In an earlier post I provided a rough snapshot of the evolving relationship between inflation and consumer demand. But here is a fresh academic paper covering the same subject:
Bachmann, Rüdiger, Tim O. Berg, and Eric R. Sims. 2015. "Inflation Expectations and Readiness to Spend: Cross-Sectional Evidence." American Economic Journal: Economic Policy, 7(1): 1-35. https://www.aeaweb.org/articles.php?doi=10.1257/pol.20130292 (h/t to @CHCEmsden for this link)
From the abstract: the authors examined "the relationship between expected inflation and spending attitudes using the microdata from the Michigan Survey of Consumers. The impact of higher inflation expectations on the reported readiness to spend on durables is generally small, outside the zero lower bound, often statistically insignificant, and inside of it typically significantly negative. In our baseline specification, a one percentage point increase in expected inflation during the recent zero lower bound period reduces households' probability of having a positive attitude towards spending by about 0.5 percentage points."
In other words, when interest rates are not close to zero, consumers expecting higher inflation do lead to a weak, statistically frequently zero, uplift in readiness to increase durable consumption (type of consumption that is more sensitive to price variation, and thus should see a significant positive increase in consumption when consumers anticipate higher inflation).
But when interest rates are at their zero 'bound', consumers expecting higher inflation in the future tend to actually cut their readiness to spend on durables. Not increase it! And this negative effect of future inflation on spending plans is "significantly negative".
Now, give it a thought: the ECB is saying they need to lift inflation to close to 2% from current near zero (stripping out energy and food). Based on the US data estimates, this should depress "households' probability of having a positive attitude towards spending" by some 1 percentage point or so. In simple terms, there appears to be absolutely no logic to the ECB concerns with deflation from consumer demand perspective.
Update: a delightful take on deflation from Colm O'Regan: http://www.bbc.com/news/blogs-magazine-monitor-31489786. And a brilliant vignette on prices, markets, consumers and ... thought for deflation too: http://www.eastonline.eu/en/opinions/hobgoblin/economics-elsewhere-three-tales-that-may-rock-the-boat by @CHCEmsden h/t above.