29. Sticky Price: New Keynesian Models (Calvo Assumption) and Productivity Shocks

29.1 New Keynesian approach and Calvo assumption

The New Keynesian approach, like the New Classical approach, assumes that households and firms have rational expectations. However, New Keynesian approach always consider market failures. In our discussion on nominal rigidities, New Keynesian models assume that there is imperfect competition in price and wage setting, which explains why prices and wages are sticky. So, it is typical in New Keynesian models that firms do not adjust prices instantaneously in response to changes in the economy.

Instead of the fixed price adjustment cost of the Caplin-Leahy model (menu cost model) in section 28, we can assume the following reason for price stickiness, which is the Calvo assumption:

29.2 Representative household's problem

As before, this model will have a representative household that has preferences over consumption and leisure. Household will choose how much to consume of each of the individual goods. We set up the household's problem in exactly the same way as in section 28, so we should obtain the same results as follows.

• Demand (optimal consumption level) for each good \(j\) (same as (27.8)):

$$ c_{j,t}(s^t)=C_t(s^t)P_t(s^t)^{\eta}p_{j,t}(s^t)^{-\eta} \tag{29.1} $$

where \(P_t(s^t)=\left(\int_0^1 p_{j,t}(s^t)^{1-\eta}dj\right)^{\frac{1}{1-\eta}}\) is the ideal consumer's price index.

• Intra-temporal indifference condition; if we plug in the specific utility functional form \(u(C,H)=\log C-v(H)\):

$$ \frac{-u_H(C_t(s^t),H_t(s^t))}{u_C(C_t(s^t),H_t(s^t))}=\frac{w_t(s^t)}{P_t(s^t)}\ \Rightarrow\ C_t(s^t)v'(H_t(s^t))=\frac{w_t(s^t)}{P_t(s^t)} \tag{29.2} $$

• Inter-temporal indifference condition (EE) (where \(\Pi_0^t(s^t)\) is the probability of having history \(s^t\) in the perspective of period 0):

$$ \frac{\Pi_0^t(s^t)u_C(C_t(s^t),H_t(s^t))}{P_t(s^t)Q_0^t(s^t)}=\beta\frac{\Pi_0^{t+1}(s^{t+1})u_C(C_{t+1}(s^{t+1}),H_{t+1}(s^{t+1}))}{P_{t+1}(s^{t+1})Q_0^{t+1}(s^{t+1})}\quad \forall t,\forall s^{t+1}>s^t \tag{29.3} $$

The household's problem doesn't have price stickiness in it, so we don't see effects of Calvo assumption yet.

29.3 Firm's problem

The new element of this model is the firm's problem.

29.3.1 The objective function

Each firm is allowed to adjust its price with probability \(1-\theta\) in each period. When the firm adjusts its price, it realizes that it won't be able to adjust its price again in each future period with probability \(\theta\). So the firm chooses its optimal adjusted price that maximizes discounted expected value until next adjustment (the firm will do the same optimization in its next adjustment; periods after the next adjustment are to be considered in the next adjustment, so the optimization now only considers the periods before next adjustment):

$$ \max_p\sum_{t'=t}^{\infty}\sum_{s^{t'}\ge s^t}\underbrace{\theta^{t'-t}}_{\text{prob. can't adjust}}\underbrace{\frac{Q_0^{t'}(s^{t'})}{Q_0^t(s^t)}}_{\text{discounted to }t}\left(p\underbrace{c_{j,t}(p;s^{t'})}_{\text{demand}}-w_{t'}(s^{t'})\underbrace{\frac{c_{j,t}(p;s^{t'})}{Z_{t'}(s^{t'})}}_{=h_{j,t'}(s^{t'})}\right) $$

where \(s^{t'}\ge s^t\) means \(s^{t'}\) is a continuation history of \(s^t\) that equals \(s^t\) when \(t'=t\), and \(c_{j,t}(p;s^{t'})\) is the amount of good demanded. By assigning measure 0 to each firm, we continue to assume that individual firm cannot affect the aggregate level.

