5/2021
Household
The household sector is made by a representative household maximising his expected lifetime utility \(U(C_t, N_t)\) at period \(t = 0\) . Assume a utility function depending only on consumption \(C_t\) and normalised leisure \(1-N_t\). Assume that regularity conditions on the utility function hold and that it is concave in its arguments. Moreover, assume a standard constant relative risk aversion (CRRA) functional form of additively separable consumption and labor.
\[MAX_{C_t, N_t, B_t} = \mathbb{E_0} \sum_{t=0}^\infty (\frac{C_t^{1-\sigma}}{1-\sigma} -\frac{N_t^{1+\phi}}{1+\phi})\] Where \(\beta\) is the intertemporal discount factor, \(\sigma\) is the coefficient of the relative risk aversion, and \(\phi\) is the inverse of the Frisch elasticity. \(\mathbb{E_0}\) is the expectation operator conditional to the information set at time zero. Also assume that there is a continuum (in the [0, 1] interval) of different goods produced with constant elasticity of substitution (CES) technology
\[ C_t = (\int_{0}^{1} C_t(i)^\frac{\epsilon-1}{\epsilon} \,di )^\frac{\epsilon}{\epsilon-1} \]
Where \(\epsilon\) is the parameter controlling the degree of substitutability among goods. The utility is maximised subject to the household’s budget constraint and a no-Ponzi game condition in the government bonds market. The period budget constraint now takes the form:
\[ \int_{0}^{1} P_t(i)C_t(i)\,di +Q_tB_t \leq W_tN_t + B_{t-1} + D_t \]
For \(t = 0, 1, 2 ...\) , where \(P_t(i)\) is the price of good i, and where \(N_t\) denotes hours of work (or the measure of household members employed), \(W_t\) is the nominal wage, \(B_t\) represents purchases of one-period bonds (at a price \(Q_t\)), and \(D_t\) is a lump-sum component of income (which may include, among other items, dividends from ownership of firms). The above sequence of period budget constraints is supplemented with a solvency condition of the form \(limT →∞ \mathbb{E_t}(B_T) ≥ 0\)
The household now must decide how to allocate its consumption expenditures among the different goods. This requires that the consumption index \(C_t\) be maximized for any given level of expenditures \(\int_{0}^{1} P_t(i)C_t(i)\,di\) . The Lagrangian of the described maximisation problem is displayed and the associate multiplier is denoted by \(\psi_t\)
\[ min_{C_t} \quad L = \int_{0}^{1} P_t(i)C_t(i)\,di - \psi_t( [\int_{0}^{1} C_t(i)^\frac{\epsilon-1}{\epsilon} \,di ]^\frac{\epsilon}{\epsilon-1}-C_t) \]
From the first order conditions (FOC) I can recover the demand schedule and the aggregate price index described:
\[ P_t(i) \equiv \psi_t = [\int_{0}^{1} P_t(i)^{1-\epsilon}\,di]^\frac{1}{1-\epsilon} \]
Intra-temporal: Given \(P_t(i)\) and \(C_t\) , choose between varieties \(C_t(i)\), ∀i.
\[ C_t(i) =C_t (\frac{P_t(i)}{P_t})^{-\epsilon} \]
Furthermore, and conditional on such optimal behavior:
\[ \int_{0}^{1} P_t(i)C_t(i)\,di = P_tC_t \]
This is, total consumption expenditures can be written as the product of the price index times the quantity index. After some algebraic manipulations and by plugging the first stage of consumers decision (intra-temporal) into the consumer’s budget constraint, it is possible to write the second-step of the household mazimisation problem as a current value Lagrangian:
\[MAX_{C_t, N_t, B_t} \quad L = \mathbb{E_0} \sum_{t=0}^\infty \beta^t(\frac{C_t^{1-\sigma}}{1-\sigma} -\frac{N_t^{1+\phi}}{1+\phi}) -\lambda_t(P_tC_t+Q_tB_t -B_{t-1}- W_tN_t - D_t)\]
By solving the system of first-order conditions I can recover the labour supply and the Euler equation
\[ Q_t = \beta \mathbb{E_t} (\frac{C_{t+1}}{C_t})^{-\sigma} \frac{P_t}{P_{t+1}} \]
\[ C_t^\sigma N_t^{\phi} = \frac{W_t}{P_t} \]
Firms
Assume a continuum of firms indexed by i ∈ [0, 1]. Each firm produces a differentiated good, but they all use an identical technology, represented by the production function.
\[ Y_t(i) = A_tN_t(i) \]
Firms operate under monopolistic competition and produce differentiated goods by using labour Nt as their only source of input. Technology At is equal among firms. All firms face an identical isoelastic demand schedule and take the aggregate price level Pt and aggregate consumption index Ct as given.
