# Recovering information on Prices from Expenditure data

this is more of a "Research Based" Question where the answer may be open ended and really am wondering from all the experts here. Feel free to hammer me on this as I'm not 100% confident, my data work looks nice so far.

I'm interested in demand system estimation, however often times we only have expenditure data to work with. This is fine for estimating the linear expenditure system, however for for estimating much more interesting and useful demand systems like the Almost Ideal Demand System and Rotterdam demand system we require information on prices.

In short my question is:how do we recover information on prices from expenditure data in a time series when price and quantity aren't separated from eachother (i.e we only have knowledge of spending not the separate quantity and price information)

The following method I'm proposing is a follows:

Step 1: Note that $$m_i=p_ix_i$$ that is the expenditure on good $$i$$ is equal to price of good $$i$$ times its quantity . most expenditure data is indexed by time so a more accurate representation of this relationship is: $$m_{it}=p_{it}x_{it}$$

Step 2:In a competitive market where prices are stable the increase in expenditure on good $$i$$ comes from an increase demand for the product.

Step 3: obtain the fitted values for the regression model $$\hat{m}_{it}=\alpha_0+\alpha_1t$$ which is a simple regression which tells us how expenditure should evolve over time ignoring information on prices because we are assuming things are stable for the time being. If prices are constant througout time this is the same as $$\hat{x}_{it}=\alpha+\alpha_1t$$ since a transformation of $$x_{it}$$ by a scaler wont affect our estimates of what $$\alpha_1$$ is:

Step 4: Obtain price data by the use of the following relationship $$\hat{p}_{it}=\frac{m_{it}}{\hat{m}_{it}}$$ or more specifically from our relationship in step 3: $$\hat{p}_{it}=\frac{m_{it}}{\hat{x}_{it}}$$

Using relationship we have a series of data relating to prices.

I'm not sure if this is silly or something that useful for someone who has a hard time finding data. Does this make sense?