28. Sticky Price: Menu Cost Model and Productivity Shocks

In this section, we will be considering the same model as in section 27. The only difference is that we now add in the price adjustment cost term. The productivity in each period will create a wedge between the optimal price and the previous period price. If the wedge is big enough to cover the flat adjustment cost, then the monopolistic firms would adjust price to the optimal level. If the wedge is not large enough, the firms won't adjust and thus prices are sticky. We will also have productivity shocks embedded in this model, which are generated randomly following some distribution.

28.1 Rewrite the problem

28.1.1 Unchanged household's problem

The household's problem has not changed from section 27:

$$ \max_{\{c_{j,t}(s^t),H_t(s^t)\}}\sum_{t=0}^{\infty}\sum_{s^t}\beta^t\Pi_t(s^t)u(C_t(s^t),H_t(s^t)) $$

$$ \text{s.t.}\quad C_t(s^t)=\left(\int_0^1 c_{j,t}(s^t)^{\frac{\eta-1}{\eta}}dj\right)^{\frac{\eta}{\eta-1}},\quad A_0=\sum_{t=0}^{\infty}\sum_{s^t}Q_0^t(s^t)\left(\int_0^1 p_{j,t}(s^t)c_{j,t}(s^t)dj-w_t(s^t)H_t(s^t)\right) $$

where \(Q_0^t(s^t)\) is the period 0 price of a state-contingent claim to a dollar (a nominal bond) in period \(t\) if history \(s^t\) is realized, and \(\Pi_t(s^t)\) is the unconditional probability of having history \(s^t\) in period \(t\).

28.1.2 Firm's new problem

Firm's problem now involves a new term for price adjustment:

$$ V_{j,0}(p_{j,-1})=\max_{\{p_{j,t},h_{j,t}\}}\sum_{t=0}^{\infty}\sum_{s^t}Q_0^t(s^t)\Big(p_{j,t}(s^t)c_{j,t}(s^t)-w_t(s^t)h_{j,t}(s^t)-\underbrace{w_t(s^t)\bar H\,\mathbf{1}\{p_{j,t}(s^t)\ne p_{j,t-1}(s^{t-1})\}}_{\text{menu cost}}\Big) $$

where \(V_{j,0}(p_{j,-1})\) is the discounted value of the firm. We need to keep track of \(p_{j,-1}\) because last period's price enters the adjustment cost term this period.

28.2 Market clearing conditions

We still have the three market clearing conditions as in section 27, but they are slightly different.

Good market:

$$ c_{j,t}(s^t)=Z_t(s^t)h_{j,t}(s^t)=\underbrace{\Phi(p_{j,t}(s^t);s^t)}_{\text{Demand Curve}}\ \text{for }\forall j,\forall t \tag{28.1} $$

Labor market:

$$ H_t(s^t)=\int_0^1\Big[\underbrace{h_{j,t}(s^t)}_{\text{Production}}+\underbrace{\bar H\,\mathbf{1}\{p_{j,t}(s^t)\ne p_{j,t-1}(s^{t-1})\}}_{\text{Adjustment Cost}}\Big]dj \tag{28.2} $$

Ownership (potential stock market):

$$ A_0=\left(\int_0^1 V_{j,0}\,dj\right) \tag{28.3} $$

28.3 Flexible price benchmark case: \(\bar H=0\)

28.3.1 Results from household's problem

Our flexible price benchmark case is when we assume that \(\bar H=0\). Then, we can get exactly the same equation as (27.8) in section 27, i.e.

$$ c_{j,t}(s^t)=C_t(s^t)P_t(s^t)^{\eta}p_{j,t}(s^t)^{-\eta} \tag{28.4} $$

where \(P_t(s^t)=\left(\int_0^1 p_{j,t}(s^t)^{1-\eta}dj\right)^{\frac{1}{1-\eta}}\). Similarly, we can get the intra-temporal indifference condition in the same way as (27.11):

$$ \frac{-u_H(C_t(s^t),H_t(s^t))}{u_C(C_t(s^t),H_t(s^t))}=\frac{w_t(s^t)}{P_t(s^t)} \tag{28.5} $$

Similarly, we can get the inter-temporal indifference condition in the same way as (27.12):

$$ \frac{\Pi_t(s^t)u_C(C_t(s^t),H_t(s^t))}{P_t(s^t)Q_0^t(s^t)}=\beta\frac{\Pi_{t+1}(s^{t+1})u_C(C_{t+1}(s^{t+1}),H_{t+1}(s^{t+1}))}{P_{t+1}(s^{t+1})Q_0^{t+1}(s^{t+1})}\quad \forall t,\forall s^{t+1}>s^t \tag{28.6} $$

(28.6) holds for each history, and it thus holds for the probability weighted sum across history, which is the expectation (if we express it as an expectation we would have our typical Euler equation). The household's problem doesn't have price stickiness in it.

