The good news is that staple foods and fruits and vegetables are still far cheaper. Peru's fertile, equatorial climate means that crops are grown year-round. When we visited a friends farm a few hours north of Lima, near Barranca, we returned laden with enough watermelons, passionfruit, mangoes and bananas to last for weeks. However, dairy is only a little cheaper, and processed and imported foods are actually more expensive, as are northern fruits like strawberries, raspberries and blueberries. This means that if you cook and eat fresh food, you'll definitely save money moving to Lima. However, if you rely on breakfast cereal and TV dinners (and who hasn't, in their heady younger days?), you'd be better off staying in Akron.
The cheapest grocery stores in Lima are generally Tottus (which I'm such a big fan of that people make fun of me for it) and Metro (often better for fresh fruits and vegetables). Geek that I am, I compared prices at these stores to data from the United States Department of Agriculture and Bureau of Labor Statistics to draw a true apples to apples comparison between food prices in Lima and the United States.
Price Comparison List
Italics for items cheaper in the US
Starches (25% cheaper)
Rice $0.36/lb $0.70/lb
Potatoes $0.29/lb $0.64/lb
Flour $0.68/lb (not sold in bulk) $0.53/lb
Lentils $0.49/lb $1.02/lb
Quinoa $1.45/lb $4.50/lb
Sliced Bread $1.26/lb $1.40/lb
Breakfast Cereal $3.26/lb $2.49/lb
Fruit/Vegetables (49% cheaper)
Tomatoes $0.32/lb $1.45/lb
Avocado $0.32/lb $2.50/lb
Broccoli $0.49/lb $1.66/lb
Red Pepper $0.49/lb $2.13/lb
Bananas $0.32/lb $0.60/lb
Mangoes $0.16/lb $1.00/lb
Strawberries $3.27/lb $1.65/lb
Meats (31% cheaper)
Chicken filet $2.28/lb $2.99/lb
Ground beef $1.95/lb $3.32/lb
White-meat fish (bonito/tilapia) $2.44/lb $3.39/lb
Dairy/Eggs (10% cheaper)
Milk $4.29/gal $3.44/gal
Soy Milk $3.66/gal $6.95/gal
Yogurt $5.41/gal $7.91/gal
Cheese (e.g. Gouda/Swiss) $4.40/lb $5.56/lb
Unsalted butter $4.06/lb $3.54/lb
Eggs $1.90/dozen $1.87/dozen
Note that for many tropical items, such as passion fruit (my girlfriends's favorite!), soursop, custard apple etc. don't even exist in the US or are incomparably expensive. New fruits I've never seen before somehow seem almost magical to me, in a way that other new foods don't. There are also infinite varieties of new potatoes, gourds and avocadoes, which can be fun as well.
Prices are also shaped by the Peruvian diet, so common foods in the US (like peanut butter) are much more expensive, and worst of all, real fresh milk is non-existent. Like in Europe, all the milk is ultra-pasteurized and sold unrefrigerated!
One adapts to the Peruvian diet over time, because it is both cheaper and easier, but I have definitely broken the bank a few times when craving strawberries, or French brie, or Nutella.