29.3.2 First-order conditions

From (29.1) in the household's problem, we know that \(c_{j,t}(p;s^{t'})=C_{t'}(s^{t'})\left(\frac{P_{t'}(s^{t'})}{p}\right)^{\eta}\). Substitute and rewrite the firm's problem, and take the f.o.c. w.r.t. \(p\), to obtain the optimal reset price set by all adjusting firms:

$$ P_t^{\star}(s^t)=\frac{\eta}{\eta-1}\frac{\sum_{t'=t}^{\infty}\sum_{s^{t'}\ge s^t}\theta^{t'-t}Q_0^{t'}(s^{t'})C_{t'}(s^{t'})P_{t'}(s^{t'})^{\eta}\frac{w_{t'}(s^{t'})}{Z_{t'}(s^{t'})}}{\sum_{t'=t}^{\infty}\sum_{s^{t'}\ge s^t}\theta^{t'-t}Q_0^{t'}(s^{t'})C_{t'}(s^{t'})P_{t'}(s^{t'})^{\eta}} \tag{29.5} $$

where \(P_t^{\star}(s^t)\) is the price that the firm would charge if it could change its price — so this is the price set by all adjusting firms. But don't confuse this \(P_t^{\star}(s^t)\) with the ideal price index \(P_t(s^t)\). In general, \(P_t^{\star}(s^t)\ne P_t(s^t)\).

29.3.3 Recursive equation for ideal price index

Note that all adjusting firms (\(1-\theta\) fraction) would charge the same \(P_t^{\star}(s^t)\). Also note that since the firms that cannot change price (\(\theta\) fraction) are drawn randomly from a continuum of firms (uncountably infinite), the price index they charge over them in period \(t\) is just \(P_{t-1}(s^{t-1})\). So, we have

$$ P_t(s^t)^{1-\eta}=\theta P_{t-1}(s^{t-1})^{1-\eta}+(1-\theta)P_t^{\star}(s^t)^{1-\eta} \tag{29.6} $$

29.3.4 Labor market clearing condition

Since we already embedded the good market clearing condition in firm's problem (i.e. \(y_{j,t}(s^t)=c_{j,t}(s^t)\)), we only need to think about the labor market clearing condition. Introduce a "Production Price Index" \(\tilde P_t(s^t)\equiv\left(\int_0^1 p_{j,t}(s^t)^{-\eta}dj\right)^{-\frac{1}{\eta}}\), then the labor market clearing condition gives the production function

$$ C_t(s^t)=Z_t(s^t)H_t(s^t)\left(\frac{\tilde P_t(s^t)}{P_t(s^t)}\right)^{\eta} \tag{29.7} $$

29.3.5 Inefficiency caused by price stickiness and dispersion

Since \(\eta>1\), we have \(-\eta<1-\eta<0\). By Jensen's inequality, one can show that \(\tilde P_t(s^t)\le P_t(s^t)\). So, \(\left(\dfrac{\tilde P_t(s^t)}{P_t(s^t)}\right)^{\eta}\le1\) and \(C_t(s^t)\le Z_t(s^t)H_t(s^t)\), with inequality strict if prices are not all the same, which shows inefficiency caused by price stickiness and price dispersion (caused by inability of adjusting price).

29.3.6 Recursive equation for production price index

By exactly the same logic as in the argument for (29.6):

$$ \tilde P_t(s^t)^{-\eta}=\theta\tilde P_{t-1}(s^{t-1})^{-\eta}+(1-\theta)P_t^{\star}(s^t)^{-\eta} \tag{29.8} $$

29.4 Simplification of the system of equations

29.4.1 The system of six equations and seven unknowns

Combining the equations above with our conditions from the household's problem, we have the following system of six equations: (1) HH intra-temporal (29.9)=(29.2); (2) HH inter-temporal EE (29.10)=(29.3); (3) adjusting firm's optimal price (29.11)=(29.5); (4) production function / labor market clearing (29.12)=(29.7); (5) recursive ideal price index (29.13)=(29.6); (6) recursive production price index (29.14)=(29.8).

