Consider the utility functionu(x1, x2) = 10x1 - x21 + x2, again with accompanying budget constraint p1x1 + p2x2 = m.
1.1 Define monotonic preferences. For what bundles (x1, x2)does this utility function represent monotonic preferences?
1.2 Form the Lagrange function associated with this utilitymaximization problem, find the first-order conditions, and solve forthe demand function for good x1. Do not worry about second-orderconditions.
1.3 Find the own price elasticity of demand for good 1. Nowassume p2 = 1. When (i.e., for what prices p1) is demand inelastic?unit elastic? elastic? With this answer in mind, draw a graph with p1 on the horizontal axis.
Identify those values of p1 for which demand isinelastic, unit elastic, and elastic. Now put the amount of money theconsumer spends on good x1 (i.e., p1x1) on the vertical axis. Usingwhat you know about the elasticity, show where the resulting functionis increasing, where it is decreasing, where it achieves its maximum,and where it hits the horizontal axis.