If a person is earning minimum wage or in some cases higher, but the cost of living is steady rising they will develop a deficit in funds. When one cannot support oneself and/or their family, worry and other anxiety sets in. All of these thing cause stress that can a drive a person into depression. Seeing ones family go without their needs while still working a back breaking job could drive anyone into a depressive state.
I don't believe that a high cost of living causes depression. I believe that a low wages can cause depression. There are people that can actually afford a high cost of living because they make a lot of money. Primarily if people don't make a lot of money in a high cost living area, they will be more under pressure for monthly expenses and such.