Problem set 1: Competitive Equilibrium Spring 2024
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Problem set 1: Competitive Equilibrium
Spring 2024
Master in Economics
Due: February 2, 2024
Each question below is worth 20 points for a total of 100 points.
1 Competitive Equilibrium
Consider the Lucas Asset Pricing Model study in class. The economy is populated by infinitely many identical consumers. Normalize the total population L = 1. Preferences are represented by the standard utility function where ρ > 0 is the coefficient of relative risk aversion and β < 1 is the discount factor.
In this economy there is only one asset which consists of a set of identical infinitely-lived trees. Normalize the stock of trees K = 1. Aggregate output is the fruit that falls from the trees, and cannot be stored, dt. These dividends are stochastic. Let Pt be the price of the asset and kt(i)+1 be agent i holdings of stocks in the tree.
1. Write the Bellman equation for the individual agent.
2. Define a competitive equilibrium for this economy
3. Solve the agent’s problem (i.e., derive the Euler equation).
4. What is the equilibrium price of the asset (hint: iterate forward the Euler equation, in- voke the transversality condition, and apply the resource constraint to replace individual consumption). What is the stochastic discount factor between t and t + n? What is the interpretation of the stochastic discount factor?
5. Assume u(c) = log(c). What is the price-dividend ratio? Does the price-dividend ratio de- pend on the expected level of dividend in the future? Why?
2024-02-02