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lecture7_467

# lecture7_467 - Monetary Macroeconomics Lecture 7 Capital...

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Monetary Macroeconomics Lecture 7: Capital and Financial Intermediation Dr. Pınar Ye¸ sin December 9, 2005 University of Zurich 1

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Including Other Assets in Our OLG Model Fiat money is not the only asset available in the real world for saving. Now we will ana- lyze how the presence of other assets affect the demand for money. 2
What is Capital? Capital is an asset that can be used for pro- duction, e.g. machinery, seeds, etc. We generally denote the production function with F ( K, L ) where K is capital, and L is labor. One usually assumes that each period capital depreciates at rate δ , such that 0 δ 1. So next period one still has (1 - δ ) K capital left after receiving the output F ( K, L ). 3

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The Model Consider our OLG model with 2 period lived agents. There is one country, and no fiat money! The young have y units of endowment and the old have no endowment. The initial old have k 0 units of capital that can be used for production of the consump- tion good. Assume that when k units are invested in the production technology then f ( k ) units of the consumption good are pro- duced next period. We assume that capital depreciates completely after one period. Thus δ = 1. Consider stationary allocations. 4
Generation t ’s budget constraints: when young: c 1 + k y . when old: c 2 f ( k ). Hence the lifetime budget constraint is: c 2 f ( y - c 1 ) So his problem is max u ( c 1 , c 2 ) s.t. c 2 f ( y - c 1 ) 5

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The Lagrangian is L = u ( c 1 , c 2 ) + λ [ f ( y - c 1 ) - c 2 ] L ∂c 1 = ∂u ( c 1 , c 2 ) ∂c 1 + λ [ f 0 ( y - c 1 )( - 1)] = 0 L ∂c 2 = ∂u ( c 1 , c 2 ) ∂c 2 + λ ( - 1) = 0 L ∂λ = f ( y - c 1 ) - c 2 = 0 So marginal utility of c 1 = λ marginal product marginal utility of c 2 = λ Thus - MRS = MP 6
At the solution ( c * 1 , c * 2 ), the marginal rate of substitution between the two period con- sumptions is equal to the marginal product of capital. Note that there is nothing special about the OLG model here. Agents are not trading with each other. 7

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Properties of the Production Function In general f 0 ( k ) > 0 and f 00 ( k ) < 0, so that we have diminishing marginal product of cap- ital. That is the added output from an extra unit of capital gets smaller as capital increases. 8
A Model of Private Debt Here we will find conditions so that more than one asset can exist in the economy. Consider an OLG model with 2-period lived consumers. There is no fiat money in the economy. Assume first that there is no capital either. There is one type of consumption good that everybody likes to consume when young and when old. 9

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There are two types of people: borrowers and lenders. Borrowers have no endowment when young and y units of the consumption good when old.
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