Consider a teenager who evaluates whether she should engage in sexual activity with her partner of the opposite sex. She thinks ahead and expects to have a present discounted level of life-time consumption of x1 in the absence of a pregnancy interrupting her educational progress. If she gets pregnant, however, she will have to interrupt her education and expects the present discounted value of her life-time consumption to decline to x0 — considerably below x1.
A. Suppose that the probability of a pregnancy in the absence of birth control is 0.5 and assume that our teenager does not expect to evaluate consumption any differently in the presence of a child.
(a) Putting the present discounted value of lifetime consumption x on the horizontal axis and utility on the vertical, illustrate the consumption/utility relationship assuming that she is risk averse. Indicate the expected utility of consumption if she chooses to have sex.
(b) How much must the immediate satisfaction of having sex be worth in terms of lifetime consumption in order for her to choose to have sex?
(c) Now consider the role of birth control which reduces the probability of a pregnancy. How does this alter your answers?
(d) Suppose her partner believes his future consumption paths will develop similarly to hers depending on whether or not there is a pregnancy — but he is risk neutral. For any particular birth control method (and associated probability of a pregnancy), who is more likely to want to have sex assuming no other differences in tastes?
(e) As the payoff to education increases in the sense that x1 increases, what does the model predict about the degree of teenage sexual activity assuming that the effectiveness and availability of birth control remains unchanged and assuming risk neutrality?
(f) Do you think your answer to (e) also holds under risk aversion?
(g) (g)Suppose that a government program makes daycare more affordable — thus raising x0. What happens to the number of risk averse teenagers having sex according to this model?
B. Now suppose that the function u(x) = ln(x) allows us to represent a teenager’s tastes over gambles involving lifetime consumption using an expected utility function. Let δ represent the probability of a pregnancy occurring if the teenagers engage in sexual activity, and let x0 and x1 again represent the two lifetime consumption levels.
(a) Write down the expected utility function.
(b) What equation defines the certainty equivalent? Using the mathematical fact that α lnx + (1−α) ln y = ln(xαy(1−α)), can you express the certainty equivalent as a function x0, x1 and δ?
(c) Now derive an equation y(x0, x1, δ) that tells us the least value (in terms of consumption) that this teenager must place on sex in order to engage in it.
(d) What happens to y as the effectiveness of birth control increases? What does this imply about the fraction of teenagers having sex (as the effectiveness of birth control increases) assuming that all teenagers are identical except for the value they place on sex?1
(e) What happens to y as the payoff from education increases in the sense that x1 increases? What does this imply for the fraction of teenagers having sex (all else equal)?
(f) What happens to y as the government makes it easier to continue going to school — i.e. as it raises xo? What does this imply for the fraction of teenagers having sex?
(g) How do your answers change for a teenager with risk neutral tastes over gambles involving lifetime consumption that can be expressed using an expected utility function involving the function u(x) = x?
(h) How would your answers change if u(x) = x2?