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I have this

Labor (L)   0   1   2   3   4   5   6   7   8   9   10  11
Output      0   32  72  114 161 204 242 272 292 304 302 288

and I need to estimate function expressions for $TC$, $AVC$, and $MC$.

I have capital $K = 10$, wage $w = 30$, and rate $r = 50$.

I think that total costs are $$ TC = wL + rK = 30 L + 50 \cdot 10 = 30 L + 500. $$ and marginal costs are $MC = TC'(Q)$ but my $TC$ doesn't depend on $Q$. Is it $MC = TC'(L) = 30$ instead?

If I'm calculating the marginal cost ($MC_t = \frac{\Delta TC}{\Delta Output}$) in discrete quantities I get

Labor (L)   0   1   2   3   4   5   6   7   8   9     10    11
Output      0   32  72  114 161 204 242 272 292 304   302   288
MC          -   25  20  19  17  19  21  27  40  67    -400  -57

but these numbers seem very wrong.

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  • $\begingroup$ First, your algebraic calculations seem wrong: for example if you employ $L=2$ instead of $L=1$, you increase total costs by $30$ and quantity by $72-32=40$. So the ratio ("marginal cost on average") would be $30/40 = 0.75$, not $20$. Etc. Second, if you go from $L=9$ to $L=10$ it appears that output starts to get reduced -you have reached the threshold of diminishing returns where marginal product becomes negative -too many seeds in the pot. What meaning would you give to "marginal cost" in such a case? $\endgroup$ – Alecos Papadopoulos Mar 8 '15 at 23:13
  • $\begingroup$ Thanks for your answer. I've by a mistake calculated it with $K=20$, $w=800$, and $r=500$. If I'm asked to estimating the function expressions, shouldn't I just find a function like $TC=ax+b$ and $MC = TC'=a$? $\endgroup$ – Jamgreen Mar 9 '15 at 6:15
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As the OP corrected in the comments, using $K=20, w=800$, and $r=500$, so $TC = 10000 + 800L$, we get , in the $\{Q,C\}$ space enter image description here

This looks pretty text-book like. The fact that there are levels of labor for which output is reduced, tells us that for these levels we have to employ more of the "fixed" factor, capital (which thus stops being fixed).
The function as is, with capital fixed at $10000$, stops having economic sense after $L=9$.

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  • $\begingroup$ Thank you! I can understand that you are plotting the MC but if $TC=10000+800L$, wont marginal cost be $MC = TC'$? But if TC doesn't depend on $Q$, how can I find the derivative? $\endgroup$ – Jamgreen Mar 9 '15 at 9:56
  • $\begingroup$ @Jamgreen The relation is not linear, so it appears you will have to approximate Total Cost as a function of quantity using some polynomial in $Q$. There is a production function implied here, and it is not Cobb-Douglas. $\endgroup$ – Alecos Papadopoulos Mar 9 '15 at 10:06
  • $\begingroup$ From my trendline I get the function $TP = 0.0329x^4 - 1.092x^3 + 8.6677x^2 + 19.626x + 5.1111$. Is this correct? But is it now $TP(L)$ or what? $\endgroup$ – Jamgreen Mar 9 '15 at 10:16
  • $\begingroup$ @Jamgreen If $x$ runs from $1$ onwards it is (approximately) - with $5.11 = K^a \implies 5.11 = 20^a \implies a = \ln (5.11)/\ln (20)$. But only for the specific range. $\endgroup$ – Alecos Papadopoulos Mar 9 '15 at 11:00
  • $\begingroup$ But how can I estimate total cost? It looks linear to me $\endgroup$ – Jamgreen Mar 9 '15 at 11:15

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