IntMacro2018springslidesTopic4_wfigFeb4.pdf

# IntMacro2018springslidesTopic4_wfigFeb4.pdf - Topic 4...

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Topic 4 Two-Period Competitive Equilibrium Model: Part 1: Investment and Saving in a Closed Economy Mark Gertler NYU Macroeconomic Theory and Analysis Spring 2018 0

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Intertemporal Competitive Equilibrium Model We move from the static model to an intertemporal one for two basic reasons: To analyze the economy’s behavior over time To study the behavior of saving and investment, as well as consumption, output and employment We consider a simple two period model The two period model delivers most of the key insights one would obtain from a more realistic infinite horizon model. As earlier, the model is useful for analyzing: The economy’s long run equilibrium Capacity output in the short run However, the economic behavior of households and firms we describe will be useful for the analysis of business cycles 1
Assumptions Assume: 1. Two periods : Both consumption and investment goods produced. 2. A ”representative” household that consumes and saves in period 1, consumes in period 2, and receives dividend income in 1 and 2. 3. A ”representative” firm that produces consumption goods in periods 1 and 2, invests in period 1, and pay dividends to households in periods 1 and 2. 4. The household and firm act competitively, i.e., take market prices (the inter- est rate) as given. In the baseline model we hold employment fixed in order to focus on how the economy divides current output between consumption and investment: We then extend the model to allow for employment determination as well. ”Extended” model is a natural blend of the one and two period models. 2

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Assumptions: Preferences C k consumption in period k = 1 , 2 : Households enjoy utility in periods 1 and 2 1 1 - σ ( C 1 ) 1 - σ + 1 1 - σ β ( C 2 ) 1 - σ : if σ 6 = 1 (1) log C 1 + β log C 2 : if σ = 1 (2) where 0 < β < 1 and σ 0 . β is the household’s ”subjective discount factor” Captures that the household is ”impatient”: everything else equal (i.e. C 1 = C 2 ), values a unit of consumption at the margin more today than in the future. With an infinite horizons, the discount factor keeps the household from putting too much weight on the future. 3
Assumptions:Technology Y k firm’s output in period k ; K k firm capital in period k ; I investment; A k total factor productivity; δ capital depreciation rate Production function for period k : Y k = A k ( K k ) α ; k = 1 , 2 (3) with 0 < α < 1 ( concavity) Law of motion for capital: K 2 = (1 - δ ) K 1 + I where 0 < α < 1 . The firm’s initial capital stock K 1 is exogenous. 4

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Assumptions:Technology (con’t) Cost of investing (adding I new units of capital) I + c 2 ( I K 1 ) 2 K 1 I cost of building I new machines (given we normalize the price of new capital, p k , at unity.) c 2 ( I K 1 ) 2 K 1 cost of installing new machines (”adjustment cost”).
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