Too often we see "efficiency" required of government, meaning government spends more money than the writer thinks appropriate. We rarely see "effective" applied, meaning that the government did something with the desired results.
This morning's Duluth News Tribune's "Our View" used "efficiency" and "effective" in the same sentence with regard to a study to be commissioned to compare a few cities' expenses and services. It can be a difficult task because of several variables, like hilly and narrow vs. flat and compact. See "Our View: Collect data but get proper perspective, too".
While thinking about the article, I came up with a pair of simile's regarding automobiles. A car sitting in the garage is efficient but not effective. It doesn't use any fuel at all, but it is not taking anyone anywhere. A four-wheel drive vehicle plowing through snow is effective but it is very inefficient. It gets people to their destination but it uses a lot more fuel than it would normally.