To make it stupidly simple, assume that the tire has a roughly rectangular cross-section. So a 1-inch tire is 1 inch across has a width of 1 inch in contact with the pavement, while a 4-inch tire is 4 inches across and has a width of 4 inches in contact with the pavement.
Also, for the sake of simplicity, assume that the tire is perfectly flexible -- that the stiffness of the tire carcass and tread is not a factor.
And assume that we have a 200 pound bike+rider, and that the weight is evenly distributed between front and rear, meaning there is 100 pounds of weight on each tire.
At 50 PSI the 1-inch tire will need to have two square inches == two linear inches in contact with the pavement, while the 4-inch tire will need to have 2 square inches == one-half linear inch in contact.
On a 24 inch wheel, the 1-inch wide tire will deflect about 0.04 inches, while the 4-inch tire will deflect about 0.0026 inches.
If you hit a bump which momentarily doubles the force on the wheel (and the size of the contact patch), the 1-inch tire will deflect about 0.168 inches, while the 4-inch tire will deflect about 0.01 inches. (Using this online calculator.)
So the narrower tire is "softer" by a factor of (0.168-0.04) / (0.01-0.0026) = 0.124 / 0.074 = 1.7.