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Consider a firm with the production function $y=CL^{a}K^{b}$, where $C>1$, $a>0$, $b>0$. Write down the conditions under which this production function exhibits: i) increasing returns to scale, ii) constant returns to scale, and iii) decreasing returns to scale. Calculate the marginal product of $L$ , the marginal product of $K$, and the marginal rate of technical substitution. Write down the conditions under which this technology exhibits diminishing marginal product in both factors. Write down the conditions under which the marginal rate of technical substitution is diminishing.

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  • $\begingroup$ Hi welcome to SE. could you provide the work you have already done so that we can help see where you might have gone wrong? What part are you stuck on? $\endgroup$ – Brennan Oct 1 at 23:08
  • $\begingroup$ I don't really know how to go about starting this problem. In examples I have substituted an a or an m in so that it is the factor and I can bring it out to see what happens to the function. $\endgroup$ – user24420 Oct 1 at 23:23
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Returns to scale are computing as: $y(\theta L, \theta K) = C(\theta L)^{a}(\theta K)^{b}$. Try manipulating this such that you get $\theta^{r}$ where $r$ is some number. Then interpret that, increasing both inputs by a factor $\theta$ leads to what kind of change? This is clear after simplifying down to $\theta^{r}CL^{a}k^{b} = \theta^{r}y$: $$y(\theta L, \theta K) = C(\theta L)^{a}(\theta K)^{b}$$ $$=\theta^{a+b}CL^{a}K^{b} = \theta^{a+b}y$$ Now, for what $a+b$ do we IRS, CRS, and DRS?

Then, what is the definition of the marginal product? Just a derivative with respect to that particular good. Take the marginal product of labour using the original production function $y(L,K) = CL^{a}K^{b}$ for example: $$\frac{\partial y}{\partial L} = aCL^{a-1}K^{b}$$ The same process follows for the marginal product of capital, $K$.

Then the marginal rate of technical substitution is the ratio of these: $$MRTS_{LforK} = \frac{MP_{L}}{MP_{K}}$$ After substituting the two marginal products into the above, when would this be diminishing? That sounds like something that the second derivative could answer for you. This would also be a function of $a$ and $b$. For what values of $a$ and $b$ would it be negative? positive? This would perhaps be the next step.

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