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Intertemporal Rate of Substitution Question

Julie Carlisle is a financial planner at a large wealth management firm. One of her
clients, Esteban Blake, just received a sizable inheritance. He invests a portion of the
inheritance in an annuity that will immediately increase his income by a substantial
amount.

Holding all else constant, the change in Blake’s income will most likely result in:
A an increase in his marginal utility of consumption.
B an increase in his intertemporal rate of substitution.
C a decrease in his required risk premium for investing in risky assets.

The correct answer is C. Not sure why B is wrong. If as wealth increases, marginal utility of current consumption decreases, then shouldn’t intertemporal rate of substitution increase? Why is B wrong in that case? Thanks!

Cash is a fact, profit is an opinion

"Using Wiley for my CFA journey was by far the best option… I was able to pass on my first attempt.”– Moe E., Canada

Remember how ITRS is calculated:

ITRS = u1 / u0

Where u1 is the marginal utility of consuming in the future and u0 is the marginal utility of consuming today.

So, as the guy has received a great inheritance, his marginal utility of consuming in the future has decreased significantly. I mean, why he would want to cut current consumption and save for future consumption when he has a fortune? He will not. So u1 has decreased.

As his marginal utility of consuming today (u0) has increased because he has now a lot of resources to spend, then u0 has increased significantly. Replace the new values of u1 and u0 in the ITRS formula:

u1 is smaller than before, u0 is bigger than before, then ITRS has decreased.

Option B is wrong.

You can check more discussions in the link below:

https://www.analystforum.com/forums/cfa-forums/cfa-level-ii-forum/91349903

Hope this helps.

Las almas de todos los hombres son inmortales, pero las almas de los justos son inmortales y divinas.
Sócrates

Thank you for responding! As per your explanation, wouldn’t that mean Option A is correct as well. Option A says that the marginal utility of consumption has increased. Please let me know. Thanks!

Cash is a fact, profit is an opinion

Good catch. But I have to say that Option A is poorly written. The creator of the vignette meant (in Option A) that the rate at which the marginal utility changes is constantly increasing. However, in reality, at some point in time, the marginal utility of consumption will start decreasing, not from an intertemporal rate of substitution stand point of view as I explained in my previous post, but from the consumption itself point of view.

Simplifying my explanation: As the guy has a fortune now, he will start consuming a lot today, but at some point in time, the marginal utility of consuming his favorite goods and services will start decreasing. This means that the guy will buy 2, 3 or 5 new cars for example, but at each purchase, his satisfaction (utility) is lower than in the previous purchase until buying a new car provides 0 utility or even negative. At this time, the guy starts doing drugs in compensation, you know.

So option A is not the choice. I consider the choices create a space for confusion and ambiguity as we found. You won’t find questions like this on the exam.

By the way, could you please share the whole answer of that question?

Las almas de todos los hombres son inmortales, pero las almas de los justos son inmortales y divinas.
Sócrates

Got it. Thank you. Here’s the solution from the book:

The additional annuity payment substantially increases Blake’s
income and wealth, which decreases his marginal utility of consumption. As a
result, the average loss of marginal utility from any risk taking decreases as his
wealth increases. Thus, he requires a lower risk premium and is willing to buy
more risky assets.

Cash is a fact, profit is an opinion