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 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 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 w2 0 vs w2 0 ii Show that the dependence of period 1 consumption on w1 w2 can be expressed in terms of permanent income 1 b Assume u c 1 1 c where 0 1 Do the same as in a In the discussion identify which results apply for all and which ones only for greater or less than one Provided online for use by UCSB students C Copyright 2014 Henning Bohn 1 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 ct wt of a change ct in consumption triggered by some change wt 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 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 ln c 2 2 ln c 3 The person has labor incomes w1 w2 w3 in periods 1 3 respectively Let at 1 1 r at w t c t 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 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 0 9 c Again assuming 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 2 1 4 Midterm 2009 Consider an individual with utility function U u c1 u c2 over consumption where 0 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 1 1 c1 is a power function with parameter 0 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 i 1 w1 c1 1 c2 1 r w1 w2 1 r dci d 1 r 1 r ci i 1 2 Show that and 2 1 1 Explain in economic terms why for small 1 0 2 whereas for large 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 3 Part 2 2 1 Suppose an economy has a production function yt kt in efficiency units a savings rate s 0 a population growth rate n and a depreciation rate of a Suppose 1 3 s 0 2 n 1 g 1 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 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 e Suppose productivity growth is increased to g 2 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 …
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