Recently, I went for a walk on a Wednesday evening at 8 PM in my neighborhood. I passed by 4 restaurants and dining rooms had lots of available tables. The best of the four was a popular chain steak house concept. Let's try to put ourselves in the shoes of management.
Last year, our example steak house had annual covers of 50,000 and sales of $2,000,000. Covers in the current month are off 30% vs. the same month last year. The current average sales per cover is $40 and management is studying options.
One manager favors a 5% across the board cut in menu prices. The hope is to keep the drop in covers at 20% vs. last year. Sales would drop to $1,520,000 and the food cost % would increase 1.84% (to 36.84% up from 35%). Labor expenses and other operating expenses are forecasted to hold at the current cost per cover.
The general manager sees danger in the drop and favors holding menu prices at the current levels.
So who has the better plan?

Across the board menu price cuts are always risky. As covers decline, we would also see a drop in the contribution per cover. This is a very difficult plan to manage.
I prefer the GM's solution. No menu reprint is required and there are plenty of tactical options available. Selective price drops may be implemented using specials to promote entrees using lower cost seasonal ingredients. Specific days of the week could be targeted for selective price cuts. As long as the covers on busy nights and busy seasons stay close to plan, the strategy will succeed.
In our example, the GM's solution would produce a higher profit despite a steeper drop in covers. A 5% cut in sales per cover would not produce a bigger profit despite 5,000 more covers. The drop in sales per cover of $2 is not a simple to implement strategy. All income statement accounts would feel the impact. Waitstaff would see their tips per cover decline. Careful study is needed to predict the impact on your operation.