Seven unknowns: \(C_t(s^t),H_t(s^t),Q_0^t(s^t),w_t(s^t),P_t(s^t),P_t^{\star}(s^t),\tilde P_t(s^t)\). It's impossible to use six equations to close a model with seven unknowns, so we need an exogenously determined variable. To make the model interesting, we can let \(i_t(s^t)=\dfrac{Q_0^t(s^t)}{\sum_{s^{t+1}>s^t}Q_0^{t+1}(s^{t+1})}\) be such that \(w_t(s^t)=\bar w\) is a constant. Then, we will have six equations with six unknowns.

State variables: the ideal price index \(P_t(s^t)\) and the production price index \(\tilde P_t(s^t)\), which pin down all other variables deterministically for given deterministic productivity shocks \(Z_t(s^t)\) or in expectation for random sequence of \(Z_t(s^t)\).

29.4.2 Define new objects for simplification

• Real marginal cost \(m_t(s^t)\):

$$ m_t(s^t)=\sum_{t'=t}^{\infty}\sum_{s^{t'}\ge s^t}\theta^{t'-t}\frac{Q_0^{t'}(s^{t'})C_{t'}(s^{t'})P_{t'}(s^{t'})^{\eta+1}}{Q_0^t(s^t)C_t(s^t)P_t(s^t)^{\eta+1}}\underbrace{\frac{w_{t'}(s^{t'})}{Z_{t'}(s^{t'})P_{t'}(s^{t'})}}_{\text{real marginal cost}} $$

• \(d_t(s^t)=\sum_{t'=t}^{\infty}\sum_{s^{t'}\ge s^t}\theta^{t'-t}\dfrac{Q_0^{t'}(s^{t'})C_{t'}(s^{t'})P_{t'}(s^{t'})^{\eta}}{Q_0^t(s^t)C_t(s^t)P_t(s^t)^{\eta}}\)
• \(x_t(s^t)=\dfrac{\tilde P_t(s^t)}{P_t(s^t)}\)
• Gross inflation rate \(\pi_t(s^t)=\dfrac{P_t(s^t)}{P_{t-1}(s^{t-1})}\)

29.4.3 Properties of the new objects

Check that \(m_t(s^t)\) and \(d_t(s^t)\) satisfy

$$ P_t^{\star}(s^t)=\frac{\eta}{\eta-1}\frac{m_t(s^t)P_t(s^t)}{d_t(s^t)} \tag{29.15} $$

29.4.4 The simplification

We simplify the system of six equations in the following steps:

1. Use the Euler equation (29.10) to eliminate the inter-temporal price, getting \(\dfrac{Q_0^{t+1}(s^{t+1})}{Q_0^t(s^t)}=\beta\dfrac{\Pi_0^{t+1}(s^{t+1})P_t(s^t)C_t(s^t)}{\Pi_0^t(s^t)P_{t+1}(s^{t+1})C_{t+1}(s^{t+1})}\) (29.16).
2. Plug (29.16) into the expressions for \(m_t\) and \(d_t\), getting \(d_t(s^t)=1+\theta\sum_{s^{t+1}>s^t}\beta\dfrac{\Pi_0^{t+1}P_{t+1}^{\eta-1}}{\Pi_0^t P_t^{\eta-1}}d_{t+1}(s^{t+1})\) (29.17).
3. Recall \(\pi_t\equiv P_t/P_{t-1}\). Substitute this into the recursive equation for ideal price index (29.13) to get an explicit expression for \(d_t\) in terms of \(m_t\) and \(\pi_t\): \(d_t(s^t)=\pi_t(s^t)\dfrac{\eta}{\eta-1}m_t(s^t)\left(\dfrac{\pi_t(s^t)^{1-\eta}-\theta}{1-\theta}\right)^{\frac{1}{\eta-1}}\) (29.18).
4. Use (29.17) to eliminate the \(d_{t+1}(s^{t+1})\) in (29.18), getting (29.19).
5. Combining the previous results gives an equation relating \(m_t,\pi_t\) to \(m_{t+1},\pi_{t+1}\):

