# Decreasing returns to scale vs downward sloping demand curve

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

• 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. Feb 26, 2022 at 7:12
• 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. Feb 26, 2022 at 7:13
• Downward sloping demand curve is implied by monopolistic competition. Feb 27, 2022 at 14:26

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.

• 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". Feb 26, 2022 at 10:16
• Perfect thanks! Feb 27, 2022 at 14:29