Trying to apply in practice the theory of finding the optimal price in a cournot competition

Question

I recently received my bachelors degree in economics. For fun I wanted to try to apply some micro-economic theory of finding the optimal price in a cournot competition.

I wanted to do this for a business my friend has. He's a goldsmith and has his own established goldsmith store in a city with a population of approximate 100.000 people. There are two more goldsmiths with their own store in this city.

My plan would be to only look at one product, suppose a simple and plain gold ring. Finding the marginal cost function for this should be easy. However, I struggle with thinking about how I would find the marginal revenue function to eventually find MR=MC. For the marginal revenue function I need to know the demand curve. In order to find the demand curve I need to know the the sum of utilities of individuals (for simplicity assuming that consumers are homogeneous), right? How could I estimate this utility of an individual?

Furthermore, I would believe that this market could be described as a cournot competetion (n=3) where we could assume, for simplicity, that each goldsmith would have the same marginal costs, since they all buy gold from the same supplier and the costs of effort and production wouldn't differ that much from each other.

So mainly I struggle with finding or estimating the utility function. Since a simplistic utility function could be U(p)=X-p, where X is the willingness to pay and p the price of a golden ring. But how do I find the willingness to pay of an individual. Is it perhaps possible to estimate deduce this from my friends previous years revenues or profits?

Last question. This optimal price would only be optimal in a cournot competition when the other two goldsmiths would also use the same price? Assuming that these firms are symmetric.

If you have critique on my process of thought of how to approach this, please do and correct me if I mentioned something wrong about the micro-economic theory I had in mind for this application.

Thank you for your attention :)

Answer 1

Suppose you are willing to believe that the demand curve has a constant elasticity $\epsilon$. Then the optimal price $P^*$ is given by

$$ P^* = \frac{|\epsilon| \hspace{0.1em}c}{|\epsilon|-1}$$

where $c$ is the marginal cost of producing gold rings (assumed constant). So if your friend can estimate the elasticity $\epsilon$ (e.g. by randomly varying the price and seeing what happens to the number of rings sold), they should be able to estimate the optimal price.

To illustrate, suppose that your friend thinks that increasing the price by 1% will decrease demand by 2%. Then $\epsilon \approx -2$, i.e. $| \epsilon | \approx 2$. The formula then proposes charging $2c/(2 - 1) = 2c$; that is, a 100% markup on the constant marginal cost.

As an aside, note that this formula holds independently of the market structure. So you don't need to worry about whether the market is described, say, by the Cournot model (except insofar as this bears on your estimate of $ \epsilon$).

Answer 2

Willingness to pay, or WTP, is the most a consumer will spend on one unit of a good or service. So if you want to find it, take the most expensive gold ring that has been bought at least once out of the 3 goldsmiths.

Hope this helps.