This question is obviously not a simple yes or no question and like many complicated questions, the answer is both yes and no. Living in a developed society requires money to have the basic needs of food, water, and shelter. The difference in the level of happiness between homeless, starving transients and people who have home cooked meals and a warm house to come home to is undeniable.
Nevertheless, the distinctness of the change in happiness becomes less noticeable as the two examples move up in the socioeconomic bracket. Is there a difference in happiness levels between the lower class worker who has a cozy apartment and eats spaghetti and garlic bread for dinner versus the extravagant lawyer who goes home to a three bedroom house in the paradise of suburbia, and eats chicken cordon bleu for his supper?
There is a difference in happiness, but it doesn't depend on their income. A large portion of the difference in their happiness comes from how they feel about their income. If a person doesn't think they will be happy making less money, well , you can bet your bootstraps that they won't be happy making less money. On the other hand, someone who is drawn to reading or writing and finds their joy through those means may not feel that making lots of money is necessary to their level of contentedness with their life. And there are many other factors that determine whether an individual is happy or not. Tribes don't have forms of currency, and if a person from a "developed" country gave them some form of currency, they would see no value in it. It would not make their lives better. The value of currency is determined by the people. So, as the tribe example demonstrates, it is what people think money is worth that determines its value. There is no concrete value of money, and there is no concrete measure of how much happiness money can bring.
Of course, there is the infamous Dreaded Lottery Curse. A lady who went by the name of Evelyn Adams won the lottery two years in a row. She won $5.4 million. Within 16 years, she was broke and living in a trailer.
Another lady, Victoria Zell, who won an $11 million lottery with her husband is now in a Minnesota prison, serving time on a vehicular collision charge involving substance abuse. One person was killed and another was left paralyzed for life. These people were her passengers, whom she had promised she would show them how to "drive crazy." Well, she did drive crazy. Perhaps a little too crazy.
Two years after a fellow named Billie Bob Harrell, Jr. won $31 million in a Texas Lottery, he decided that committing suicide was what he needed.
These are only a few of the examples of changes in happiness when someone comes into contact with large sums of money. I only focus on the negative aspects of winning large sums of money, because so many people are familiar with the positive aspects of it. The real aspects of change in happiness through money is based on the outlook of the individual. Some people have total control over their happiness regardless of money.