chap17.pdf - Lecture notes on capital

# chap17.pdf - Lecture notes on capital - Capital Capital...

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1 1 Capital Capital accumulation relies on foregoing some current consumption in exchange for future gain The key attribute of a capital asset is that the asset is durable and contributes to production for several periods 2 Rate of Return single period rate of return: r 1 =(X-S)/S=X/S-1, where S is savings (foregone consumption) and X is extra consumption Example: forego 50 in consumption for one period and have 55 extra next period, so r = (55-50)/50 = .10 We expect that X>S, so the rate of return is positive Consumption per Period time S X

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2 3 Perpetual Rate of Return r = Y/S, where S is savings and Y is extra income in each subsequent period Example: if S=10000 and Y=500, then r =0.05 Consumption per Period time S Y 4 Two-Period Model of Savings and Consumption Current and future consumption are defined as C 0 and C 1 : r=X/S - 1 = Δ C 1 / Δ C 0 - 1 Rewriting, we have Δ C 1 / Δ C 0 = r+1 and Δ C 0 / Δ C 1 = 1/(1+r) = P 1 - This shows how much current consumption must be given up to increase future consumption on one unit - P 1 is therefore the relative price of future good, where the numeraire P 0 = 1
3 5 Demand for Future Goods Budget constraint: I = C 0 + P 1 C 1 , where I is the initial income endowment and P 0 is normalized to 1 If C 1 = 0, then C 0 = I If C 0 = 0, then C 1 = I/P 1 = I(1+r) U C 0 I C 1 I/P 1 6 Individual Chooses Consumption Levels in Each Period to Maximize Utility Standard utility maximization problem L = U(C 0 , C 1 ) + λ (I - C 0 - [1/(1+r)]C 1 ) L/ C 0 = U/ C 0 - λ = 0 L/ C 1 = U/ C 1 - λ [1/(1+r)] = 0 Optimality requires equating MRS and price ratio as usual - P 0 is normalized to 1 as the numeraire 1 0 1 1 0 P P r MRS C U C U = + = =

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## This note was uploaded on 05/26/2008 for the course ECON 101 taught by Professor Buddin during the Winter '08 term at UCLA.

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chap17.pdf - Lecture notes on capital - Capital Capital...

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