Note the useful relationship between marginal and average products: where Average Product is at a maximum, the Marginal Product will be equal to this Average Product, with the Marginal curve crossing the Average curve from above. Has the further useful property that a = elasticity of K i.e. But since the second partial derivative ofĮach input is log(Q) = log( K^a*L^b) = a*log(K) + b*log(L) If the power terms add to 1 as above, there are constant returns to scale, i.e.,ĭoubling the inputs doubles the output. Take, for example, the Cobb-Douglas function Returns to a single input will be increasing, decreasing, orĬonstant according to whether the exponent on that input is greater than, less Returns to scale says nothing about whether or not there may be diminishing The fact that a production function shows constant or increasing Increase in Q, when L goes from 100 to 101, is an increase not by 300, as That the elasticity prediction is for a 1.5% It will be seen that theĮlasticity prediction is a little too pessimistic in this case. Thus if L is increased by 10%, Q is increased by approximately 15%. This implies that,įor any level of K, if L is increased by 1%, Q is increased by approximatelyġ.5%. (Note how taking logs allows the non-linearĬobb-Douglas form to be linearized, and thus be one that we can estimate byĬoefficients for a linearized Cobb-Douglas function are also its estimated Represents the productive elasticity of that input. The exponent for any input term in a Cobb-Douglas function Time from 2 to 4, then Q again more than doubles, from 5.657 to 16. Note in the above that when L and K double from 1 to 2, that Now, we can verify that when b = 0.7 and c = 0.8, so that b +Ĭ = 1.5 > 1, that we will have increasing returns. We will find that the value of Q also doubles it goes to 8. Now let us double the inputs, so that K = L = 4. Since b + c = 1, this implies constant returns To 1, this implies Increasing, Decreasing, or Constant returns to scale,Įxample: Q = a*K^b*L^c = 2*K^(0.5)*L^(0.5) According to whether the Sum ofĮxponent terms in a Cobb-Douglas function is Greater Than, Less Than, or Equal The advantages of using the Cobb-Douglas function forĮstimates of production are many - and it is widely used in empiricalĬobb-Douglas # 1. This sectionconcentrates more on theory than on estimation, but we
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