$$ \pi_t(s^t)m_t(s^t)\left(\frac{\pi_t(s^t)^{1-\eta}-\theta}{1-\theta}\right)^{\frac{1}{\eta-1}}=\frac{\eta-1}{\eta}+\theta\beta\mathbb{E}\left[\pi_{t+1}(s^{t+1})^{\eta}m_{t+1}(s^{t+1})\left(\frac{\pi_{t+1}(s^{t+1})^{1-\eta}-\theta}{1-\theta}\right)^{\frac{1}{\eta-1}}\,\Big|\,s^t\right] \tag{29.20} $$

1. Substitute \(x_t,\pi_t\) and (29.17) into the recursive equation for production price index (29.14) to get an equation relating \(x_t,\pi_t\) to \(x_{t+1},\pi_{t+1}\):

$$ \big(\pi_{t+1}(s^{t+1})x_{t+1}(s^{t+1})\big)^{-\eta}=\theta x_t(s^t)^{-\eta}+(1-\theta)\left(\frac{\pi_{t+1}(s^{t+1})^{1-\eta}-\theta}{1-\theta}\right)^{\frac{\eta}{\eta-1}} \tag{29.22} $$

1. Eliminate \(C_t(s^t)\): combining household's intra-temporal indifference condition (29.9) and the production function (29.12), get \(\dfrac{w_t(s^t)}{Z_t(s^t)P_t(s^t)}=H_t(s^t)x_t(s^t)^{\eta}v'(H_t(s^t))\) (29.23).
2. Use (29.23) to eliminate the real wage \(\dfrac{w_t(s^t)}{Z_t(s^t)P_t(s^t)}\) in (29.21):

$$ m_t(s^t)=H_t(s^t)x_t(s^t)^{\eta}v'(H_t(s^t))+\theta\beta\mathbb{E}\left[\pi_{t+1}(s^{t+1})^{\eta}m_{t+1}(s^{t+1})\,\Big|\,s^t\right] \tag{29.24} $$

29.4.5 The system of three equations and four unknowns

We finally have the following system of three equations: (29.22), (29.20), and (29.24). Four unknowns: \(H_t,m_t,x_t,\pi_t\). We need an exogenously determined variable. To make the model interesting, we can let the gross inflation rate be a constant, i.e. \(\pi_t(s^t)=\bar\pi\). Then, we will have three equations with three unknowns, which is solvable.

State variable: now we have the minimum of only one state variable \(x_t(s^t)=\dfrac{\tilde P_t(s^t)}{P_t(s^t)}\le1\) (which stands for the degree of price dispersion), with inequality strict for price dispersion. So the only state variable is non-degenerate because of price dispersion. In the case of zero inflation rate, there is no price dispersion in optimal solution, which means that we will have zero state variable in that case (i.e. the only state variable degenerates to 1). If we do log-linearization around the zero inflation steady state, we will have a deterministic sequence of \(x_t(s^t)\), i.e. \(\hat x_{t+1}=\theta\hat x_t\), so any deviation away from the steady state of no price dispersion will deterministically converge back to the steady state price homogeneity.

29.5 Three equation system: New Keynesian Phillips Curve, Dynamic IS Curve and Taylor's Rule

Now we want to rearrange the system of three equations in subsection 29.4.5 and do some log-linearizations to obtain another system of three equations for the economy, which is more famous. The three equations are called New Keynesian Phillips Curve, Dynamic IS Curve and Taylor's Rule respectively.

