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: problem sets 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 (w1,w2). They maximize utility U = u(c1) +βu(c2). 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 withw2= 0 vs. w2> 0.] ii. Show that the dependence of period-1 consumption on (w1,w2) can be expressed in terms of permanent income. b. Assume u(c) =11−γc1−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 2014 Henning Bohn2 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 Δct/Δwt of a change Δct in consumption triggered by some change Δwt 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 holds 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 across cases? Do you find them surprising? Why or why not? What is the economic intuition? 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. Discuss: How do the results compare across cases? Do you find them surprising? Why or why not? What is the economic intuition? 1.3. [Midterm 2008] Consider an individual with utility function U = ln(c1) +βln(c2) +β2ln(c3). The person has labor incomes w1, w2, w3, in periods 1-3, respectively. Let at +1= (1+ r)at+ wt− ct denote assets carried into the next period. The interest rate r is constant. Assets a1>0 are given. a. Specify the budget equations, derive the intertemporal budget constraint, and derive first order conditions for optimality. Be explicit about any terminal conditions that may be required, and make sure the number of optimality conditions and constraints matches the number of variables. b. Suppose β=1/(1+r), wt=w is constant for all t. Solve for period-1 consumption as function of the labor incomes and of initial assets. Define the individual’s permanent income. Determine the marginal propensity to consume (MPC) in period 1 from period-1 labor income; compute the MPC value for β=0.9. c. Again assuming β=1/(1+r)<1 and constant w, show that assets at are a declining sequence (that is: a3<a2<a1) and that at/at-1 is also decreasing over time. Can you explain why this makes sense economically? Use the same economic argument to make a conjecture about the behavior of at/at-1 in problems with more than T=3 periods, notably for T → ∞.3 1.4. [Midterm 2009] Consider an individual with utility function U = u(c1) +βu(c2) over consumption, where 0 <β< 1 and u(⋅) is increasing and concave. Labor incomes w1 and w2 are given. Initial assets are zero. The individual can save or borrow between periods 1 and 2 at a given interest rate r. a. Specify the budget equations, derive the intertemporal budget constraint, and derive first order conditions for optimal consumption. b. Suppose u(c) =11−θc1−θ is a power function with parameter 0 <θ≠ 1. Derive equations for c1 and c2 as functions of the preference parameters and the exogenous variables. c. Define the elasticity of ci with respect to (1+r) by εi=dcid (1+ r )⋅(1+ r )ci (i=1,2). Show that ε1=w1− c1−1θc2/(1+ r )w1+ w2/(1+ r ) and ε2=ε1+1θ. Explain in economic terms why for small θ, ε1< 0 <ε2, whereas for large θ both elasticities are positive for savers and negative for borrowers. [Hint: If you have trouble deriving the formulas, take them as given and focus on the interpretation.]4 Part 2: 2.1. Suppose an economy has a production function yt = ktα (in efficiency units), a savings rate s>0, a population growth rate n, and a depreciation rate of δ. a. Suppose α=1/3, s=0.2, n=1%, g=1%, δ=4%. What are the steady state value of the capital-labor ratio, output per efficiency unit, and consumption per efficiency unit? For parts b-e, assume the economy starts in the steady state derived in (a). b. Suppose an earthquake destroys 10% of the capital stock. Sketch the time paths of the capital-labor ratio and of per-capita consumption. Compare to (a). [To clarify: Per-capita means per actual worker, not in efficiency units.] c. Suppose savings are increased to s=0.22. What is the impact effect on consumption? What are the new steady state values of the capital-labor ratio, output per efficiency unit, and consumption per efficiency unit? Sketch the time paths of the capital-labor ratio and of per-capita consumption. Compare to (a). d. Suppose population growth is increased to n=2%. What are the new steady state values of the capital labor
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