27. Monopolistic Competition

27.1 Set-up

• Continuum of goods: good \(j\in[0,1]\), each good is monopolized by one firm; price of good \(j\) in period \(t\) is \(p_{j,t}\).
• Nominal bond that pays 1 dollar in period \(t\) is sold at price \(Q_0^t\) in period 0. Household and firms face the same bond market. By the homogeneity of this model, the equilibrium condition must be that no one actually trade the bond (otherwise all agents want to do the same direction trade), but the equilibrium prices of the bond still inform agents about the discounting factor of each period.
• Representative household: lives infinitely long, the utility discounting factor is \(\beta<1\); initial wealth \(A_0\); preference is defined over the compound consumption bundle \(C_t\) and labor supply \(H_t\), i.e. utility function is \(u(C_t,H_t)\).
  • the compound consumption bundle \(C_t=\left(\int_0^1 c_{j,t}^{\frac{\eta-1}{\eta}}dj\right)^{\frac{\eta}{\eta-1}}\) where \(\eta>1\) and \(c_{j,t}\) is the consumption of good \(j\) in period \(t\). (Here \(\frac{\eta-1}{\eta}\in(0,1)\) and \(\frac{\eta}{\eta-1}\in(1,\infty)\). Think about an easy example: \((a^{0.5}+b^{0.5})^2=a+b+2(ab)^{0.5}\), so this design makes the compound bundle greater than the simple sum of its component goods but greater by a concave pack.)
  • the representative household decides the labor supply to all firms at posted nominal wage \(w_{j,t}\). By symmetry (symmetry comes from the construction of compound consumption bundle) of the model, wages are the same across firms, so we denote wage by \(w_t\).
• Firms: firm \(j\)'s output is good \(j\), which in period \(t\) is denoted by \(y_{j,t}\); firm \(j\) observes the wage \(w_t\), and decides the amount of labor \(h_{j,t}\) to hire in period \(t\); firm \(j\) faces demand curve \(\Phi_t(p_{j,t})\) (by symmetry, no \(j\) subscript in demand function) and also decides price \(p_{j,t}\); even though firms don't own capital, it still has discounted value \(V_{j,0}\) in period \(t\) because it can earn profits in this monopolistic set-up.

27.2 Market clearing conditions

27.2.1 The three conditions

$$ \text{[Good market]}\quad y_{j,t}=c_{j,t}=\Phi_t(p_{j,t})\ \text{for }\forall j,\forall t \tag{27.1} $$

$$ \text{[Labor market]}\quad H_t=\left(\int_0^1 h_{j,t}\,dj\right) \tag{27.2} $$

$$ \text{[Ownership (potential stock market)]}\quad A_0=\left(\int_0^1 V_{j,0}\,dj\right) \tag{27.3} $$

27.2.2 Walras law

By Walras law, one market clearing condition, e.g. (27.3), which is redundant, can be obtained by rewriting the other two, e.g. (27.1) and (27.2): start with the definition of \(V_{j,0}\):

$$ \begin{aligned} \int_0^1 V_{j,0}\,dj&=\int_0^1\left(\sum_{t=0}^{\infty}Q_0^t(p_{j,t}y_{j,t}-w_t h_{j,t})\right)dj\overset{\text{good market}}{=}\sum_{t=0}^{\infty}Q_0^t\int_0^1(p_{j,t}c_{j,t}-w_t h_{j,t})\,dj\\ &=\sum_{t=0}^{\infty}Q_0^t\left[\int_0^1 p_{j,t}c_{j,t}\,dj-w_t\underbrace{\int_0^1 h_{j,t}\,dj}_{=H_t\text{ by labor market}}\right]=\sum_{t=0}^{\infty}Q_0^t\left[\int_0^1 p_{j,t}c_{j,t}\,dj-w_t H_t\right]=A_0 \end{aligned} $$

where the last line is true by the assumption of binding budget constraint (27.4) of household.

