Your first example has a noun, and the second has a verb. The basic meaning is "go up" but there is a difference. "A rise" means an increase. "A raise" means an increase in salary. "To rise" is what the subject does. "To raise" is what the subject does to another object. In your second example, we can say "sellers raise prices" (sellers make prices go up) or "prices rise" (prices go up).

Both sentences are perfectly correct. Both raise and rise can be used with prices. The difference here is that in the first sentence 'rise' is being used as a noun, and in the second sentence 'raise' is being used as a verb. Raise wouldn't generally be used as a noun when talking about prices. It is more commonly used as a noun when talking about your salary. Eg: "My boss gave me a raise this month!"

you are going to come across subtle differences here between American and British usage. although generally they are the same between American and English, you will come across speakers who disagree slightly. ----------- Usage note Dictionary.com Raise and rise are similar in form and meaning but different in grammatical use. Raise is the causative of rise; to raise something is to cause it to rise. Raise is almost always used transitively. Its forms are regular: Raise the window. The flag had been raised before we arrived. Raise in the intransitive sense “to rise up, arise” is nonstandard: Dough raises better when the temperature is warm. Rise is almost exclusively intransitive in its standard uses. Its forms are irregular: My husband usually rises before seven. The earliest I have ever risen is eight. The sun rose in a cloudless sky. The dough is rising now. Both raise and rear are used in the United States to refer to the upbringing of children. Although raise was formerly condemned in this sense (“You raise hogs but you rear children”), it is now standard. In American English, a person receives a raise in salary. In British English it is a rise.

in your example food prices rise, they cost more are more expensive. the food prices rise (have risen) because of (x) factor(s). The people supplying the shops/markets raise their costs (ask for more money). this caused prices to rise in the shops. prices rises are the effect upon customers after suppliers have raised their prices.