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Chapter_6_Malthus_to_Solow

# Chapter_6_Malthus_to_Solow - 1 Malthus to Solow We begin...

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Unformatted text preview: 1 Malthus to Solow We begin this chapter by combining the Malthusian model with the Solow model. By this we mean that we give firms in the economy the choice of technology which to use. We then investigate whether the combined model can account for the evolution of international income levels. I. Population Growth Function Before doing this there is the matter of specifying the population growth function for the unified theory. For the Malthusian era, we need the production function to have the property that it is increasing and sufficiently large in the region of the Malthusian living standard. For the post 1900 period of modern economic growth, we need the population growth function to have a slope equal to zero for large living standards. The population growth function for the combined model thus looks as follows: G(c) c t c m N t+1 /N t 2 II. The Combined Theory Let us assume that the economy begins on a steady state corresponding to the Malthus- only model. Namely, consumption is at c m and the economy’s capital stock and population are given by (1) α φ ε φ α δ δ γ / 1 1 / ) 1 ( ) )( 1 ( ) 1 ( +- + + =--- g s sc g sA L N ss t m ss t (2) ss t ss ss t N g s s c K ) )( 1 ( δ +- = In the steady state, the rental price of capital and the wage rate are both constant. This follows from the profit maximizing conditions were each factor is paid its marginal product. Namely, (3) mt mt t N Y w ) 1 ( φ α-- = (4) mt mt kt K Y r φ = . Let us consider the problem of a firm that is considering using the Solow technology. This firm will want to maximize profits. This firm can hire capital and labor at the Malthusian steady state values, w m and m k r . Thus, the firm will want to maximize its profits as (5) st m k st m st t st s K r N w N K A-- Γ- θ θ 1 ) ( Maximizing profits required differentiating by K st and N st and setting the derivatives equal to zero. These are just the usual marginal cost = marginal product results. (6) st st st st st s t N Y N K A w ) 1 ( ) 1 ( 1 θ θ θ θ θ- = Γ- =-- 3 (7) st st st st st s t K Y N K A r θ θ θ θ = Γ =-- 1 1 ) ( We can use (6) and (7) to solve for the ratio of the prices as a function of the labor and capital stocks. This is (8) st st k K N w r θ θ- = 1 Equation (8) allows us to solve for the capital stock as a function of the labor input. (9) st k st N r w K θ θ- = 1 . We can now substitute for K st in the firm’s profits using equation (9)....
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Chapter_6_Malthus_to_Solow - 1 Malthus to Solow We begin...

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