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Unformatted text preview: nt ( ) ( ) PV Inter-temporal Budget Constraint The solution below is in terms of FV (answers in PV terms is OK). The UMP is: ( ) ( ) ( ) s.t. ( ) s.t. ( ( ) ) Setting up the Lagrangian gives: [( ) The first order conditions (FOCs) are: ( ( [( ) ) ) And the Kuhn-Tucker conditions are: According KT conditions we have four cases: 14 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Inter-temporal Consumption in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Case-I: From first and second FOC we have Solving the system of two equations gives ( ) Substituting this into the third FOC: ( ) ( ) ( [ ) ]( ) Hence case-I is a solution only if ( ( ) ) And ( [ Therefore, case-I satisfies when ( ) ) ]( ) ( ) ⁄. Case-II: From FOCs one and three we have And substituting For into FOC three to be true { ⁄( )} {⁄} must satisfy; therefore, case-II is a solution only if ( ) Case-III: From FOC two and three we have Solving the system of two equations and substituting for ( For to be true {[( )( )⁄ ] FOC one gives )( ) } must satisfy; therefore, case-III is a solution only if Case-IV: From FOC three we can see that case-IV cannot be a solution, since the same time. cannot be equal to zero and greater than zero at 15 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Inter-temporal Consumption in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. For the rest of solutions we assume that there exists an interior solution. That is ( ( (b) When will this consumer borrow in ? As )( ) ) will she continue to be a borrower? Show all calculations. Answer: In the consumer eats: If she is a borrower in , it must be that: That is, she will borrow in if her income tomorrow is below a threshold level. Observe that as the right hand side of this equation becomes smaller and may eventually reverse so that she switches from being a borrower to a saver. (c) Assume the consumer is a borrower in Answer: Assuming she is a borrower in . If inflation rises, will she borrow more or less? she will borrow: Now, either from the exact formula for real interest rates: Or the Fisher approximation: We see that as inflation goes up, falls so that goes up, implying that she will borrow more in . 16 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Inter-temporal Consumption in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (d) If the nominal interest rate equals inflation, when will this consumer exhibit consumption smoothing? Show all calculations. Answer: From above: [ If then so that: [ For consumption smoothing: [ [ (6.5) Consider a 3 period economy ( ) where at the beginning of each consumer receives (individualspecific) real incomes (in units of corn) respectively. In the first two periods, consumers can “borrow” or “save” corn at the prevailing real interest rate. Suppos...
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