29.5.1 New Keynesian Phillips Curve

The first equation is obtained from the log-linearization around the steady state with \(\hat x_t=0\). Subtract (29.28) from (29.29) (using \(\hat x_t\approx0\)) to get

$$ \hat\pi_t=\kappa\hat H_t+\beta\mathbb{E}\left[\hat\pi_{t+1}\,\Big|\,s^t\right] \tag{29.30} $$

where

$$ \kappa=\frac{1-\theta}{\theta}(1-\theta\beta)\left(1+\frac{1}{\varepsilon}\right)>0,\qquad \varepsilon=\frac{v'(H)}{Hv''(H)}\ \text{is the Frisch elasticity of labor supply} $$

(29.30) is the New Keynesian Phillips Curve (this equation is not exactly the Phillips curve, but since it displays negative correlation between inflation and unemployment rate, it is appropriate to name it Phillips curve).

29.5.2 Dynamic IS Curve

This system of three equations won't involve \(m_t(s^t)\). Equivalently, the system involves two exogenous variables: productivity \(Z_t(s^t)\) and (gross) nominal interest rate \(i_t(s^t)\), which are more intuitively relevant. Starting from household's inter-temporal indifference condition (Euler equation (29.10)) and the definition of (gross) nominal interest rate \(i_t(s^t)\), combined with the production function (29.12), we get

$$ i_t(s^t)=\frac{1}{\mathbb{E}\left[\frac{Z_t(s^t)H_t(s^t)}{\pi_{t+1}(s^{t+1})Z_{t+1}(s^{t+1})H_{t+1}(s^{t+1})}\,\big|\,s^t\right]} \tag{29.31} $$

Denote \(\hat\imath_t=\ln i_t(s^t)-\ln i\) and \(\hat Z_t=\ln Z_t(s^t)-\ln Z\). Log-linearize (29.31) (around the steady state \(x_t=x_{t+1}=1\)) to get the Dynamic IS Curve:

$$ \hat H_t=\mathbb{E}\left[\hat\pi_{t+1}\,\big|\,s^t\right]-\hat\imath_t+\mathbb{E}\left[\hat Z_{t+1}\,\big|\,s^t\right]-\hat Z_t+\mathbb{E}\left[\hat H_{t+1}\,\big|\,s^t\right] \tag{29.33} $$

(29.33) is the Dynamic IS Curve (this equation is not exactly the IS (investment-saving) curve, but since it displays negative correlation between interest rate and hours worked, it is appropriate to name it IS curve).

29.5.3 Taylor's Rule

Previously, we have imposed assumptions on the path of nominal interest rate \(i_t(s^t)\). Here, we also make the path exogenous but more explicit, i.e.

$$ \hat\imath_t=\phi_{\pi}\hat\pi_t+\phi_H\hat H_t+\nu_t \tag{29.34} $$

which is called Taylor's Rule.

29.5.4 Solve the system of three equations

Now, we have reached the system of three equations as follows:

$$ \hat\pi_t=\kappa\hat H_t+\beta\mathbb{E}\left[\hat\pi_{t+1}\,\big|\,s^t\right] \tag{29.35} $$

$$ \hat H_t=\mathbb{E}\left[\hat\pi_{t+1}\,\big|\,s^t\right]-\hat\imath_t+\mathbb{E}\left[\hat Z_{t+1}\,\big|\,s^t\right]-\hat Z_t+\mathbb{E}\left[\hat H_{t+1}\,\big|\,s^t\right] \tag{29.36} $$

$$ \hat\imath_t=\phi_{\pi}\hat\pi_t+\phi_H\hat H_t+\nu_t \tag{29.37} $$

First, substitute (29.37) into (29.36) to obtain (29.38). Shut down the productivity shock \(\mathbb{E}[\hat Z_{t+1}|s^t]=\hat Z_t=0\) (this assumption is made to explore the stability of the system when the central bank uses Taylor's rule, so the system thus does not involve any productivity shock), and assume the monetary policy shocks satisfy \(\mathbb{E}[\nu_{t+1}|s^t]=\rho_{\nu}\nu_t\) with \(|\rho_{\nu}|<1\). Rewrite to get

$$ \mathbb{E}\left[\hat\pi_{t+1}\,\big|\,s^t\right]+\mathbb{E}\left[\hat H_{t+1}\,\big|\,s^t\right]=\hat H_t(1+\phi_H)+\phi_{\pi}\hat\pi_t+\nu_t \tag{29.39} $$

