I try to build a theoretical model to support my empirical analysis.
I have 3 industries (let's say, agriculture, manufacture, service). The objective function for each industry is a CES function of capital and labour.
What I want to do is to derive a relationship between the growth rate of productivity and labour. From other articles I know that under general equilibrium, capital-labour ratio must be equal across all industries (and did not give explanations). Using this assumption, I can obtain my results.
However, some people asked me why is that? That is to say, i.e. what if the capital-labour ratio in manufacture is larger than that in agriculture?
Kaizhao,
Assuming mobility of factor in a two sector Ricardian model, if the two profit rates are different say between the Agriculture and an another industry, then the entrepreneur will react to create a uniform rate of profit. The competitive mechanism is the need to equalize the profit rates. What the entrepreneur will do is to move capital to where the rate of return is higher, and slow down capital where the return is lower.
For a three sector model say of corn, gold and machines, one needs a uniform capital-labor ratio if the labor theory of value should hold. If you are interested in this type of argument, then a good reference is “Ricardo and the Theory of Value, Distribution, and Growth,” by Giovanni A. Caravale and D. Tosato, Routledge and Kegan Paul, 1980.
Hello,
I will refer only on your last statement:
" However, some people asked me why is that? That is to say, i.e. what if the capital-labour ratio in manufacture is larger than that in agriculture? "
If capital labor ratio is larger in manufacture then agriculture it means that manufacture is more capital intensive then agriculture. Meaning, that due to technological advance machines are substituting men for equal or greater productivity result.
Look up Solow-Romer Model of endogenous growth it may hold the answers you seek.
Lall B. Ramrattan Thank you.for your comments. I will try to read the articles you recommended.
I also obtain the derivatives with respect to capital and labour to get the labour wage and capital rents, then I can know some information about profits across each industry.
However, to derive (dy/dL) and (dy/dk) is already a kind of optimisation within a specific industry. For example, I can use such method to get labour wage and capital rents in agriculture and manufacture, then I find the profits in manufacture is higher than agriculture. Since they are already optimum point, is there any forces which could promote the flow of labour and capital?
Damir Magaš Thank you for your comments.
If the capital-labour ratio in manufacture is higher than that in agriculture, then manufacture will become more capital intensive. Then more and more capital are substituting labour, do you mean that then the gap of capital-labour ratio between manufacture and agriculture should be larger?
The gap will depend on the marginal rate of substitution of the factors involved. General equilibrium is pareto optimal meaning there is no better way to redistribute the factors. A change in prices of labour or capital will redistribute them equally across all industries in a perfectly competitive market. This is never the case in reality. The force that will move capital or labour from one sector to another depends on the elasticity of substitution of the factors, will one unit of labour added result in a 1+n of output or will it result in 0,75+n of output. The factors will alway seek an increasing return to scale i.e. one unit of labour resulting in a 1,3 +n increse in output. Your CES model presumes that labour and capital are perfect substitutes and thus they are equaly distributed throught the economy i.e. add one labour, now you need 1 capital less for constant output of K, while in reality it is not so.
At least this is the way I understand it.
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