Later in the interview, Warren responds to a critique with the following:
But I get your point, maybe it’s only four cents on $7.19. But if your entrees are $14.40 we’ll see how fast I can do the math — are you telling me you can’t raise your prices by eight cents?
Warren's quick math implies that an entrée that costs twice as much now would still cost twice as much after an increase in minimum wage — this implies that it scales with entrée price. Where the specific four cents comes from is somewhat of a mystery but it amounts to a roughly 0.6% increase in retail price.
(As an aside, it is clear is that this system is not completely accurate due to differing overhead costs on various entrées. The costs of producing a $7.19 entrée are not half of the costs of producing a $14.40 entrée. So a literal interpretation of adding a 0.6% increase to each meal wouldn't be strictly accurate. But I suspect Warren's point is that there is a buffer in the profit margin to fully absorb a wage increase.)
This 0.6% increase is hotly contested and various other numbers have been suggested by articles that dig into the topic. BusinessWeek has this statement:
A look at company-operated McDonald’s and Burger King outlets shows profit margins above 10 percent. Still, labor expenses at these locations exceed profits, so suddenly raising the hourly wage to $15 would slam margins. Based on recent restaurant financials, if payroll costs doubled and other expenses didn’t decrease, menu prices at McDonald’s would have to go up about 25 percent to offset the increase. That would mean paying up to an extra $1 for a Big Mac, likely sending price-sensitive consumers elsewhere.
Heritage.org ran a similar article also looking at a minimum wage increase to $15/hr:
The Bureau of Labor Statistics reports the average cook in a fast-food restaurant earned $9.04 an hour in 2013. The SEIU’s push for $15 an hour would consequently raise fast-food wages by at least 66 percent. Paying $15 an hour would raise fast-food restaurants’ total costs by approximately 15 percent.
Fast-food restaurants could not pay this additional amount out of their profits. The typical restaurant has a profit margin of just 3 percent before taxes. That works out to approximately $27,000 a year — less than the annual cost of hiring one full-time employee at $15 an hour. In order to raise wages, fast-food restaurants must raise prices.
The article later concludes that the price increase would be 38%.
Warren's suggestion of $10.10 would obviously have a much smaller impact than the $15 discussed above. The BusinessWeek article does address this briefly:
Much of the public debate, however, is focused on raising wages to considerably less than the much-hyped $15 an hour. Wicks-Lim and 99 other economists signed a petition in July to raise the federal minimum wage to $10.50. They say the increase in costs for restaurants would equal about 2.7 percent of sales. Wicks-Lim adds that companies could then make up the difference through price increases (say, a nickel more for a burger), reduced employee turnover, productivity gains, and slower raises for the highest-paid employees.
This "nickel more" sounds suspiciously close to Warren's four cent increase. Thankfully, BusinessWeek provided a link to the petition (PDF warning). Here is the greater context for the nickel increase:
On average, even fast-food restaurants, which employ a disproportionate share of minimum wage workers, are likely to see their overall business costs increase by only about 2.7 percent from a rise today to a $10.50 federal minimum wage. That means, for example, that McDonalds could cover fully half of the cost increase by raising the price of a Big Mac, on average, from $4.00 to $4.05. The remaining half of the adjustment could come through small productivity gains or a slightly more equal distribution of companies’ total revenues.
This petition is the most authoritative claim I've seen on the subject and it makes me feel rather confident that Warren's claim is completely off-base. The menu price increase is over twice as high (1.25% instead of 0.6%) but the petition explicitly notes that this only covers half of the cost. They estimate the total cost would be 2.7 percent. To completely cover that cost with menu price increases, a $4.00 Big Mac would cost $4.10 and Warren's $7.19 combo meal would cost $7.38 — an increase of 19 cents.
Although Warren is proposing a smaller minimum wage, her example also cuts the menu price increase in half and completely ignores the other half of the impact. While it is possible that Warren understands that her four cent suggestion is only covering half of the needed increase, the claim as represented in this question seems to have no hint of this extra requirement. It means the $10.10 minimum wage + 0.6% price increase is supposed to accomplish the same effect as a $10.50 minimum wage + 2.7% price increase.
For those curious, the proposal linked to a more thorough document that reveals the source for all of their calculations (PDF warning). A quick snippet:
We use the numbers in Table A1 to produce a scatterplot with the size of the minimum wage increase on the x-axis, and the size of the business cost increase to sales on the y-axis. We find that the curve with the following equation best fits the data points (with an R2 of 0.73):
y = 0.0454 x^0.6363
We use this equation to extrapolate that the business cost increase relative to sales figure given a 44.8-percent minimum wage hike would be 2.7 percent (
0.0454 x 0.448^0.6363 = 0.027). Therefore, a price increase sufficient to cover half of this rise in costs would amount to a 1.35% (2.7%/2). For a $4.00 Big Mac, a 1.35% increase equals about 5 cents.
The 44.8-percent minimum wage hike is simply
10.50 / 7.25 = 1.448.
In order to hit Warren's numbers, you would need this equation to result in 0.6% instead of 2.7%.
y = 0.0454 x^0.6363
0.006 = 0.0454 x^0.6363
WolframAlpha reveals that this solves at 0.042 or 4.2%. A minimum wage hike of 4.2 percent would be $7.55/hr. This means Warren is claiming that a $10.10 minimum wage is doing the work of $7.55/hr. It isn't even close.
So let's be charitable to Warren and assume she knew this was only supposed to cover half of the extra costs.
0.012 = 0.0454 x^0.6363
This results in a minimum wage hike of 12.4% or $8.15/hr. Still not quite right. Unless Warren's suggestion has some other way for McDonald's to cope with the price increase, this still isn't very plausible.
To run the numbers the other way, a minimum wage hike to $10.10 is a 39.3% increase:
y = 0.0454 0.393^0.6363 = 0.0250592...
Warren's 0.6% price hike leaves 1.9% of the price increase remaining. I'm sure if push comes to shove, McDonald's could figure something out but the implication that adding four cents to their meals would solve the issue is very misleading.