# Price Change Analysis with multiple factors

The following are my thought process in price change analysis:

Price Change analysis is to find the impact of the price change on sales. If a product’s selling price is 50, and the quantity sold is 100, then the sales if 5000.

If the price increase from 50 to 60, then the price change on sales of this product is: $$(60-50)x100 = 1000$$ This means that the sales have increased by 1000, due to the price increase from 50 to 60.

The term Average Selling Price(ASP) is to simply take the total sales divided by total quantity within a period. In this case: 5000/100=50

Normally, when free goods are provided to a customer, the Average Selling Price (ASP) sold to this customer will decrease. Because free goods are offered to a customer at 0 prices. For example, if there are 10 units of free goods, the ASP is: $$5000 / (100+10) = 45.45$$

Rebate is a deduction from an amount paid from a customer. It is usually calculated at a given percentage to sales. For example, if the Rebate% is 10%, then the rebate amount: 5000 x 10% = 500. And sales after rebate deduction: $$5000 – 500 = 4500$$

The calculation of price change is the difference between the ASP of the two periods and multiplies by the least of the quantities in two periods. In Period 1: $$Price = 110$$ $$Quantity = 25$$ $$Sales = 2750$$ $$Units Free Goods = 10$$ $$Rebate Amount = -110$$ $$ASP = (2750-110)/(25+10)=75.43$$ In Period 2: $$Price = 120$$ $$Quantity = 35$$ $$Sales = 4200$$ $$Units Free Goods = 5$$ $$Rebate Amount = -120$$ $$ASP = (4200-120)/(35+5)=102$$ And the price change is: $$(102-75.43) x min((25+10),(35+5)) = 930$$

Now I would like to calculate the price change effect in three different scenarios: price change without the effect of free goods and rebate, the effect of price change from free good only, and the effect of price change from rebate only. Assume that the three calculations of price change will add up to be 930, which is the total price change.

The price change without the effect of free goods and rebate is to simply calculate the two ASPs without considering free goods and rebates and multiplies the least of the two period’s quantity:$$(120-110)xmin(25,35)=250$$

The price change from free goods is to calculate price change two times: one time counting free goods quantity, and another time excludes free good quantity, then subtract the two results. The first step is to calculate the two period’s ASP with free goods. ASP with free goods in period 1: $$2750/(25+10)=78.57$$. ASP with free goods in period 2:$$4200/(35+5)=105$$. Price change with free goods: $$(105-78.57)xmin((25+10),(35+5))=925$$.

The price change effect from free goods is 925 – 250 = 675.

For price change impact from rebate, the method is to calculate the percentage of rebate to sales from the two periods, subtract them and multiplies by the least of the two period’s sales: $$(((-120)/4200)-((-110)/2750))xmin(2750,4200)=31.43$$

Finally, I add the three separate types of price changes together:250 + 675 + 31.43 = 956.43. This does not equal to 930, which is the price change calculated with both free goods and rebate combined. How do I to reconcile this difference? Have I missed anything in my thought process?