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Allen elasticities assumed assumption capita terms capital stock Cobb-Douglas costate variables country's current consumption demand functions depreciation derive developing countries developing economy domestic labor duality theory dynamic shadow prices elasticities of intensity elasticities of substitution Equations 3.54 extraction rule fixed inputs foreign company foreign inputs functional forms given homogeneous commodity Hotelling 1931 imported inputs initial resource conditions intertemporal consumption intertemporal model intertemporal problem inverse elasticities investment in mining investment in non-mining long-run marginal revenue marginal utility mineral exports mineral revenues mineral-exporting mining and non-mining mining demand mining industry mining structures mining technology natural capital non-mining capital non-mining technology non-renewable objective function obtain opportunity cost optimal rate output planners planning horizon price of non-mining priori restrictions production technologies rate of extraction rate of interest reproducible capital resource depletion short-run social welfare function static shadow prices terminal stocks utility function variable cost function variable inputs world price Zambia zero profit conditions