In this particular example, yes. But every item or service is going to see a variable effect, depending on what happens to it before it's consumed. Items with many stages of production, such as computers or cars, will realize a greater price drop, while pure services such as the nanny will not. There are estimated effects for different industries on the FairTax site.
And in every other example you can think of. You're mixing apples and ringshank nails. The Income Tax taxes income; the FairTax taxes consumption ... regardless of what industry, price, or other factors.
Anyway, what does price drop have to do with whether the dollars escape the tax authority? My response that you quoted addresses the claim that under the FairTax, the underground economy is supposed to grow the tax base thereby reducing the effective tax rate.
I will stipulate that yes, different aspects of the economy will be affected in different ways. More correctly, goods and services with high LABOR content will have the potential for the greatest price impacts since wage taxes and payroll taxes make up the lions share of tax collected under the current scheme.
But again, what does that have to do with whether there FairTax can eliminate the underground economy?