Production and Cost in the Short Run

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Today's Agenda

Part 1: Total and Unit Cost Curves

Part 2: Relationship of Production to Cost

Expansion paths and total cost in the SR

Fixed and variable costs

Average costs

Marginal cost

Relationship between Average & Marginal

Production and Cost with One Input

Returns to Scale with Two Inputs

Economies and Diseconomies of scale

Producer Theory

L^*(w,r,y)
K^*(w,r,y)

Exogenous Variables

Endogenous Variables

technology, f()

level of output, y

factors used

total cost, c(y)

Cost Minimization

Production Set

Choice

Rule

factor prices (w, r)

y^*(w,r,p)

optimal level of output, y*

Profit Maximization

output prices

Total Revenue R(y)

Total Cost, c(y)

Last Time

Today

Wednesday

Example

f(L,K) = L^\frac{1}{4}K^\frac{1}{4}

We found before that

L^*(y) = \sqrt{\frac{r}{w}}y^2
K^*(y) = \sqrt{\frac{w}{r}}y^2
c(y) = wL^*(y) + rK^*(y)
=w\sqrt{\frac{r}{w}}y^2 + r\sqrt{\frac{w}{r}}y^2
=2\sqrt{wr}y^2
=\sqrt{wr}y^2 + \sqrt{wr}y^2

The Firm in the Short Run

In the "short run" at least one input is fixed.

We'll use the metaphor that that input is capital (K).

Short-Run Costs

L^*(w|y,\overline{K})

Exogenous Variables

Endogenous Variables

technology, f()

level of output, y

fixed inputs

prices of variable inputs

Optimal Choice of Inputs

Production Set

Choice

Rule

c_s(y|\overline{K},w)

FOR VARIABLE INPUT(S)

COST FUNCTION

CONDITIONAL DEMAND

Simplifying Assumption: If there are only two inputs,
there there's no "optimal choice" to be made:
there's only one choice of labor if y and capital are both fixed.

Conditional Demand
in the Short Run

If there's only one variable input,
it's perfectly inelastic -- there's only one choice!

Our continuing example

f(L,K) = L^\frac{1}{4}K^\frac{1}{4}

If K is fixed at \(\overline K\), then

L^*(y) = \frac{y^4}{\overline K}
y = L^\frac{1}{4}\overline K^\frac{1}{4}

Solve for L to get

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Short-Run Total Cost of y Units

c_s(y,\overline{K}) = wL(y,\overline{K}) + r\overline{K}

Variable cost

"The total cost of producing y units in the short run
is the variable cost of the required amount of the input that can be varied,
plus the fixed cost of the input that is fixed in the short run."

Fixed cost

Rejoining our friend

f(L,K) = L^\frac{1}{4}K^\frac{1}{4}

We found before that

c(y) = wL^*(y) + r\overline K
L^*(y) = \frac{y^4}{\overline K}
c(y) = \frac{wy^4}{\overline K} + r\overline K

Therefore SR total costs are

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Relationship between
Short-Run and Long-Run Costs

c_s(y,\overline{K}) = \text{Cost if capital is fixed at }\overline{K}
c_s(y,K^*(y)) = \text{Cost if capital is fixed at optimal }K \text{ for producing }y

What conclusions can we draw from this?

c_s(y,\overline{K}) \ge c(y)
= c(y) \text{ -- i.e., the long-run total cost of }y
c(y) \text{ is the lower envelope of the family of }c_s(y)\text{ curves}
\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c(w,r,y) = 2\sqrt{wr}y^2
c_s(w,r,y) = w\frac{y^4}{\overline{K}} + r\overline{K}
\text{Long-run and Short-Run Total Cost Functions: }
\text{Now let's fix }w=r=10\text{ and }\overline K = 25:
c(y) = 20y^2
c_s(y) = 250+0.4y^4

LONG RUN

SHORT RUN

Fixed and Variable Costs

Fixed and Variable costs

c_s(y,\overline{K}) = wL(y,\overline{K}) + r\overline{K}

Cost of variable inputs

Cost of fixed inputs

c(y) = c_v(y) + F

From input decision ("Cost Minimization" chapter):

When thinking about output ("Cost Curves" chapter):

Fixed Costs (F): All economic costs that don't vary with output.

