# Estimate total variable cost function (TVC) and average variable cost function (AVP) from marginal cost function (MC)

I need to estimate the function expressions for $TVC$ and $AVC$ from the $MC$ function for different amounts of quantity $(Q)$

From the solutions I have, that

For $Q \leq 50$

$MC1(Q) = -Q + 100$

$TVC1(Q) = -0.5 Q^2 + 100Q$

$AVC1(Q) = -0.5Q+100$

$\\$

For $50 < Q \leq 100$

$MC2(Q) = 50$

$TVC2(Q) = 50Q + 1250$

$AVC2(Q) =50 + \frac{1250}{Q}$

$$\\$$

For $Q > 100$

$MC3(Q) = 0.5Q$

$TVC3(Q) = 0.25 Q^2 + 3750$

$AVC3(Q) = 0.25Q + \frac{3750}{Q}$

How can I estimate the constants in $TVC2(Q)$ and $TVC3(Q)$?

You lost a constant at $$TVC(Q)=\int{MC(Q)}dQ$$ because for any $\alpha \in \mathbb{R}$ the derivative of $TVC(Q) + \alpha$ will be $MC(Q)$.
If there is no quasi-fixed cost then $TVC(0) = 0$. From this and by calculating $TVC(Q)$ for $Q \leq 50$ you will get the value for $TVC(50)$. (Seems to be 3750.) Because $TVC(Q)$ is a continuous function $$\lim_{Q \to 50^{+}} TVC(Q) = TVC(50)$$ where by $Q \to 50^{+}$ I mean that $Q$ converges to 50 from above (you take the right hand side limit). This, together with $$TVC(Q)=\int{MC(Q)}dQ$$ should give you $TVC(Q)$ for $Q > 50$.
• I am not sure what you denote by $TVC1$ and I don't know where 100 came from. – Giskard Jul 7 '15 at 4:43