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204A-problems - Collection of Practice Problems Econ 204A...

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1 Collection of Practice Problems Econ 204A Henning Bohn * In previous years, students have often asked me about practice problems in addition to the problem sets. Here is a collection. Some will be assigned for the weekly problem sets. I hope the others are useful for practice. Request : Please tell me about errors or ambiguities. Many of the questions below are old exam problems. As I am updating the course over the years, the notation, references, and sequencing has changed, which means that some old problems may have lost educational value without me noticing immediately. So if a problem seems obscure to you, please let me know. (Your incentive: your problem set might shrink if you convince me that a problem is unclear.) Part 1 : 1.1. Consider the two-period consumption model: Individuals have initial assets A, earn interest r on assets, and earn wage income ( w 1 , w 2 ) . They maximize utility U = u ( c 1 ) + β u ( c 2 ) . a. Assume u ( c ) = ln( c ) . i. Solve for optimal consumption and period-1 asset holdings as functions of wage income, the interest rate, and the time-discount factor β . Discuss under what conditions a marginally higher interest rate reduces consumption. [Discuss means: Interpret the solution. Conditions may be exact, or necessary, or sufficient. Hint: Distinguish cases with w 2 = 0 vs. w 2 > 0 .] ii. Show that the dependence of period-1 consumption on ( w 1 , w 2 ) can be expressed in terms of permanent income. b. Assume u ( c ) = 1 1 γ c 1 γ where γ > 0, γ 1 . Do the same as in (a). In the discussion, identify which results apply for all γ , and which ones only for γ greater or less than one. * Provided online for use by UCSB students. (C) Copyright 2011 Henning Bohn
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2 1.2. Economists sometimes use the marginal propensity to consume (MPC) to express the effects of income changes on consumption: MPC is defined as the ratio Δ c t / Δ w t of a change Δ c t in consumption triggered by some change Δ w t in the current wage income that may or may not be accompanied by changes in future wages. This question will ask you to compute the MPC for several scenarios. (Hint: Use a spreadsheet for calculation.) Assume the permanent income model with planning horizon of n periods. Assume β =1/(1+r) with r=3% per year. Assume zero initial assets (A=0). a. Assume the time horizon is n=50 years (interpretation: roughly the life expectancy at age 30). Determine the MPC from i. a one-year wage increase; ii. an increase in wages that lasts 5 years; iii. a wage increase that last for 35 years [intuition: until about retirement]; iv. a tax reduction for one year followed by a tax increase of the same size in the next year. Discuss: How do the results compare? Do you find them surprising? Realistic? Why or why not? b. Determine the impact of a one-year wage increase for consumers with alternative planning horizons of, respectively: n=1 year; n=2 years; n=10 years; n=50 years; the limiting case of n=infinity.
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