If good 1 is a “neutral,” what is its marginal rate of substitution for good 2?

If we observe a consumer choosing (x1, x2) when (y1, y2) is available one time, are we justified in concluding that (x1, x2) (y1, y2)?

Think of some other goods for which your preferences might be concave.

What is your marginal rate of substitution of $1 bills for $5 bills?

Can you explain why taking a monotonic transformation of a utility function doesn’t change the marginal rate of substitution?

Consider the utility function u(x1, x2) = √x1x2. What kind of preferences does it represent? Is the function v(x1, x2) = x2 1x2 a monotonic transformation of u(x1, x2)? Is the function w(x1, x2) = x2 1x2 2 a monotonic transformation of u(x1, x2)?

The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case f(u) = u2.)

We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indifference curve twice?)

What kind of preferences are represented by a utility function of the form u(x1, x2) = x1 + √x2? Is the utility function v(x1, x2) = x2 1 + 2x1 √x2 + x2 a monotonic transformation of u(x1, x2)?

What kind of preferences are represented by a utility function of the form u(x1, x2) = √x1 + x2? What about the utility function v(x1, x2) = 13x1 + 13x2?