What ought economists to measure? We want to know why people produce and trade scarce resources, and we want to know what are the most efficient ways to do those things. Some have noted that the traditional method - summing the price of all the traded goods and services produced by an economy - is a poor method, since it is measuring production instead of utility. This is an especially strong critique of Keynesian policy, which seeks to raise production even if that means lowering consumption. Philosophical types, myself included, also note that consumption seems to have very little to do with happiness (that is, utility) in real lives.
It should not be lost that economists study economy. Non-scarce goods, such as friendship or righteousness, may be vital to happiness but fall outside the realm of economy. What economy does contribute to people's lives is, essentially, trade. I have something that is of more value to you than it is to me, and if we can trade, we can increase world happiness. In order to measure the contribution of economy to happiness, we would like to measure the gross surplus in all exchanges.
Obviously, that is impossible. We can often measure price, which must fall between the two valuations in an exchange, but gives us no information on the size of the surplus. The convenient thing about measuring surplus, like consumption or utility, is that it can only accrue to individuals. Firms or governments don't feel and cannot enjoy surplus (although they can help create or destroy it). So, focusing on individuals, we should look at all the trades that they make. The most obvious are the sale of labor and the purchase of goods and services. There are others: taxes and government services, for instance. In addition, the screwy issue of externalities arises when an exchange reduces a third party's happiness.
Nonetheless, if we are to measure surplus as the contribution of economy to happiness - specifically in its fluctuations or in the differences between countries - we do well to focus on the biggest components. Those, in turn, are well-measured by the traditional measures of consumption and income. In order to fully justify classical economic analysis, we need the additional assumptions that the surplus from each type of exchange is roughly constant across time and location and that surpluses across different types of exchanges are roughly proportional to their total value.
I would be interested to know if these assumptions have been tested anywhere; lacking evidence for or against them, I will proceed using classical economic analysis - but remembering that this approximates not total happiness but the contribution of economy to happiness.