28.3.2 Results from firm's problem

If \(\bar H=0\), the firm's problem implies the same equation as (27.15) in section 27, which is

$$ p_{j,t}(s^t)=\frac{\eta}{\eta-1}\frac{w_t(s^t)}{Z_t(s^t)}=P_t(s^t) \tag{28.7} $$

Suppose that we have the balanced growth preferences, i.e. \(u(C,H)=\ln C-v(H)\). Then, intra-temporal indifference condition (28.5) becomes

$$ \begin{aligned} \frac{-u_H}{u_C}=\frac{w_t(s^t)}{P_t(s^t)}&\Rightarrow v'(H_t(s^t))C_t(s^t)=\frac{w_t(s^t)}{P_t(s^t)}\\ \overset{C_t=Z_t H_t}{\Rightarrow}v'(H_t(s^t))Z_t(s^t)H_t(s^t)=\frac{w_t(s^t)}{P_t(s^t)}&=\frac{\eta-1}{\eta}Z_t(s^t)\ \Rightarrow\ v'(H_t(s^t))H_t(s^t)=\frac{\eta-1}{\eta} \end{aligned} \tag{28.8-28.9} $$

where the second last line is true by (28.7). Note that as long as we assume \(v'(\cdot)>0\) and \(v''(\cdot)>0\) (as we always do), the LHS of (28.9) is increasing in \(H_t(s^t)\). So, (28.9) pins down a particular value of \(H_t(s^t)\) that is constant across periods and histories, i.e. \(H_t(s^t)=H\) for \(\forall t,\forall s^t\).

28.3.3 Nominal interest rate

Go back to the Euler Equation, i.e. (28.6).

Alternatively, we can find the gross-nominal interest rate from the Euler equation (28.6), which is equivalent to (28.10):

$$ i_t(s^t)=\left(\sum_{s^{t+1}>s^t}\beta\frac{\Pi_{t+1}(s^{t+1})u_C(C_{t+1}(s^{t+1}),H_{t+1}(s^{t+1}))P_t(s^t)}{\Pi_t(s^t)u_C(C_t(s^t),H_t(s^t))P_{t+1}(s^{t+1})}\right)^{-1} $$

28.4 Sticky price case: \(\bar H>0\)

28.4.1 Some assumptions

Going forward, we make the following assumptions:

1. \(u(C,H)=\log C-\gamma H\).
2. \(\log Z_{t+1}(s^{t+1})=\Delta s_{t+1}+\log Z_t(s^t)\) with \(s^{t+1}\in\{-1,0,1\}\) i.i.d. (that i.i.d. is saying that productivity is a random walk; we will also exploit the discrete values of \(s^{t+1}\) to ensure that the problem is well-behaved).
3. \(\{i_t(s^t)\}_{t=0}^{\infty}\) are set such that \(w_t(s^t)=w\) (constant wage).
4. \(\bar H>0\) (sticky prices). This leads you to not want to change your price unless your price is very far off from the optimum. Once the firm changes its price, we make no restrictions on what the price will adjust to, so it would always adjust to the optimal value.

28.4.2 Conjectured equilibrium: discrete uniform distribution of log mark-ups

More formally, we assume there are two thresholds characterized by the values \(y,n\). These thresholds govern whether or not a firm adjusts its price, in particular:

• When the log mark-up is between the two thresholds, i.e.

$$ y-n\Delta<\underbrace{\log p_{j,t}(s^t)+\log Z_t(s^t)-\log w}_{\text{log mark-up}}

the firm won't adjust the price, i.e. \(p_{j,t}(s^t)=p_{j,t-1}(s^{t-1})\).

• Otherwise, firm would adjust the price to \(p_{j,t}(s^t)=e^y\dfrac{w}{Z_t(s^t)}\).
• \(n\) is an integer that denotes how many \(\Delta\) steps a firm can go — i.e. how many productivity shocks in a row a firm can receive — before the firm adjusts price.

