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https://www.chegg.com/homework-help/questions-and-answers/given-long-run-total-cost-function-tc-q-wrq-w-r-w-wage-r-rental-rate--studying-total-cost--q65688612

Given

$TC (Q) = \frac{wrQ}{w+r} \ \ \ \ \ ....... (1)$

Average total cost is given by

$ATC (Q) = \frac{wr}{w+r} \ \ \ \ \ .......... (2)$

Since average total cost is independent of Q. It means that by increasing the output the average cost of production remains constant. This means that the production function has a constant return to scale.

Part B

Using Shephard’s Lemma, conditional input demand is given by

$L(w,r,Q)=\frac{\partial TC}{\partial w}$

and $K(w,r,Q)=\frac{\partial TC}{\partial r}$

so taking first partial differentiation of eq (1) w.r.t w

$L(w,r,Q)=rQ(w+r)^{-}-wrQ(r+w)^{-2} \\ \\ \Rightarrow L(w,r,Q)= \frac{r^2}{(w+r)^2} \ \ \ \ \ .......... (3)$

Similarly, conditional input demand of input K

$K(w,r,Q)=wQ(w+r)^{-}-wrQ(r+w)^{-2} \\ \\ \Rightarrow K(w,r,Q)= \frac{w^2}{(w+r)^2} \ \ \ \ \ .......... (4)$

Part C

Production function Q= F(L,K)

We have a a ch? 2 L = - L = 2. (w +90) لا w + 1)² 9с. and 0 a a 2 را W+)2 (1+2 Put 31% - V So L = @ (v + 1)² => (1+v)? lot ol

K 2 V = 2 60 Ferom egr G and & 2 @ k)? -> - ) Jo - SK - VI L & SK an beth side we get squering a = (JL + FK)? Hence 0 = F (L,

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Link BD consists of a single bar 30 mm wide and 12 mm thick. Knowing that each pin has a10 mm diameter, determine the maximum value of the average normal stress in the link BD if θ = 15 degrees.

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