# Estimate function expression for TC and MC

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.

• 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? – Alecos Papadopoulos Mar 8 '15 at 23:13
• 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 $MC = TC'=a$? – Jamgreen Mar 9 '15 at 6:15

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
The function as is, with capital fixed at $10000$, stops having economic sense after $L=9$.
• 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? – Jamgreen Mar 9 '15 at 9:56
• @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. – Alecos Papadopoulos Mar 9 '15 at 10:06
• 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? – Jamgreen Mar 9 '15 at 10:16
• @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. – Alecos Papadopoulos Mar 9 '15 at 11:00