The second piece of this assumption is that at any point in time, the distribution of firms is a discrete uniform in log markup space. (Notice that the statement "any point in time" includes the period $-1$ and so a restriction on the initial condition.) The lowest log markup is denoted by \(x_t(s^t)\in\{y-(n-1)\Delta,\ldots,y\}\). There are \(n\) firms in total, and their log markups (i.e. \(\ln e^{y+k\Delta}=y+k\Delta\), where \(k\in\{-(n-1),\ldots,(n-1)\}\)) are distributed uniformly on \(\{x_t(s^t),x_t(s^t)+\Delta,\ldots,x_t(s^t)+(n-1)\Delta\}\).

28.5 Solve for the model in 28.4

Such design of the distribution of the log markups guarantees that the firms always have their log markups follow a discrete uniform distribution, which is true because the shock in productivity is systematic to all firms, not idiosyncratic, so all the firms will move together one step upwards or one step downwards until the one touches the edge and returns to the center (the length the interval between two thresholds is \(2n-1\) steps and the interval of firms takes up \(n-1\) steps, so it visually looks like a elevator moving up and down with no change in its shape). Then, such unchanging uniform distribution of markups is consistent with a particular distribution of prices. We will solve the household's problem for this distribution which will allow us to characterize the price index. Because of the cost on price adjustment, the model will deliver a region of the parameter space where firm's don't change their price in response to productivity shocks, so there is some inefficiency involved with sticky prices.

28.5.1 Revisit the household's problem

• Intra-temporal condition: in this sticky price case, the intra-temporal condition does not change from (28.8) in the flexible price case. Plug in \(v(H)=\gamma H\) and \(w_t(s^t)=w\) to (28.8):

$$ v'(H_t(s^t))C_t(s^t)=\frac{w}{P_t(s^t)}\ \Rightarrow\ \gamma C_t(s^t)=\frac{w}{P_t(s^t)} \tag{28.11} $$

(28.11) implies that when no firm changes price, \(P_t(s^t)\) doesn't change, so \(C_t(s^t)\) doesn't change, i.e. consumption doesn't change for productivity shock as long as no firm adjusts price.

• Inter-temporal condition (EE): in this sticky price case, the inter-temporal condition does not change from (28.6) in the flexible price case. Plug in \(u(C,H)=\log C-\gamma H\) to (28.6), and use \(C_t(s^t)P_t(s^t)=w/\gamma\) (constant by (28.11)):

$$ \frac{\Pi_t(s^t)}{P_t(s^t)Q_0^t(s^t)C_t(s^t)}=\beta\frac{\Pi_{t+1}(s^{t+1})}{P_{t+1}(s^{t+1})Q_0^{t+1}(s^{t+1})C_{t+1}(s^{t+1})}\ \Rightarrow\ Q_0^{t+1}(s^{t+1})=\beta\frac{\Pi_{t+1}(s^{t+1})}{\Pi_t(s^t)}Q_0^t(s^t) \tag{28.12} $$

(28.12) pins down the path of nominal prices, i.e. \(\{Q_0^t(s^t)\}_{t=0}^{\infty}\) for \(\forall s^t\). By (28.12), we can characterize the gross nominal interest rate:

$$ i_t(s^t)=\frac{Q_0^t(s^t)}{\sum_{s^{t+1}>s^t}Q_0^{t+1}(s^{t+1})}=\frac{Q_0^t(s^t)}{\sum_{s^{t+1}>s^t}\beta\frac{\Pi_{t+1}(s^{t+1})}{\Pi_t(s^t)}Q_0^t(s^t)}=\frac{1}{\beta\sum_{s^{t+1}>s^t}\frac{\Pi_{t+1}(s^{t+1})}{\Pi_t(s^t)}}=\frac{1}{\beta} $$

So, the gross nominal interest rate in this equilibrium is \(\frac{1}{\beta}\).

• Ideal price index:

$$ \begin{aligned} P_t(s^t)&=\left(\int_0^1 p_{j,t}(s^t)^{1-\eta}dj\right)^{\frac{1}{1-\eta}}=\left[\frac{1}{n}\sum_{i=0}^{n-1}\left(e^{x_t(s^t)+i\Delta}\frac{w}{Z_t(s^t)}\right)^{1-\eta}\right]^{\frac{1}{1-\eta}}\\ &=e^{x_t(s^t)}\frac{w}{Z_t(s^t)}\left[\frac{1}{n}\sum_{i=0}^{n-1}e^{i\Delta(1-\eta)}\right]^{\frac{1}{1-\eta}}=e^{x_t(s^t)-\log Z_t(s^t)}w\left[\frac{1}{n}\sum_{i=0}^{n-1}e^{i\Delta(1-\eta)}\right]^{\frac{1}{1-\eta}} \end{aligned} \tag{28.13} $$

We can observe the following results from (28.13): the only part of \(P_t(s^t)\) that changes along with time is \(x_t(s^t)-\log Z_t(s^t)\).

