FE II – Midterm Questions (Due 10AM, Friday, April 8.)
1 Question #1 (50 Marks)
Consider the following lottery. With probability ⇡H the individual receives cH units of the consumption good and with probability ⇡L = 1 ⇡ the individual receives cL units of the consumption good, cH > cL. Let the individual’s preferences be represented by a utility function u(c) such that more is better, u0(c) > 0.
Suppose the individual is risk-averse. Using a diagram, illustrate the concept of the certainty equivalent level of consumption cce and the risk premium of the lottery. (15 Marks)
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Identify the certainty equivalent level of consumption for a risk-neutral individual (which you can denote by cce) and the risk-premium for this individual using a diagram. (15 Marks)
From your answers above, what property of the individual’s utility function (in addition to u0(c) > 0) is being exploited in modelling risk aversion and how does this property of the utility function result in the individual’s distaste of risk? (20 Marks)
Question #2 (50 Marks)
Consider an individual facing the prospect of having high income, yH > 0, with probability ⇡H, medium income of yM with probability ⇡M and low income, yL, with probability ⇡L, yH > yM > yL. Let the individual’s preferences be represented by a utility function that is defined on consumption of a consumption good, u(c). The utility function represents preferences such that more is better, u0(c) > 0, features diminishing marginal utility of consumption, u00(c) < 0 and satisfies the Inada condition limc!0 u0(c) = 1. Assume, for simplicity, that income is paid in units of consumption goods.
Prior to learning the realization of income, the individual has w units of wealth to spend on aquiring assets. This wealth cannot be stored nor consumed to provide utility. There are two types of assets that the individual can buy or sell. Each unit of Asset 1 costs q1 units of wealth and pays a single unit of the consumption good per asset purchased only if income is realized to be high. In contrast, the individual must pay q2 units of wealth per unit of Asset 2. Each unit of Asset 2 held by the individual yields R units of the consumption good if income is realized to be yH, yM or yL. Let a1 denote the units of Asset 1 purchased by the individual while a2 denotes the number of Asset 2 purchased by the individual. Negative values of a1 or a2 mean that the individual is selling rather than purchasing assets.
Formally, the individual’s problem is to maximize the expected utility from consump- tion subject to the constraints that consumption must be financed out of income and the realized return from the asset portfolio as well as a constraint that spending on the asset portfolio must be financed out of the non-storable endowment wealth w.
subject to
max {⇡H u(cH ) + ⇡M u(cM ) + ⇡Lu(cL)} cH ,cM ,cL ,a1 ,a2
cH =yH+a1+Ra2 (1) cM =yM+Ra2 (2) cL =yL+Ra2 (3) w = q1a1 + q2a2. (4)
Denote the optimal allocation by (c⇤H , c⇤M , c⇤L, a⇤1, a⇤2).
✓ q 1 ◆ ⇡ H u 0 ( c ⇤H ) = ✓ q 1 ◆ R [ ⇡ H u 0 ( c ⇤H ) + ⇡ M u 0 ( c ⇤M ) + ⇡ L u 0 ( c ⇤L ) ] . (5)
1. Using words, interpret the optimal trade-o↵ condition of equation (5) and why the optimal portfolio plan must satisfy this condition. Make sure that you explain the economic trade-o↵, don’t simply list the terms on both sides of the equality sign. You might need to make reference to equations (1) through equation (4) in describing the costs and benefits of any feasible portfolio reallocations. (25 Marks)
2. Do the asset return structures give the individual the opportunity to fully insure against consumption risk? Explain why (or why not). (25 Marks)
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