A baby's skin is so sensitive that using the wrong products on their tiny bodies can cause them unnecessary discomfort and possible harm. As such, limited financial resources is never a good excuse for using whatever is within reach to care for a young baby.
When it comes to bathing your infant, care should be taken to source a mild soap that won't irritate their skin. Soaps formulated for babies are non-toxic and don't usually contain the dyes and alcohol that soaps for adults do.
If your baby has sensitive skin and eczema, then you should use emulsifying ointment, emulsifying soap or something hypoallergenic instead of regular soap.
As it relates to caring for the baby's hair, you should use mild shampoos.
For cradle cap, the mother can use mineral oil or olive oil in the baby's hair overnight and then wash it out the next day with baby shampoo.
Lotions to be applied to the baby's skin should be mild as well, and free of some of the harsher chemicals used in those produced for general skin care.
By age one, the child would have acquired a level of immunity to cope better with products made for the entire family.