Rewrite (29.35) as \(\mathbb{E}[\hat\pi_{t+1}|s^t]=\frac{1}{\beta}\hat\pi_t-\frac{\kappa}{\beta}\hat H_t\) (29.40), and subtract (29.40) from (29.39) to get the expression for \(\mathbb{E}[\hat H_{t+1}|s^t]\). In matrix form, the two difference equations become

$$ \begin{pmatrix}\mathbb{E}[\hat\pi_{t+1}\mid s^t]\\ \mathbb{E}[\hat H_{t+1}\mid s^t]\end{pmatrix}=\underbrace{\begin{bmatrix}\frac{1}{\beta} & -\frac{\kappa}{\beta}\\ \phi_{\pi}-\frac{1}{\beta} & 1+\phi_H+\frac{\kappa}{\beta}\end{bmatrix}}_{\mathbf{A}}\begin{pmatrix}\hat\pi_t\\ \hat H_t\end{pmatrix}+\begin{pmatrix}0\\ \nu_t\end{pmatrix} \tag{29.41} $$

Two eigenvalues and global instability: decompose \(\mathbf{A}=\mathbf{E}\,\text{diag}(\lambda_1,\lambda_2)\,\mathbf{E}^{-1}\), where \(\lambda_1,\lambda_2\) solve \(|\lambda\mathbf{I}-\mathbf{A}|=0\), i.e.

$$ f(\lambda)=\lambda^2-\left(1+\phi_H+\frac{\kappa}{\beta}+\frac{1}{\beta}\right)\lambda+\frac{1}{\beta}(1+\phi_H+\kappa\phi_{\pi})=0 $$

• \(|\lambda_1|>1\) and \(|\lambda_2|>1\) is global instability: there is only one pair of \((\hat\pi_t,\hat H_t)\) (of measure zero) such that (29.41) is satisfied without exploding; for any other \((\hat\pi_t,\hat H_t)\), the system will explode.
• \(|\lambda_1|<1\) and \(|\lambda_2|>1\) is local stability: there is a subspace \(\mathbb{S}\) of \((\hat\pi_t,\hat H_t)\) such that the system won't explode; for any \((\hat\pi_t,\hat H_t)\in\mathbb{S}\) the system is stable, and for any \((\hat\pi_t,\hat H_t)\notin\mathbb{S}\) it will explode.
• \(|\lambda_1|<1\) and \(|\lambda_2|<1\) is global stability: the whole space is stable.

Solving for the unique non-exploding \((\hat\pi_t,\hat H_t)\): denote \(\mathbf{q}_t=(\hat\pi_t,\hat H_t)'\) and \(\boldsymbol\nu_t=(0,\nu_t)'\) (with \(\mathbb{E}[\boldsymbol\nu_{t+1}|s^t]=\rho_{\nu}\boldsymbol\nu_t\)). From \(\mathbf{q}_{t+1}=\mathbf{A}\mathbf{q}_t+\boldsymbol\nu_t\) and guessing \(\mathbf{q}_t+g\boldsymbol\nu_t=0\), we get \(g=(\mathbf{A}-\rho_{\nu}\mathbf{I})^{-1}\), so the unique non-exploding solution is

$$ \begin{pmatrix}\hat\pi_t\\ \hat H_t\end{pmatrix}=-\mathbf{E}\begin{bmatrix}\frac{1}{\lambda_1-\rho_{\nu}} & 0\\ 0 & \frac{1}{\lambda_2-\rho_{\nu}}\end{bmatrix}\mathbf{E}^{-1}\begin{pmatrix}0\\ \nu_t\end{pmatrix} \tag{29.43} $$