Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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The individual's problem is to choose the consumption bundle (cı, C2) optimally. Specifically, max {u(с1) + u(с2)} C1,C2,81 subject to the two budget constraints above. Using the Method of Lagrange, let d, be the Lagrange multiplier on the period 1 budget constraint and A, be the Lagrange multiplier attached to the period 2 budget constraint. The Lagrangean can be written as L(c1, C2, 81, A1, A2) = u(c1) + u(c2) + d1 [Y – P;c1 – s1] + A2 [(1+r)s1 – Pąc2] . - Take the first order-derivative of the Lagrangean function with respect to c1, c2, s1, d1 and A2. Set each of them equal to zero in order provide the equations that can be used to identify the Lagrangean functions "critical points".
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Transcribed Image Text:The individual's problem is to choose the consumption bundle (cı, C2) optimally. Specifically, max {u(с1) + u(с2)} C1,C2,81 subject to the two budget constraints above. Using the Method of Lagrange, let d, be the Lagrange multiplier on the period 1 budget constraint and A, be the Lagrange multiplier attached to the period 2 budget constraint. The Lagrangean can be written as L(c1, C2, 81, A1, A2) = u(c1) + u(c2) + d1 [Y – P;c1 – s1] + A2 [(1+r)s1 – Pąc2] . - Take the first order-derivative of the Lagrangean function with respect to c1, c2, s1, d1 and A2. Set each of them equal to zero in order provide the equations that can be used to identify the Lagrangean functions "critical points".
Consider the problem of an individual that has Y dollars to spend on consuming over two periods. Let c1 denote the amount of consumption that the individual would like to purchase in period 1 and c2 denote the amount of consumption that the individual would like to consume in period 2. The individual begins period 1 with Y dollars and can purchase c, units of the consumption good at a price P1 and can save any unspent wealth. Use s, to denote the amount of savings the individual chooses to hold at the end of period 1. Any wealth that is saved earns interest at rate r so that the amount of wealth the individual has at his/her disposal to purchase consumption goods in period 2 is (1+r)s1. This principal and interest on savings is used to finance period 2 consumption. Again, for simplicity, we can assume that it costs P, dollars to buy a unit of the consumption good in period 2.
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Transcribed Image Text:Consider the problem of an individual that has Y dollars to spend on consuming over two periods. Let c1 denote the amount of consumption that the individual would like to purchase in period 1 and c2 denote the amount of consumption that the individual would like to consume in period 2. The individual begins period 1 with Y dollars and can purchase c, units of the consumption good at a price P1 and can save any unspent wealth. Use s, to denote the amount of savings the individual chooses to hold at the end of period 1. Any wealth that is saved earns interest at rate r so that the amount of wealth the individual has at his/her disposal to purchase consumption goods in period 2 is (1+r)s1. This principal and interest on savings is used to finance period 2 consumption. Again, for simplicity, we can assume that it costs P, dollars to buy a unit of the consumption good in period 2.
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