<p>A simple case to show these two are isomorphic in terms of the firm problem.</p>
<p><strong>Perfect Competition:</strong></p>
<p>Each firm <span class="math-container">$i$</span> produces a homogeneous good with a decreasing returns to scale production technology <span class="math-container">$f(i) =  (s(i) n(i))^{\theta}$</span>, where <span class="math-container">$s$</span> is productivity, <span class="math-container">$n$</span> is the only input labor, and <span class="math-container">$\theta&lt;1$</span>. Each firm maximizes their static profit <span class="math-container">$\max_{n_i} pf(s_i, n_i)-wn_i$</span> and use <span class="math-container">$n_i^*(s_i,p,w)$</span> labor.</p>
<p><strong>Monopolistic Competition with Constant Return:</strong></p>
<p>Each firm <span class="math-container">$i$</span> produces a differentiated good with linear production function, <span class="math-container">$f(i)=s(i) n(i)$</span>. Assume representative agent has preferences over different goods <span class="math-container">$U=\left(\int c(i)^{\theta} d i\right)^{1 / \theta}$</span>. (Alternatively assume a final good produced by perfectly competitive firms with the production function of this.)</p>
<p>From the utility maximization problem we obtain <span class="math-container">$U^{1-\theta} c(i)^{\theta-1}=\lambda p(i)$</span> , where <span class="math-container">$\lambda$</span> is the multiplier of consumer's budget constraint (or <span class="math-container">$\lambda^{-1}$</span> is the aggregate price index).
Each firm's revenue is thus <span class="math-container">$p(i) c(i)=U^{1-\theta} \lambda^{-1} c(i)^{\theta}= U^{1-\theta} \lambda^{-1}(s(i) n(i))^{\theta}$</span> where we assume the agent has measure of 1 and thus <span class="math-container">$c(i)=f(i)$</span>. Each firm thus maximize <span class="math-container">$\max_{n_i} p(s(i) n(i))^{\theta}-wn(i)$</span> where <span class="math-container">$p \equiv  U^{1-\theta} \lambda^{-1}$</span> and use <span class="math-container">$n_i^*(s_i,p,w)$</span> labor.</p>
<p><strong>In summary</strong>, both decreasing return to scale and monopolistic competition provide some curvature  in firm objective function to pin down the firm size.</p>

CC BY-SA 4.0
2022-02-26T10:16:43.130Z
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".
Giskard
CC BY-SA 4.0
2022-02-26T09:14:26.120Z
2022-02-26T09:27:30.147Z
2022-02-26T09:27:30.147Z
Alalalalaki
<p>How can one show that decreasing returns + perfect competition is equivalent to monopolistic competition + constant returns?</p>

CC BY-SA 4.0
2022-02-26T07:12:34.933Z
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.
Giskard
CC BY-SA 4.0
2022-02-26T07:13:26.303Z
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.
Giskard
CC BY-SA 4.0
2022-02-27T14:26:11.287Z
Downward sloping demand curve is implied by monopolistic competition.
Luca Gi
<microeconomics>
Decreasing returns to scale vs downward sloping demand curve
Luca Gi
# Decreasing returns to scale vs downward sloping demand curve

(Asked by: username_0 on 2022-02-26)

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

**Tags:** microeconomics
> **Comments:**
> **username_1**: 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.  
> **username_1**: 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.  
> **username_0**: Downward sloping demand curve is implied by monopolistic competition.  

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## 1 answers

### Answer by username_2 on 2022-02-26. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_1**: 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".  

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