27.3 Representative household's problem

27.3.1 The problem

(This model is not solving a social planner's problem, but a decentralized problem since household and firms solve their own problems separately taking some variables decided by the other side as given. The reason for using the concept of representative household is for convenience as we are assuming homogeneous households.)

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

$$ \text{s.t.}\quad A_0=\sum_{t=0}^{\infty}Q_0^t\left(\int_0^1 p_{j,t}c_{j,t}\,dj-w_t H_t\right) \tag{27.4} $$

given \(A_0,\{Q_0^t,p_{j,t},w_t\}_{t=0}^{\infty}\), where \(C_t=\left(\int_0^1 c_{j,t}^{\frac{\eta-1}{\eta}}dj\right)^{\frac{\eta}{\eta-1}}\).

27.3.2 First-order conditions

The Lagrangian is \(\mathcal{L}=\sum_{t=0}^{\infty}\beta^t u(C_t,H_t)+\lambda\left[A_0-\sum_{t=0}^{\infty}Q_0^t\left(\int_0^1 p_{j,t}c_{j,t}\,dj-w_t H_t\right)\right]\). The f.o.c. w.r.t. \(c_{j,t}\), by chain rule, can be rewritten as

$$ \beta^t u_C(C_t,H_t)\underbrace{\left(\int_0^1 c_{j,t}^{\frac{\eta-1}{\eta}}dj\right)^{\frac{1}{\eta-1}}}_{=C_t^{1/\eta}}c_{j,t}^{-\frac{1}{\eta}}=\lambda Q_0^t p_{j,t}\ \Rightarrow\ \beta^t u_C(C_t,H_t)C_t^{\frac{1}{\eta}}c_{j,t}^{-\frac{1}{\eta}}=\lambda Q_0^t p_{j,t} \tag{27.5} $$

(Note that the f.o.c. (27.5) needs to hold almost everywhere instead of actually everywhere as usual due to the technical issue that each good has measure 0.)

Using the f.o.c. (27.5) for good \(j\) to compare against good 0, we get the demand relationship

$$ c_{j,t}=c_{0,t}\left(\frac{p_{0,t}}{p_{j,t}}\right)^{\eta} \tag{27.6} $$

Define the price index for the compound consumption bundle \(P_t\equiv\left[\int_0^1 p_{j,t}^{1-\eta}dj\right]^{\frac{1}{1-\eta}}\), then rewrite \(C_t\) to get

$$ C_t=c_{0,t}p_{0,t}^{\eta}P_t^{-\eta} \tag{27.7} $$

By (27.6) and (27.7), obtain the expression for the demand function \(\Phi_t(p_{j,t})\):

$$ c_{j,t}=c_{0,t}\left(\frac{p_{0,t}}{p_{j,t}}\right)^{\eta}=\underbrace{C_t P_t^{\eta}p_{j,t}^{-\eta}}_{\equiv\Phi_t(p_{j,t})} \tag{27.8} $$

Note that since there is a continuum of firms, each firm has measure 0 impact on the overall price index \(P_t\) and compound consumption bundle \(C_t\), so firms take \(C_t\) and \(P_t\) as given. Rewrite (27.5) by plugging in \(c_{j,t}=C_t P_t^{\eta}p_{j,t}^{-\eta}\):

$$ \beta^t u_C(C_t,H_t)C_t^{\frac{1}{\eta}}\big(C_t P_t^{\eta}p_{j,t}^{-\eta}\big)^{-\frac{1}{\eta}}=\lambda Q_0^t p_{j,t}\ \Rightarrow\ \beta^t u_C(C_t,H_t)P_t^{-1}=\lambda Q_0^t\ \Rightarrow\ \beta^t u_C(C_t,H_t)=\lambda Q_0^t P_t \tag{27.9} $$

