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How can one show that decreasing returns + perfect competition is equivalent to monopolistic competition + constant returns?

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  • $\begingroup$ 1. It is not; though some variables do have the same equilibrium values. Please provide context/source for your claim, otherwise it is hard to tell what exactly you are asking. $\endgroup$
    – Giskard
    Feb 26, 2022 at 7:12
  • $\begingroup$ 2. Your title includes "downward sloping demand curve", but then it does not come up in the body of your question; please edit to clarify. $\endgroup$
    – Giskard
    Feb 26, 2022 at 7:13
  • $\begingroup$ Downward sloping demand curve is implied by monopolistic competition. $\endgroup$
    – Luca Gi
    Feb 27, 2022 at 14:26

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A simple case to show these two are isomorphic in terms of the firm problem.

Perfect Competition:

Each firm $i$ produces a homogeneous good with a decreasing returns to scale production technology $f(i) = (s(i) n(i))^{\theta}$, where $s$ is productivity, $n$ is the only input labor, and $\theta<1$. Each firm maximizes their static profit $\max_{n_i} pf(s_i, n_i)-wn_i$ and use $n_i^*(s_i,p,w)$ labor.

Monopolistic Competition with Constant Return:

Each firm $i$ produces a differentiated good with linear production function, $f(i)=s(i) n(i)$. Assume representative agent has preferences over different goods $U=\left(\int c(i)^{\theta} d i\right)^{1 / \theta}$. (Alternatively assume a final good produced by perfectly competitive firms with the production function of this.)

From the utility maximization problem we obtain $U^{1-\theta} c(i)^{\theta-1}=\lambda p(i)$ , where $\lambda$ is the multiplier of consumer's budget constraint (or $\lambda^{-1}$ is the aggregate price index). Each firm's revenue is thus $p(i) c(i)=U^{1-\theta} \lambda^{-1} c(i)^{\theta}= U^{1-\theta} \lambda^{-1}(s(i) n(i))^{\theta}$ where we assume the agent has measure of 1 and thus $c(i)=f(i)$. Each firm thus maximize $\max_{n_i} p(s(i) n(i))^{\theta}-wn(i)$ where $p \equiv U^{1-\theta} \lambda^{-1}$ and use $n_i^*(s_i,p,w)$ labor.

In summary, both decreasing return to scale and monopolistic competition provide some curvature in firm objective function to pin down the firm size.

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  • $\begingroup$ I would not call the two cases "isomorphic". Yes, the firms' optimization problem is identical given a specific functional forms and a parameter $\theta$ that is used in different aspects of the models but just happens to coincide, but I don't see how this is "isomorphic". $\endgroup$
    – Giskard
    Feb 26, 2022 at 10:16
  • $\begingroup$ Perfect thanks! $\endgroup$
    – Luca Gi
    Feb 27, 2022 at 14:29

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