29 32 A Benchmark Model of Growth and Structural Transformation We use the

29 32 a benchmark model of growth and structural

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3.2 A Benchmark Model of Growth and Structural Transformation We use the model of the previous section as the starting point for our analysis of structural transformation in the context of the growth model. 3.2.1 Set up of the Benchmark Model As in the previous section, we assume an infinitely lived stand–in household that has prefer- ences characterized by (1) and is endowed with one unit of time and an initial capital stock. Di ff erent than in the previous section, we now assume that C t is a composite of agricultural consumption ( c at ), manufacturing consumption ( c mt ) and service consumption ( c st ): C t = ω 1 ε a ( c at - ¯ c a ) ε - 1 ε + ω 1 ε m ( c mt ) ε - 1 ε + ω 1 ε s ( c st + ¯ c s ) ε - 1 ε ε ε - 1 (12) where ¯ c i , ω i 0 and ε > 0. The functional form (12) is a parsimonious choice that allows us to capture two features on the demand side that are potentially important for understanding the reallocation of activity across these three sectors: how the demand of the household reacts to changes in income and in relative prices. In particular, the presence of the two terms ¯ c a and ¯ c s allows for the period utility function to be non–homothetic and therefore the possibility that changes in income will lead to changes in expenditure shares even if relative prices are constant. The parameter ε influences the elasticity of substitution between the three goods, and hence the response of nominal expenditure shares to changes in relative prices. Note, however, that in the above specification the elasticity of substitution is not equal to ε because it also depends on the non–homotheticity terms. We generalize the previous model to allow for four Cobb–Douglas production functions, one for each of the three consumption goods and one for the investment good. Formally, the 30
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production functions are given by: 20 c it = k θ it ( A it n it ) 1 - θ , i ∈ { a , m , s } (13) X t = k θ xt ( A xt n xt ) 1 - θ (14) There is a tradition in the literature of working with only three production functions, with the assumption that all investment is produced by the manufacturing sector. Under this assumption, the output of the manufacturing sector can be used as either consumption or investment whereas the output of the other two sectors can only be used as consumption. We have not adopted this specification for two reasons. First, despite the apparent reasonableness of the claim that in- vestment is to first approximation produced exclusively by the manufacturing sector, it turns out that this is not supported by the data. Moreover, such an assumption is becoming increasingly at odds with the data over time, due at least in part to the fact that software is both a sizeable and increasing component of investment, and most software innovation takes place in the ser- vice sector. In fact, total investment has exceeded the size of the entire manufacturing sector in the US since 2000. The second reason for considering a separate investment sector derives
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