Then, the f.o.c. w.r.t. \(H_t\):

$$ -\beta^t u_H(C_t,H_t)=\lambda Q_0^t w_t \tag{27.10} $$

Intra-temporal indifference condition (use (27.9) and (27.10)):

$$ \frac{-u_H(C_t,H_t)}{u_C(C_t,H_t)}=\frac{w_t}{P_t} \tag{27.11} $$

which looks like the intra-temporal indifference condition in the model without heterogeneity in goods (i.e. only one good \(C_t\)). Inter-temporal indifference condition (EE) (use (27.9) for \(t\) and \(t+1\)):

$$ u_C(C_t,H_t)=\beta\frac{Q_0^t P_t}{Q_0^{t+1}P_{t+1}}u_C(C_{t+1},H_{t+1}) \tag{27.12} $$

27.4 Firm's problem

27.4.1 The problem

Take firm \(j\) for example (by symmetry, solutions to other firms only change in subscript \(j\)):

$$ V_{j,0}^{\star}=\max_{\{p_{j,t},y_{j,t},h_{j,t}\}}\sum_{t=0}^{\infty}Q_0^t(p_{j,t}y_{j,t}-w_t h_{j,t}) $$

$$ \text{s.t.}\quad y_{j,t}=c_{j,t}=\Phi_t(p_{j,t})\ \text{for }\forall t,\qquad y_{j,t}=Z_t h_{j,t} $$

given \(\{Q_0^t,\Phi_t(\cdot),w_t\}_{t=0}^{\infty}\), where \(Z_t\) is the productivity of labor in period \(t\) that is the same across firms.

27.4.2 Rewrite the problem

By market clearing conditions for goods \(y_{j,t}=c_{j,t}\) and the results in (27.8): \(y_{j,t}=c_{j,t}=C_t P_t^{\eta}p_{j,t}^{-\eta}\) and \(h_{j,t}=\dfrac{y_{j,t}}{Z_t}=\dfrac{C_t P_t^{\eta}p_{j,t}^{-\eta}}{Z_t}\). Plug in these two conditions and rewrite the firm's problem, taking \(\{C_t,P_t\}\) as given:

$$ V_{j,0}^{\star}=\max_{\{p_{j,t}\}}\sum_{t=0}^{\infty}C_t P_t^{\eta}Q_0^t\left(p_{j,t}^{1-\eta}-w_t\frac{p_{j,t}^{-\eta}}{Z_t}\right) \tag{27.13} $$

27.4.3 First-order condition

The f.o.c. w.r.t. \(p_{j,t}\) for problem (27.13) is

$$ (1-\eta)p_{j,t}^{-\eta}+\eta w_t\frac{p_{j,t}^{-\eta-1}}{Z_t}=0\ \Rightarrow\ (1-\eta)+\eta w_t\frac{p_{j,t}^{-1}}{Z_t}=0\ \Rightarrow\ \eta w_t\frac{p_{j,t}^{-1}}{Z_t}=\eta-1\ \Rightarrow\ p_{j,t}=\frac{\eta}{\eta-1}\frac{w_t}{Z_t} \tag{27.14} $$

which is the same across firms. Again, this feature comes from the symmetry of the problem. Thus,

$$ P_t=\left[\int_0^1 p_{j,t}^{1-\eta}dj\right]^{\frac{1}{1-\eta}}=p_{j,t}=\frac{\eta}{\eta-1}\frac{w_t}{Z_t} \tag{27.15} $$

27.5 Wedge accounting

(27.14) and (27.11) gives us

$$ p_{j,t}=\frac{\eta}{\eta-1}\frac{w_t}{Z_t}\ \Rightarrow\ \underbrace{Z_t}_{MPL_t}=\underbrace{\frac{\eta}{\eta-1}}_{\text{Wedge}}\underbrace{\frac{-u_H(C_t,H_t)}{u_C(C_t,H_t)}}_{MRS_t} \tag{27.16} $$

