shortcut - Derivation of the "Shortcut...

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Unformatted text preview: Derivation of the "Shortcut Method" for Econ 102 McDevitt 1 Introduction The purpose of this short note is to derive what Professor McDevitt terms the "shortcut method" for the consumption-savings model when preferences are Cobb-Douglas. We &rst treat the model without labor, then the model with a government, and &nally the model with labor but without government. 2 The Base Model In this model there is no government and no labor. 2.1 The Model In this case consumers face the following problem max C 1 ;C 2 C & 1 C 1 & & 2 where & 2 (0 ; 1) subject to the present value budget constraint C 1 + C 2 1 + r = Y 1 + Y 2 1 + r We can write the Lagrangian L = C & 1 C 1 & & 2 & & Y 1 + Y 2 1 + r & C 1 & C 2 1 + r where is the Lagrange multiplier associated with the budget constraint. 2.2 The FOC The FOC are C 1 : &C & & 1 1 C 1 & & 2 = (1) C 2 : (1 & & ) C & 1 C & & 2 = 1 1 + r (2) Dividing (1) by (2) we have &C & & 1 1 C 1 & & 2 (1 & & ) C & 1 C & & 2 = = (1 + r ) 1 which simpli&es to & 1 & & C 2 C 1 = 1 + r (3) Which is the statment that the MRS is equated to the interest rate. Recall from graphical micro theory that this is just the standard optimality condition that the slope of the budget line, 1+ r , must equal the slope of the indi/erence curve, the MRS. As the interest rate increases, the agent will consume relatively more C 2 and higher values of & induce the agent to consume more C 1 . 2.3 Solving the Model To solve for the consumption levels, rewrite (3) to get an expression for C 2 in terms of C 1 . C 2 = (1 + r ) 1 & & & C 1 (4) and substitute this into the budget constraint as follows C 1 + 1 1 + r C 2 = Y 1 + Y 2 1 + r C 1 + 1 1 + r & (1 + r ) 1 & & & C 1 = Y 1 + Y 2 1 + r C 1 + 1 & & & C 1 = Y 1 + Y 2 1 + r 1 + 1 & & & C 1 = Y 1 + Y 2 1 + r 1 & C 1 = Y 1 + Y 2 1 + r C 1 = & Y 1 +...
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shortcut - Derivation of the "Shortcut...

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