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I have one year sales data of a retail company and lets say I am forecasting the next month sales for the product. I have got the sales using time series in R. Now I want to forecast the price as well. I want to build a model using both the demand and price. My function should be f(DP) where D is demand and P is price. I want to maximize Revenue or Profit where Revenue=DP.

As of now, I forecasted sales and did a linear programming on Price itself to get optimal price of a single product. I want my model to be robust but this is chicken - egg theory where demand forecast is dependent on the price and price is dependent on demand. Please help by sharing some theories which would better fit for this or some R models.

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First, You may model the sale or demand as a function of price. You can use a regression model to control for controls. This regression will provide you with price elasticity of demand which show how your demand is sensitive to a change in price. Then, you may use the above relationship to plug in for sale in the optimization step. This way you only solve for the optimum price.

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For a retailer in a competitive market, it may be better to first calculate the unconstrained demand of that product in the market and then see whether it is above the supply constraints of the retailer. If not, then see at what price can you meet that demand and it may be that only some demand will be met if the price is high. If yes, then you can maximise the price at which demand would equal the supply constraint.

On a personal note, being in the retail industry myself, forecasting a product category may be more useful than a single product as there are many factors involved and there could be high forecast inaccuracies in the latter.

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