Following the formalism proposed in Calvo (1983), each fifirm may reset its price only with probability \(1-\theta\) in any given period, independent of the time elapsed since the last adjustment. Thus, each period a measure \(1-\theta\) of producers reset their prices, while a fraction \(\theta\) keep their prices unchanged. As a result, the average duration of a price is given by \(1/(1-\theta)\). In this context, \(\theta\)becomes a natural index of price stickiness. This assumption is extremely useful because it allows to compute a proxy for the potential output of the economy by setting to zero the fraction of non-re-optimizing firms. Imposing \(\theta =0\) allows removing the only source of inefficiency in the production sector by releasing the price rigidity assumption. Next displays the aggregate price index under the Calvo price assumption:
\[ P_t = (\theta P_{t-1}^{1-\epsilon} + (1-\theta ) P_t^{*1-\epsilon}) ^\frac{1}{1-\epsilon} \]
Where \(P_t^*\) is the optimal price chosen by the optimizing firms. As \(θ → 0, P_t = P_t^∗\) implying that all the firms can reset their prices as in a flexible price economy. By dividing both sides by \(P_t\) equation can also be rewritten in terms of gross inflation.
\[ \Pi_t^{1-\epsilon}= \theta+ (1-\theta)(\frac{P_t^*}{P_{t-1}})^{1-\epsilon} \]
Notice that, as shown below, all firms will choose the same price because they face an identical problem. The problem of a firm setting its price in period t is:
\[MAX_{C_t, N_t, B_t} \quad \mathbb{E_t} \sum_{t=0}^\infty \theta^t Q_{t,t+k}[P_t^*Y_{t+k|t}- (W_{t+k}/A_{t+k})Y_{t+k|t}]\]
\[ s.t. \quad Y_{t+k|t}= (\frac{P_t^*}{P_{t+k}})Y_{t+k} \quad for \quad all \quad k=0,1, ... \]
Where \(X_{t+k|t}\) denotes a variable in period \(t+k\) for a firm that last reset its price in period \(t\). Where \(Q_{t+k|t}=\beta \mathbb{E_t} (\frac{C_{t+1}}{C_t})^{-\sigma} \frac{P_t}{P_{t+1}}\) is the stochastic discount factor. At optimum:
\[ \sum_{k=0}^\infty \theta^k \mathbb{E_t}[Q_{t+k|t} Y_{t+k|t}(P^*_t-\mathcal{M} \frac{W_{t+k}}{A_{t+k}})]=0 \]
Where \(\frac{W_{t+k}}{A_{t+k}}\) is the marginal cost and \(\mathcal{M}=\frac{\epsilon}{\epsilon-1}\) is the firms’s markup. Note that in the limiting case of no price rigidities \(\theta=0\) the previous condition collapses to the familiar optimal price-setting condition under flexible prices, which allows us to interpret \(\mathcal{M}=\frac{\ epsilon}{\epsilon-1}\)as the desired markup in the absence of constraints on the frequency of price adjustment. Henceforth, \(\mathcal{M}\) is referred to as the desired or frictionless markup.
Key idea: Firms set prices as a weighted average of current and future marginal cost
The equilibrium
Aggregate Price Dynamics
Denoting with \(Θ_t ∈ [0, 1]\) the set of fifirms NOT adjusting prices in period t we have
Market clearing conditions
The Competitive Equilibrium
Definition:
Given initial conditions, \(\{B_{-1}=0, S_{-1} \}\) and the histories of the exogenous process for \(\{ A_t\}_{t=0}^\infty\) a competitive equilibrioum is given by an allocation \(\{C_t, N_t, B_t \}_{t=0} ^\infty\), prices \(\{ \frac{P^*_t}{P_{t-1}}, S_t, \frac{W_t}{P_t}, Q_t, \Pi_t\}\)and policies such that for every moment in time and any history:
- Given prices and policies,\(\{C_t, N_t, B_t \}\) solves the households’ problem.
- Given prices and policies, \(\{\frac{P^*_t}{P_{t-1}}, N_t\}\) solves the firms’ problem.