(using \(\frac{w_t}{P_t}=\frac{-u_H}{u_C}\) from (27.11) and \(P_t=p_{j,t}\) from (27.15)). Here the wedge doesn't matter because in the disutility of labor function that we always use, i.e. \(v(H)=\frac{\gamma\varepsilon}{1+\varepsilon}H^{\frac{1+\varepsilon}{\varepsilon}}\) (\(\varepsilon>0\)), there is a parameter \(\gamma\) that can be adjusted freely, so as long as the wedge \(\frac{\eta}{\eta-1}\) is fixed, the model still has the f.o.c. that looks like the one without this wedge.

27.6 Close the model

To close the model, we need to make an assumption on the utility functional form to pin down \(H_t\) and thus pin down everything else. Same as before, we assume the balanced growth preference and use the following particular utility function for the household: \(u(C,H)=\dfrac{(Ce^{-v(H)})^{1-\sigma}}{1-\sigma}\). In particular, assume \(\sigma=1\) and then \(u(C,H)=\ln C-v(H)\).

27.6.1 Rewrite the intra-temporal indifference condition: pin down \(H_t\) and \(C_t\)

Then, intra-temporal indifference condition (27.11) becomes

$$ \begin{aligned} \frac{-u_H(C_t,H_t)}{u_C(C_t,H_t)}=\frac{w_t}{P_t}&\Rightarrow v'(H_t)C_t=\frac{w_t}{P_t}\\ \overset{C_t=Z_t H_t}{\Rightarrow}\ v'(H_t)Z_t H_t=\frac{w_t}{P_t}&=\frac{\eta-1}{\eta}Z_t\ \Rightarrow\ v'(H_t)H_t=\frac{\eta-1}{\eta} \end{aligned} \tag{27.17} $$

where the second last line is true by (27.15). Note that as long as we assume \(v'(\cdot)>0\) and \(v''(\cdot)>0\) (as we always do), the LHS of (27.17) is increasing in \(H_t\). So, (27.17) pins down a particular value of \(H_t\) that is constant across periods, i.e. \(H_t=H\) for \(\forall t\).

Then, by (27.16), \(C_t\) is pinned down as a function only of \(Z_t\).

27.6.2 Rewrite the inter-temporal indifference condition: discussion on nominal rigidity

With the particular utility functional form, inter-temporal indifference condition (27.12) becomes

$$ \begin{aligned} u_C(C_t,H_t)&=\beta\frac{Q_0^t P_t}{Q_0^{t+1}P_{t+1}}u_C(C_{t+1},H_{t+1})\\ \Rightarrow u_C(Z_t H,H)&=\beta\frac{Q_0^t P_t}{Q_0^{t+1}P_{t+1}}u_C(Z_{t+1}H,H)\ \Rightarrow\ \frac{Q_0^{t+1}P_{t+1}}{Q_0^t P_t}=\beta\frac{u_C(Z_{t+1}H,H)}{u_C(Z_t H,H)} \end{aligned} \tag{27.18} $$

Note that the RHS of (27.18) is a function of \(Z_t\) and \(Z_{t+1}\). Suppose we have nominal rigidity \(\frac{P_{t+1}}{P_t}\) on the LHS of (27.18) cannot change. Then \(\frac{Q_0^{t+1}}{Q_0^t}\), i.e. the nominal interest rates that relates to inflation, will change accordingly to satisfy EE (27.18).

Nominal rigidity could come from several sources:

• Physical: there might exist physical costs of updating prices such as printing menus and changing price tags, etc., which is called menu cost.
• Psychological: customers may be used to the old prices, so frequent changes in prices may make the firm lose its customers.

Even though the monopolistic competition model gives firms the pricing power and allows us to discuss nominal rigidity, it is still not satisfactory enough because it assumes too much on symmetry and homogeneity. To further relax these conditions and to actually have dispersion of prices of heterogeneous goods, we need some other modeling set-up.