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There is a large empirical literature on this topic, with fairly inconclusive results. In the theoretical literature, for a skeptical view on the effects of stimulus through government spending see Taylor and Wieland (2008) and Cogan et al. (2009) and for a more positive view see Christiano et al. (2009). Freedman et al. (2009) considers a much larger range of fiscal instruments and monetary responses, and obtains a broad range of possible multipliers.
The contributions on fiscal policy under fixed exchange rates include Beetsma and Jensen (2005) and Gali and Monacelli (2008). Muscatelli, Tirelli and Trecroci (2004) examine the closed economy case. All of these papers assume that households face no constraints on smoothing consumption intertemporally.
As summarized in Leeper, Walker and Yang (2009), the empirical literature has a wide range of values for this elasticity. At one extreme, Holtz-Eakin (1994) and Evans and Karras (1994) use state-level data and find that public-sector capital has negative or no effect on private sector productivity. At the other extreme, Aschauer (1989) and Pereira and de Frutos (1999) obtain significant productive effects from public capital, with elasticities in the range of 0.24 and 0.39. In this paper.
The implied shock autocorrelations are ρa = 0.68, ρc = 0.6 and ρΙ = 0.4, and shock standard deviations are σa = 0.018, σc = 0.015 and σΙ = 0.018. The monetary rule parameters required to match the data are δi = 0.7, δπ = 2.0 and ι = 0.5.
Of course this abstracts entirely from the question of how this should be done without inviting corruption and work disincentives.