# Practical advice for calculating optimal prices

The company I work for is interested in making the pricing for its products more quantitative. I researched the topic and I found some theory for calculating optimal pricing in Microeconomics texts and the "Pricing and Revenue Optimization" text by Phillips.

I have an engineering background and the theory makes sense to me, but I can't find any good material about how to implement these calculations in practice.

Here are some more specific questions

1. Calculating the optimal price requires a price-response function (seems very similar to a demand curve) associating price with demand. The material I've read assumes that this function is known, but I can't find any good material about how to fit a price-response function in practice. I've found some vague advice about using survey data, but it doesn't seem that practical or specific to me. Are there some more specific and established methods for fitting a price-response function? Maybe from historical transaction data?

2. Calculating the optimal price requires a cost function associating production quantity with cost. I can imagine calculating this by accounting for raw material and labor costs in the manufacturing plant, but would appreciate if there's a more formal way to do this. Maybe a complete tally of the costs I can sum up.

3. I imagine that any estimates of the price-response function and the cost function will have error. Is there an established way to quantify the error and its impact on my pricing calculations? This isn't addressed by the resources I've found. Pricing of my company's products will have a large impact on revenue and I don't want to just fly by the seat of my pants.

4. Are there established methods for simultaneously optimizing the prices of a portfolio of products given a manufacturing quantity constraint? The plants are constrained by the quantity of product they can produce. The Phillips text has some theory on constrained pricing for one product, but not a portfolio of products.

Any advice is greatly appreciated! Or even simply some insight as to whether optimal pricing theory is useful in practice, or sharing any war stories of actually implementing optimal pricing.

A price-response function differs from a demand curve. A price-response function appears in a model where prices are selected by the firm, instead of output. A demand function appears when firms are price takers and cannot influence the price of the product on the market. They will have to sell it at whatever the going rate is.

I will try to answer the questions but they will be very general answers, hopefully they are useful.

1. The price-response function can be estimated from historical data. I don't know what you mean by transaction data. But if your company is a manufacturer of stuff then transactions could mean the cost of your raw materials and intermediate goods purchases. Then it is possible to estimate a price-response function from this. But it is important to remember that the price-response you estimate is based on your own companies pricing strategy in the past. If your company has made mistakes in it's pricing before, how will estimating this price-response function help that? I think it is better to use a method that takes data from the entire industry instead and benchmark your company against that.

2. You don't need to do this in very much detail. Just setup your problem generally. Such as, Production Function = Y = F(x1, x2, ...) and cost function = C = G(x1, x2, x3, p1, p2, p3). Solve that problem, what do you end up with? Focus on estimating what is needed in that function in reduced form. I would think what you need is a good measure of your marginal costs.

3. Estimating the price-response function will certainly have error. But that depends on a lot of things: data availability, model specification, measurement error, etc. You can benchmark against other studies in your field or outside your field instead of trying to removing error completely. You can eventually compare the model's predictions against out-of-sample values.

4. Unless there is some interdependency between the production of these products within the factories, this should not be a problem I think. For instance, are these products made in the same factories? Does the production of one affect the other? If not, you can get probably get away with solving the problem for each product's price response separately.

• (-1) "A demand function appears when firms are price takers and cannot influence the price of the product on the market." This is not true, see any monopoly model or the Bertrand model. – Giskard Feb 16 at 8:53