- Markets clear
| Equation | Equation Description |
|---|---|
| \(Q_t = \beta \mathbb{E_t} (\frac{C_{t+1}}{C_t})^{-\sigma} \frac{P_t}{P_{t+1}}\) | Euler Equation |
| \(C_t^\sigma N_t^{\phi} = \frac{W_t}{P_t}\) | Labor Supply |
| \(P^*_t = \mathcal{M}[ \mathbb{E_t} \sum_{k=0}^\infty \frac{\theta^k [Q_{t |t+k} Y_{t|t+k}}{\mathbb{E_t}\sum_{k=0}^\infty \theta^k [Q_{t+k|t} Y_{t+k|t}}\frac{W_{t+k}}{A_{t+k}}]\) | Firms’ Pricing |
| \(\theta \Pi_t^{\epsilon-1}=1-(1-\theta) (\frac{P^*_t}{P_t})\) | Aggregate Pricing |
| \(C_t= Y_t = \frac{A_t}{S_t}N_t\) | Feasibility |
| \(B_t=0\) | Equilibrium in Bonds |
| (…) | Monetary Policy Rule |
Solving the model
Analyzing the model is quite complicated
- One can obtain a numerical solution (e.g. with DYNARE)
- Or study a (log) linear approximation around a steady state
Two steps procedure
1. Solve for a steady state with zero inflation
2. Solve for the approximated (log-linear) dynamics around that steady state
The steady state with zero inflation
| Equation | Equation Description |
|---|---|
| \(Q_t = \beta \Pi^{-1}\) | Euler Equation |
| \(C^\sigma N^{\phi} = \frac{W}{P}\) | Labor Supply |
| \(\frac{W}{P} = \frac{1}{\mathcal{M}}\) | Firms’ Pricing |
| \(1- \theta =(1-\theta) (\frac{P^*}{t})^{\epsilon-1} \rightarrow \frac{P^*}{P}=1\) | Aggregate Pricing |
| \(C= Y = AN\) | Feasibility |
Log-linearization
We solve the model by calculating a log-linear approximation around the model’s non-stochastic zero inflation steady state. To do so, the method uses a first order Taylor expansion, thus it reduces to compute a bunch of first order derivatives,
A log-linear approximation to the function \(Y_t\) around \(Y\) is given by:
\[ Y_t = e^{log(Y_t)} \approx Y + Y[log(Y_t) - log(Y)] \] Another example assuming \(Y_t = F(X_t, Z_t) = F(e^{logX_t}, e^{logZ_t})\)
\[ Y_t \approx F(X,Z)+ F_x(X,Z)X[logX_t-logX] + F_z(X,Z)Z[logZ_t - logZ] \] Nevertheless, it is sensible to ask ourselves, what exactly are we doing. Let’s start remarking that the reason why we use the logarithm is to interpret parameters as elasticity (percentage change). However, ysing the log and the Taylor expansion leads to two different approximations; the first coming from the fact that log-differences are approximations of the percentage change (negligible when the percentage change is small), while the second coming from the fact that we are using a linear function to approximate a non-linear one (at least using a first order Taylor expansion). This is negligible only in a small interval around the steady-state.
The equation below shows a first order Taylor expansion with n variables. Notice that the function is approximated in a particular point x0 which is a vector in R . Thus the function will be approximated to its value computed at that particular point plus the slope in the different dimensions represented by the first order partial derivatives computed at that particular point. In a two dimesion simple case this means that I am approximating a non-linear function with a line, while in a higher dimensional space I am approximating it with some planes or hyperplanes.
\[ f(x_1, x_2,..., x_n) \approx f(x_{1,0}, x_{2,0}, ... , x_{n,0}) + \frac{\partial f(x_1, x_2,..., x_n)}{\partial x_1}|_{x_1 = x_{1,0}}(x-x_{1,0}) + ... \\ ...+ \frac{\partial f(x_1, x_2,..., x_n)}{\partial x_n}|_{x_n = x_{n,0}}(x-x_{n,0}) \] Now, I would like to express my function as a log-deviation from the steady state. Recalling the properties of the log, for our proof is useful to highlight that the log-deviation is an approximation of the percentage change
\[ \hat{x}= log(x) -log(x_0) = log(\frac{x}{x_0}) \approx \frac{x-x_0}{x_0} \] The basic idea behind the prof is to rewrite the Taylor expansion in percentage change and use the log-deviation to approximate it. For instance, let’s consider the Euler Equation
\[ Q_t = \beta \mathbb{E_t} [\frac{C_{t+1}^{-\sigma}}{C_t^{-\sigma}} \frac{P_t}{P_{t+1}}] = \beta \mathbb{E_t} G_t \]
Log-linearization yields:
\[ Q + Q(log(Q_t)- log(Q)) = \beta \mathbb{E_t} \{G +G (log(G_t)-log(G) \}\underbrace{(Q-\beta G)}_\text{0} - Q \underbrace{( log(Q) - log(G))}_\text{log(B)} + Q(log(Q_t) - \mathbb{E_t} log(G_t) = 0 \]
