ZARASZÓW, Poland—Krystyna Trzcińska, 68 years old, has farmed a strip of land in this corner of eastern Poland for more than four decades. Retired now, she grows clover between neat rows of raspberry bushes to feed her rabbits. The rabbits she eats, the berries she sells.

The berries bring in the equivalent of about $1,300 a year. To survive, she and her husband depend on pensions provided by Poland’s government.

State-funded...