• Production labor: in this sticky price case, (28.4) in the flexible price case still holds, i.e. \(c_{j,t}(s^t)=C_t(s^t)P_t(s^t)^{\eta}p_{j,t}(s^t)^{-\eta}\) (28.14). Use (28.14) and write the production labor by

$$ \int_0^1 h_{j,t}(s^t)\,dj=\frac{1}{Z_t(s^t)}\int_0^1 c_{j,t}(s^t)\,dj=\frac{C_t(s^t)}{Z_t(s^t)}\left[\frac{1}{n}\sum_{i=0}^{n-1}e^{i\Delta(1-\eta)}\right]^{\frac{\eta}{1-\eta}}\left[\frac{1}{n}\sum_{i=0}^{n-1}e^{-i\Delta\eta}\right] \tag{28.15} $$

Define \(f(x)=x^{\frac{\eta}{\eta-1}}\). Since \(\eta>1\), \(f\) is convex. By Jensen's inequality,

$$ \frac{1}{n}\sum_{i=0}^{n-1}f\big(e^{i\Delta(1-\eta)}\big)\ge f\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{i\Delta(1-\eta)}\right)\ \Rightarrow\ \left[\frac{1}{n}\sum_{i=0}^{n-1}e^{-i\Delta\eta}\right]\left[\frac{1}{n}\sum_{i=0}^{n-1}e^{i\Delta(1-\eta)}\right]^{\frac{\eta}{1-\eta}}\ge1 $$

with inequality strict when \(n>1\). So,

$$ \int_0^1 h_{j,t}(s^t)\,dj\ge\frac{C_t(s^t)}{Z_t(s^t)} \tag{28.16} $$

with inequality strict when \(n>1\). (28.16) shows that there is productive inefficiency due to price stickiness (i.e. \(n>1\)).

28.5.2 Revisit the firm's problem

Now, the firm's problem becomes

$$ V_{j,0}(p_{j,-1})=\max_{\{p_{j,t},h_{j,t},a_t\}}\sum_{t=0}^{\infty}\sum_{s^t}Q_0^t(s^t)\Big(p_{j,t}(s^t)c_{j,t}(s^t)-\frac{w\,c_{j,t}(s^t)}{Z_t(s^t)}-\underbrace{w\bar H a_t(s^t)}_{\text{menu cost}}\Big) \tag{28.17} $$

where \(a_t(s^t)=0\) or \(1\) is another choice variable. If \(a_t(s^t)=1\), the firm adjusts its price. If \(a_t(s^t)=0\), the firm doesn't adjust price. We want to solve this problem to check that our conjecture of the discrete uniform distribution in the log mark-up space is actually possible to maintain. By the EE (28.12) from the household problem,

$$ Q_0^{t+1}(s^{t+1})=\beta\frac{\Pi_{t+1}(s^{t+1})}{\Pi_t(s^t)}Q_0^t(s^t)\ \Rightarrow\ \frac{Q_0^t(s^t)}{\Pi_t(s^t)}=\beta^t\frac{Q_0^0(s^0)}{\Pi_0(s^0)}=\beta^t\ \Rightarrow\ Q_0^t(s^t)=\beta^t\Pi_t(s^t) \tag{28.18} $$

Plug in (28.18) and (28.14) to rewrite (28.17), then we can get the problem as

$$ V_{j,0}(p_{j,-1})=\max_{\{p_{j,t},h_{j,t},a_t\}}\sum_{t=0}^{\infty}\sum_{s^t}\beta^t\Pi_t(s^t)\left(C_t(s^t)P_t(s^t)^{\eta}\left(p_{j,t}(s^t)^{1-\eta}-\frac{w\,p_{j,t}(s^t)^{-\eta}}{Z_t(s^t)}\right)-w\bar H a_t(s^t)\right) $$

We can also write firm's problem recursively to solve for the threshold. The solution in this part is only to make sure that such a special case of equilibrium can be sustainable, i.e. firms are optimizing when they follow the conjectured \(\pm(n-1)\Delta\) threshold rule.