As I have posted on here before, a general rule of thumb is that the cost to run a household is proportional to the square root of the number of people in it. So, assume that it costs 1 person $1000 per month to live, then it will cost 2 people living together about $1414, a family of 3 about $1732 and a family of 4 about $2000. And to look at it from the reverse angle, if 2 people can live on $1000, 1 person can live on $707. Since we're talking about social security, I think we can safely ignore everything but the 1 person versus 2 person household.
Let's throw out the edge cases and special situations** and say that we have a couple living solely on social security. She gets $X per month of social security based on her work record and he gets 1/2 of her $X as a spousal benefit. When either one of them dies, the survivor will get $X per month -- that is, if he dies first, she will just continue on with her own social security of $X per month, and if she dies first, he will switch from his 1/2 of $X spousal benefit to the $X per month survivor benefit. In either case, the survivor will be left with 67% of their prior combined social security. (1/1.5). So it will be close (67% versus 71%) to what one would expect the survivor's reduced living expenses would be. So, yes, a little belt tightening may be in order.
Of course, one could be in a couple where both partners get social security on their own record, because the second partner's own social security is more than the spousal benefit. In that case, when one partner dies, the reduction in household income would be more than 33%. I believe that the worst case is where both partners each get the same social security based on their own records. In that case, the household income will drop by 50% when one of them dies (from 2 $X to $X per month). A much more difficult accommodation may be necessary.
What this points out is the importance of modeling out what happens to the survivor when making retirement plans. The reduction in social security income will range from 33% to 50% upon death of one partner.
* In my own case, the young wife is not entitled to any own social security based on her own record (because she never paid into SS) and, due to the GPO, will not receive a survivor benefit based on my social security. So, when I predecease her, as is the more likely scenario, household social security income will drop by 100%. I have taken that into account in my plan with a paid up whole life policy.