What is the relationship between total utility and marginal utility? How do they differ?
Total utility (TU) is, well, the amount of utility from all the goods you consume.Marginal utility (MU) is the difference/delta/addition/reduction in total utility when you consume one additional unit of good.To illustrate, suppose that when you consume one glass of milk, you get 10 utils (utility isn’t really quantifiable but for the sake of theory we use “util” to measure it.) That means the TU is 10 util. The MU is also 10 util because the difference in util you get between one glass of milk and no glass of milk is 10 util. Now say that you drink another glass of milk and now the TU from drinking those two glasses of milk is 18 util. That would mean that the second glass of milk is worth or adds 8 util, hence the MU is 8 util. If you drink a third glass of milk and the TU becomes 24 util, that would mean that the MU is 6 util.Notice that marginal utility tends to decrease as you drink more glasses of milk. That’s simply because another glass of milk isn’t as good when you’ve had one or two already, perhaps because you’re starting to get full. This is the law of diminishing marginal utility. There are exceptions, though, usually addictive goods like drugs.Now as for the relationship, a positive marginal utility means that the total utility is increasing. The marginal utility will keep diminishing. When it is zero, the total utility is no longer increasing and reaches the maximum point. After that, the marginal utility is negative which means that the total utility is decreasing with more consumption. That means that you get less utility by consuming one more good than if you do not consume it. This graph helps illustrate it.
What is the pointwise maximum(minimum) of two convex functio?
The pointwise maximum of two functions f and g is the function h with rule given for all x by h(x) = max(f(x), g(x)) or, if you do not like that notation, h(x) = the larger of the two values f(x), g(x), or their common value of f(x) = g(x). Similarly the pointwise minimum of two functions f and g is the function whose rule sends x to the smaller of f(x) and g(x) (or their common value, if these two numbers are equal). The pointwise maximum of convex functions is always a convex function. Unfortunately, how you prove this depends on how "convex function" is defined. Sadly, there are many different (but logically equivalent) ways to define the concept, and whichever one you pick makes some proofs easy and other proofs more difficult. One way to convince yourself why this ought to be true is to note that (1) a function f is a convex function if and only if the set of pairs {(x,y): x in the domain of f, and y >= f(x)} [graphically, the set of all points lying on or above the graph of f] is convex in the geometrical sense, (2) an intersection of convex sets is convex, (3) for any functions f and g, the set {(x,y): y >= max(f(x),g(x)} is the intersection of the two sets {(x,y): y >= f(x)} and {(x,y): y >= g(x)}. So if f and g are convex functions, from (3) and (1) you know that {(x,y): y >= max(f(x),g(x)} is an intersection of convex sets, and hence convex by (2), and hence by (1), you know the pointwise maximum is convex. [This is a fully rigorous proof if the condition in (1) is taken as the *definition* of what it means for a function to be convex.] There isn't much you can say about the pointwise minimum of two convex functions. It need not be convex. As a simple example, consider f(x) = |x| and g(x) = |x - 1|. Draw their graphs on the same axes, and use that to draw the graph of the pointwise minimum functin. You'll see at once that it isn't convex