What is the relation between the jeweller and the Duchess?
The relationship is fundamentally one of mutual need. The Duchess needs some ready cash to pay off the enormous gambling debts she's accrued. For his part, Oliver Bacon needs to gain an entree into the upper echelons of high society. Though fabulously wealthy, Bacon is still looked down on by the upper-classes as an upstart and a parvenu because of his humble origins. Each one has something the other so desperately lacks, money in one case, high social status in the other.
On the face of it, it would appear that the Duchess gets the better deal out of the relationship. She sells what she knows to be worthless gems to Bacon in return for a weekend invitation. Yet Bacon knows full well that the jewels are fake. He also knows that the Duchess's reckless gambling habits are likely to continue. This will make her even more in Bacon's debt, changing the whole nature of their relationship. Bacon already has his eye on the Duchess's daughter, and who's to say he won't some day win her hand in marriage, with the reluctant blessing of her morally and financially compromised mother?
The relationship between these two characters is somewhat complex. Oliver Bacon is in awe of the Duchess. So in that way, he is subordinate to her. The Duchess looks down on Oliver and people like him yet, at the same time, she needs him badly. So each is superior in a way to the other.
Oliver comes from a poor, lower-class background. Because of this, he wants badly to be accepted by people of the Duchess's class and status. This is why he allows himself to be fooled by the Duchess's pearls. Even though he is materially much better off than the Duchess and even though she needs him, he still feels inferior to her.
The Duchess is an aristocrat and so she thinks she is better than everyone else. But she has no money and so she needs Oliver. She is trying to ingratiate herself with him, trying to buy him off by giving him access to her social class. In this way, she is begging from him even though she looks down on him.
So there is something of a complicated relationship between the two with each being both superior to and subordinate to the other.