## Cost Curves - Average Costs

• Total cost: $c(q) = c_v(q) + F$

• Average Cost: $AC(q) = \frac{c(q)}{q} = \frac{c_v(q)}{q} + \frac{F}{q}$

• That is: $AC(q) = AVC(q) + AFC(q)$

• Notice FC does not depend on $q$, but AFC does depend on $q$.

• Can AFC increase in $q$

## Marginal Cost

• Marginal cost is the change in cost due to change in output

• $c’(q) = \frac{ ∂c(q) }{ ∂q } = \frac{ ∂c_v(q) }{ ∂q }$

## Some relations of Cost Curves

• Marginal cost equals AVC at zero units of output

• Because AVC starts at the origin.
• MC: crosses at minimum points of AC and AVC.

• MC < AC when AC is decreasing and MC > AC when AC is increasing.

## Cost Curves - Example:

• $c(q) = 1 + q^2$

• $c_v(q) = ?$

• $FC(q) = ?$

• $AVC(q) = ?$

• $AFC(q) = ?$

• $AC(q) = ?$

• $MC(q) = ?$

## Cost Curves - do-at-home examples:

• $c(q) = 10 - 0.5(q-2)^2 + (q-2)^3$

• Try with all kinds of functions...

## Cost minimization in two plants

• Possibly two technologies, therefore two cost functions $c_1(q_1) \text{ and } c_2(q_2)$

• Graphical approach

• Use a production-requirement "box"
• Mathematical approach

• $\text{minimize} (C = c_1(q_1) + c_2(q_2))$ subject to: $q_1 + q_2 = q$

• replace $q_2$ by $q-q_1$ and solve $dC/dq_1 = 0$

• Solution: $q_1 \text{ and } q_2$ such that $MC_1(q_1) = MC_2(q_2)$

## Long-run and short-run cost function - Example.

• $q = 50 L^{0.5} K^{0.5}$

• Find long run cost function:

• Total Cost (TC): $c(w,r,q) = (q/25)(wr)^{0.5}$

• $AC(w,r,q) = (1/25)(wr)^{0.5}$

• $MC(w,r,q) = (1/25)(wr)^{0.5}$

## Long-run and short-run cost function - Example.

• $q = 50 L^{0.5} K^{0.5}$

• Find short-run cost function ($K = \bar{K}$):

• $L^{SR} = \frac{ q^2 }{ 50^2 \bar{K} }$

• $c^{SR}(w,r,q) = w \frac{ q^2 }{ 50^2 \bar{K} } + r \bar{K}$

• ${AC}^{SR}(w,r,q) = w \frac{ q }{ 50^2 \bar{K} } + r \frac { \bar{K} }{ q }$

• ${MC}^{SR}(w,r,q) = 2 w \frac{ q }{ 50^2 \bar{K} }$

If for example w = 25 and r = 100:

## Short-run AC (SAC) and long-run AC (SAC)

• Cost curves coincide if fixed level of capital is also LR solution.

## Short-run AC (SAC) and long-run AC (SAC)

• More generally: LR cost curves envelope from below the SR ones.