Intermediate Microeconomics UA10

Practice Questions for Second Mid-Term Exam Fall 2025

1. Assume that ≿ is a complete and transitive preference order defined over lottery space X, the set of all possible lotteries over a prize space Z with N prizes Z = {z1, ....., zN}.

(a) Carefully state what it means for preferences ≿ to satisfy the Continuity Axiom.

(b) Carefully state what it means for preferences ≿ to satisfy the Substitution Axiom.

(c) Carefully state the expected utility theorem.

2. Consider a compound lottery that occurs in two stages. In the first stage, an initial lottery determines which lottery you get in the second stage. The first stage lottery produces lottery A with 40% probability, lottery B with 40% probability, and lottery C with 20% probability. The second stage lotteries A,B,C are as follows:

• Lottery A places 50% probability on winning $16 and 50% prob-ability on winning $36.

• Lottery B places 65% probability on winning $16 and 35% prob-ability on winning $36.

• Lottery C places 20% probability on winning $16 and 80% prob-ability on winning $36.

(a) Draw the probability tree for the full compound lottery.

(b) Identify the simple one-stage lottery to which this compound lot-tery corresponds: specifically what is the probability that it offers each of the possible prizes, $36 and $16.

(c) Draw a risk averse expected utility function on a graph with con-sumption on the horizontal axis and expected utility on the verti-cal axis. Label the two prizes ($16 and $36) and their correspond-ing expected utilities on the graph.

i. On your graph, approximately mark the expected utility of the full compound lottery as well as the expected payout in dollars.

ii. Using your graph, approximately mark the certainty equiva-lent of this compound lottery and indicate how it compares to the expected dollar payout on the lottery (more than, equal to, or less than).

3. Suppose ≿ over lotteries over possible consumption levels c ∈ R+ has an expected utility representation and that the expected utility function is given by the equation u(c) = √c.

(a) Illustrate (in an approximate way) this expected utility function in a graph with consumption from $0 to $25 on the horizontal axis, and expected utility on the vertical axis. Mark on the vertical axis the expected utility of lottery C, which puts probability 1/2 on winning $0, and probability 1/2 on winning $25.

(b) Write out and solve the equation defining the certainty equivalent of lottery C and mark both the expected dollar payout and the certainty equivalent of lottery C on the horizontal axis.

(c) Now consider a new lottery, called D, which puts probability 1/4 on winning $0, probability 1/4 on winning $25, and probability 1/2 on winning $16. Calculate the expected utility of this lottery, and determine whether a person with these preferences would prefer lottery D or receiving $9 for sure. Illustrate your answer on the same graph drawn for part (a).

4. A consumer has wealth W = $50, 000 and faces possible loss of L = $30, 000 in state 1 with probability p1 = 0.15, and otherwise no loss in state 2. Any amount of insurance paying off $Y > 0 in case of loss is available for payment of premium $0.20Y up front: so that the premium rate is π1 = 0.2. Suppose that the utility function of the consumer is:

u(c) = ln c,

and that they spend all available resources in either state.

(a) Is the insurance actuarially fair or unfair?

(b) Write consumption in state 1 and in state 2 as a function of Y given that all available resources are spent in each state.

(c) Draw the corresponding budget set assuming that you can buy any amount of insurance Y ≥ 0.

(d) Find the optimal amount of insurance bought, Yˆ , as well as con-sumption in both states.

(e) Illustrate the solution and the indifference curves that illustrate it on the budget set you drew above (no need to be precise).

5. Consider a two period model in which an individual has lifetime utility,

u(c1, c2) = ln c1 + β ln c2.

Suppose further that this individual gets income m1 = 100 in period 1 with m2 = 0, and that the individual has a discount factor of β = 0.5. Suppose there are perfect capital markets and that the individual may save from period 1 income at an interest rate of r = 0.25 (such that $1 saved today becomes $1.25 in period 2).

(a) Write c2 as a function of c1 using the fact that all wealth in hand in period 2 is spent.

(b) Solve for the optimal consumption in each period. Be sure to explain how you arrived at your results.

6. Consider a two period model with no discounting so that an individual has lifetime expected utility,

U(c1, c2) = v(c1) + v(c2).

with the period expected utility function v(c) strictly increasing and strictly concave. Suppose that in the first period income is m1 = 3 while period 2 income is either m2 = 3 with probability 2/3 or m2 = 0 with probability 1/3. The individual may save from period 1 income (at interest rate 0) for second period consumption.

(a) Write total expected utility as a function of c1 using the fact that all available resources will be spent in period 2.

(b) Write the first order condition for the optimal level of period 1 consumption and interpret this equation.

(c) Suppose further that the expected utility function is logarithmic

v(c) = ln c,

and confirm that optimal period 1 consumption satisfies ˆc1 = 2 using the first order condition you derived in (b). Write down the pattern of optimal consumption (ˆc1, cˆ2) in the two periods as it depends on the actual level of income in period 2.

7. Consider an individual with the following period 1 utility functions over consumption in three periods c1, c2, c3:

U1(c1, c2, c3) = ln c1 + 0.6 ln c2 + 0.4 ln c3                 (1)

Suppose that their period 2 utility function over consumption in periods c2, c3 looking forward in period 2 is:

U2(c2, c3) = ln c2 + 0.6 ln c3               (2)

Suppose they have wealth $120,000 to divide between the three periods, and there is no interest on savings.

(a) Explain why this individual is not time consistent.

(b) Suppose that this individual is fully aware of their time incon-sistency. Write c1, c2 as a function of c3 taking account of what period 2 self will choose to do with remaining wealth in period 2.

(c) Solve for optimal levels of consumption in all periods in this case.

(d) Now consider a setting in which the individual in period 1 could commit to a maximum level of consumption in period 2. In this case, what would their optimal level of consumption be in each period?

8. A decision maker (DM) has wealth W to spend over two periods of life. There is no discounting, no return on assets, and log utility,

U(c1, c2) = ln c1 + ln c2.              (3)

There is probability pD = 0.25 of cognitive decline in the second period. The DM has an agent available in both periods who wastes 20% of the wealth (tA = 0.2). If cognitively declined the DM wastes 50% (tD = 0.5). The DM believes that they have a 75% chance of recognizing that they are declined in period 2 if they are (pR = 0.75). Suppose that the DM believes that if they do not recognize decline in period 2 they will believe that they are certainly not declined. Is it better in this case to hand over to the agent in period 1 or to hand over only if cognitive decline is recognized in period 2? (You can use a calculator if needed).