Monopolistic Competition in General Equilibrium

I like this model. It’s cute. Unrealistic, but cute.

Environment:

• Intermediate sector. A unit measure of firms $i \in [0,1]$. Firm $i$ produces differentiated variety $Y_i$ with linear technology: $$ Y_i = A_i L_i $$ We assume $A_i\equiv A$ for now.

• Final sector. A competitive firm aggregates varieties via CES: $$ Y = \left(\int_0^1 Y_i^{\frac{\sigma-1}{\sigma}}, di\right)^{\frac{\sigma}{\sigma-1}}, \qquad \sigma > 1. $$

• Consumer. Representative agent with preferences: $$ U = \frac{C^{1-\gamma}}{1-\gamma} - \frac{L^{1+\varphi}}{1+\varphi}. $$

Equilibrium

Prices $({P_i}_i, W, P)$ and allocations $({Y_i, \Pi_i, L_i}_i, C, Y, L)$ such that:

(i) Consumer solves: $$ \begin{align*} & \max_{C,L} \frac{C^{1-\gamma}}{1-\gamma} - \frac{L^{1+\varphi}}{1+\varphi} \cr & \text{s.t.} \quad PC = WL + \int_0^1 \Pi_i, di \end{align*} $$ (ii) Final good firm takes ${P_i}i, P$ as given and solves: $$ \begin{align*} & \max{{Y_i}_i,, Y} ; PY - \int_0^1 P_i Y_i, di \cr & \text{s.t.} \quad Y = \left(\int_0^1 Y_i^{\frac{\sigma-1}{\sigma}}, di\right)^{\frac{\sigma}{\sigma-1}} \end{align*} $$ (iii) Intermediate firm $i$ takes $W$ and a demand function $\mathcal{Y}i(P_i, Y, P)$ as given and solves: $$ \begin{align*} & \max{P_i, Y_i, L_i} ; P_i Y_i - W L_i \cr & \text{s.t.} \quad Y_i = \mathcal{Y}_i(P_i, Y, P),\quad Y_i = A L_i \end{align*} $$ (iv) Markets clear: $$ C = Y, \qquad \int_0^1 L_i, di = L $$ Price $P$ is numeraire-free; we eventually set $P=1$.

Solving the Eqm.

• Final good firm’s FOC yields a demand function for each firm $i$: $$ Y_i = Y\left(\frac{P_i}{P}\right)^{-\sigma}, \forall i. $$

• Then for each intermediate firm, they price optimally according to the above demand curve $$ P_i = \frac{\sigma}{\sigma - 1}\frac{W}{A}. $$

• Consumer’s FOC gives a labor-leisure substitution condition $$ C^\gamma L^\varphi = \frac{W}{P}. $$

• Symmetric Eqm + Zero profit in the final sector (since it has CRS production) implies: $$ P \equiv P_i \quad \forall, i $$

Equilibrium allocations

Combining $C^\gamma L^\varphi = W/P = W/P_i = \frac{\sigma-1}{\sigma}A$ with $C = Y = AL$ (market clearing + technology):

$$\boxed{C = \left(\frac{\sigma-1}{\sigma}\right)^{!\frac{1}{\gamma+\varphi}} A^{\frac{1+\varphi}{\gamma+\varphi}}, \qquad L = C/A}$$

Real profits per intermediate firm: $$\frac{\Pi_i}{P} = Y_i - \frac{W}{P}L_i = \frac{1}{\sigma}C$$

Efficiency

Consider the planner’s problem $$ \begin{align*} & \max_{C,L,{Y_i,L_i}_i}; \frac{C^{1-\gamma}}{1-\gamma} - \frac{L^{1+\varphi}}{1+\varphi} \cr & \text{s.t.}\quad C = \left(\int Y_i^{\frac{\sigma-1}{\sigma}} di\right)^{\frac{\sigma}{\sigma-1}},\cr & \qquad Y_i = AL_i, \forall i.; \int L_i, di = L \end{align*} $$ By symmetry, the solution is: $$C^* = A^{\frac{1+\varphi}{\gamma+\varphi}}, \qquad L^* = C^*/A$$

The equilibrium differs from the planner’s solution by the factor $\left(\frac{\sigma-1}{\sigma}\right)^{1/(\gamma+\varphi)} < 1$. The monopolistic equilibrium under-produces relative to the efficient allocation.

Optimal Policy

Introduce a labor income tax $\tau$ and lump-sum transfer $T$. The consumer’s budget becomes: $$PC = (1-\tau)WL + \int \Pi_i, di + T, \qquad T = \tau WL$$

Modified FOC: $C^\gamma L^\varphi = (1-\tau)\frac{W}{P} = (1-\tau)\frac{\sigma-1}{\sigma}A$.

Setting $(1-\tau)\frac{\sigma-1}{\sigma} = 1$ yields:

$$\boxed{\tau^* = -\frac{1}{\sigma-1}}$$

The optimal policy is a labor subsidy ($\tau < 0$), financed by lump-sum taxation. It exactly offsets the markup distortion. Note $\tau^*$ is independent of $\varphi$; when $\varphi = \infty$ the subsidy is irrelevant since no labor reallocation occurs.

Heterogeneous Firms ($A_i$ varies across $i$)

The markup is unchanged: $P_i = \frac{\sigma}{\sigma-1}\frac{W}{A_i}$. Cross-firm comparisons for $A_i > A_j$:

More productive firms charge lower prices, sell more, hire more, and earn higher profits.

Price index

With heterogeneous $P_i$, the zero-profit condition in the final sector pins down the ideal CES price index: $$P = \left(\int_0^1 P_i^{1-\sigma}, di\right)^{\frac{1}{1-\sigma}}$$

This equals the marginal (= average) cost of the final good firm, since its technology is CRS. The efficiency and optimal-policy results carry over: $\tau^* = -1/(\sigma-1)$ restores the first best.