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 $A_i > A_j$:
| Variable | Ratio |
|---|---|
| Prices | $P_i/P_j = A_j/A_i$ |
| Output | $Y_i/Y_j = (A_i/A_j)^\sigma$ |
| Employment | $L_i/L_j = (A_i/A_j)^{\sigma-1}$ |
| Revenue | $P_iY_i / P_jY_j = (A_i/A_j)^{\sigma-1}$ |
| Profits | $\Pi_i/\Pi_j = (A_i/A_j)^{\sigma-1}$ |
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.