Variable Costs (VC): All economic costs that vary with output

Opp. costs, sunk costs, "quasi-fixed" costs

\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c(y) = 20y^2
c_s(y) = 250+0.4y^4

LONG RUN

SHORT RUN

F
c_v(y)

Average Costs

\text{Total Cost (TC)}: c(y) = c_v(y) + F
\text{Average Cost (AC)}: \frac{c(y)}{y} = \frac{c_v(y)}{y} + \frac{F}{y}

Fixed Costs (F)

Variable Costs (VC)

Average Fixed Costs (AFC)

Average Variable Costs (VC)

Average fixed, variable, and total costs

\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c(y) = 20y^2
c_s(y) = 250+0.4y^4

LONG RUN

SHORT RUN

Comparing Total and Average Costs

Fixed costs are constant;

average fixed costs start at infinity and decrease

Variable costs start from zero and increase;
so do average variable costs.

Short-run total costs start at F and increase by VC;
SAC start out like AFC (since AVC = 0)
and end up like AVC (as AFC approaches 0).

Long-run total costs start from zero and increase;
so do long-run average total costs. This is because we're assuming there are no other fixed costs (e.g. opp. costs)

Marginal Costs

\text{Total Cost (TC)}: c(y) = c_v(y) + F
\text{Marginal Cost (MC)}: c'(y) = c_v'(y) + 0

Fixed Costs (F)

Variable Costs (VC)

(Marginal cost  = marginal variable cost)

\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c(y) = 20y^2
c_s(y) = 250+0.4y^4

LONG RUN

SHORT RUN

LMC(y) = c'(y) = 40y
SMC(y) = c'_s(y) = 1.6y^3

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Relationship between Average and Marginal Costs

Marginal cost tends to "pull" average cost toward it:

MC > AC \Rightarrow AC \text{ increasing}
MC = AC \Rightarrow AC \text{ constant}
MC < AC \Rightarrow AC \text{ decreasing}

Marginal grade = grade on last test, average grade = GPA

\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c_s(y) = 250+0.4y^4

SHORT RUN

SMC(y) = c'_s(y) = 1.6y^3
SAC(y) = \frac{c_s(y)}{y} = \frac{250}{y} + 0.4y^3

Production and Cost with One Input

The Curvature of the Production Function affects the Curvature of the Total Cost Curve.

Decreasing marginal returns => increasing marginal cost

Constant marginal returns => constant marginal cost

Increasing marginal returns => decreasing marginal cost

How does this work with more than one input?

Think returns to scale.

Returns to a Single Input

  • Increasing marginal product: MPL is increasing in L
  • Constant marginal product: MPL is constant in L
  • Diminishing marginal product: MPL is decreasing in L

Returns to Scale (Scaling all inputs.)

  • Increasing returns to scale: doubling all inputs more than doubles output.
  • Constant returns to scale: doubling all inputs exactly doubles output.
  • Decreasing returns to scale: doubling all inputs less than doubles output.

How do the attributes of the production function 
(diminishing marginal product, returns to scale)
affect average costs?

Intuitively: if you have diminishing productivity, you'll have increasing marginal costs and eventually increasing average costs.

Relationship between Production Function and the Curvature of Long-Run and Short-Run Costs

  • If the production function has diminishing \(MP_L\), the short-run cost curve will get steeper as you produce more output (have increasing marginal costs)
  • If the production function has decreasing returns to scale, the long-run cost curve will get steeper as you produce more output.
  • What about for constant returns to scale? Increasing returns to scale? 

Economies and Diseconomies of Scale

Returns to Scale

Has to do with the production function

Economies of Scale

Has to do with cost curves

Increasing Returns to Scale:
double input => more than double output

Decreasing Returns to Scale:
double input => less than double output

Always deals with the long run

Can occur in both the long run and short run

Economies of Scale:
increasing output lowers average costs

Diseconomies of Scale:
increasing output raises average costs

In the short run, the cost of producing y units of output is the sum of the variable cost wL(y) and the fixed cost rK.

Unit costs (average costs and marginal costs) are direct derivations from the total cost functions.

The curvature of cost curves is related to the nature of the production function: diseconomies of scale and rising marginal costs come from diminishing marginal returns to inputs.

Conclusion and Next Steps

Next: profit maximization, output supply, and input demands for competitive firms.

Econ 50 | Production and Cost in the Short Run

By Chris Makler

Econ 50 | Production and Cost in the Short Run

Having derived the total cost of producing y units in the short run and long run, we take a closer look at total cost (fixed and variable) and unit cost curves (average and marginal). We then examine the relationship between the nature of production processes and their associated cost curves.

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