Lecture 15: Cost Minimization

TOPIC 1) Isocost Curve

Cost function

• c(q)= rK(q) + wL(q)

• Note c(q), K(q), and L(q) do not mean multiply by q but rather that these variables are a function of output.

Accounting cost versus economic cost

• Accounting cost refers to explicit out-of-pocket expenses.

• Economic cost includes both explicit expenses and implicit opportunity cost (what you give up to run a business)

• Economic cost > accounting cost

• Economic profits < accounting profits

Why would a firm remain even if there economic profits are 0

Because it means the owners are covering all production costs and earning exactly what they could have made at the next best alternative venture.

Do all points on the isoquant curve (various combinations of K & L that yield the same output) have equal costs associated with them?

No! Take a look at this example, note we know W and R are fixed (perfect competition)

Isocost Curve

Curve showing all possible combinations of K and L that can be purchased at a given cost.

Principles of isocost curve

• there are many isocost clines, one for each value of overall cost

• higher lines reflect higher costs

• lower lines reflect lower costs

• Not a constraint! Just measures cost

Scale expansion path

Isocosts and Isoquants (fitting them together)

we can combine them to determine the cost of production for a firm. The cost will depend on whether we are considering the short run (K fixed, L variable) or the longrun (both variable.)

Topic 2: Short Run Cost

• K is fixed let K= K*

• At point A, this lower-cost option would be more optimal for the firm, but K is their only option in the SR so instead they produce at point b

Short run total cost function

• The relationship between TC & q depends on costs arising from fixed and variable inputs.

• SATC= rk* + wL(q)= SRFC + SRVC

EXAMPLE: Calculate SRTC given SRTC given that q= f(K,L)= K x L, Ket k*=8, let w= $4, let r=$1. And illustrate this on a graph

Topic 3: Long-run Costs

What differentiates LR from SR

In this instance, firms can change both K & L. THe firm will end up choosing the lowest cost (K,L) combination to produce q*.

Long run cost optimal point graphically

Choose K and L to be on lowest isocost line that produces q*.

Lung run cost optimum mathematically

Minimize cost subject to the production constraint so..

slope of isocost= slope of isoquant

-w/r=-MRTS= -MP_L / MP_K

Rule for minimizing profit (alternate equimarginal principle and intuition behind it.)

Why is it that SRTC > LRTC

• Short un K is usually not the cost-minimizing K.

• Optimal K < K*, cost at optimal K <cost at K*

Graphical depiction of above

How do you find the Long run total cost function

Use lagrandian

LRTC function example: min rK + wL st. q= f(K,L)

EX2: Longrun cost function K* q= K x L and W= $4 and r= $1

Using the above equation compare SRTC to LRTC for q=9, 25, q= 16 with a focus on k