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Theory of the Firm

- Firms want to maximize profit
- This implies minimizing cost
- Need to identify underlying technological relationships between inputs and outputs.

Factors of ProductionInputs

- Broadly – labor, land, raw materials, capital
- Motion picture studio:
- producers, directors, actors, lots, sound stages, equipment, film.
- Electricity Generator
- Managers, technicians, coal, generation equipment, (clean air)

Specification of Technology – one output

- Production function: f(x) = {y such that y is the maximum output associated with –x}
- Cobb Douglas f(x1,x2) = x1ax2(1-a)
- Leontiff f(x1,x2) = min(ax1,bx2)
- CES f(x1,x2) = [ax1r + bx2r]1/r
- Linear f(x1,x2) = x1 + x2

Sketch the Cobb Douglas, Leontiff, and Linear Isoquants

Specification of Technology – one output

slope a/b

input 2

Q(y2)

Q(y1)

input 1

- Leontiff isoquant
- f(x1,x2) = min(ax1,bx2)

Technical Rate of Substitution

- The technical rate of substitution is the amount that you need to adjust one input in order to keep output constant for a small change in another output. This is equivalent to the slope of the isoquant.

If the production function is f(x1,x2) what is the technical rate of substitution?

Elasticity of substitution

- The elasticity of substitution measures the curvature of the isoquant:

s = d(x1/x2)/(x1/x2)

d(TRS)/TRS

The higher s, the less curved the isoquant is, and easier it is to substitute between inputs.

Elasticity of substitution

s = ∞

Input 2

s = 1 (Cobb Douglas)

s = - ∞

Input 1

Constant Elasticity of Substitution

f(x1,x2) = [ax1r + bx2r]1/r

s = 1/(1-r)

Example

- Let f(x1,x2) =
- Find the TRS
- Then, try to find the elasticity of substitution, using the ln formulation.

Returns to Scale

- Constant returns to scale: a doubling of inputs will result in a doubling of output.
- f(tx) = tf(x) the production function is homogeneous of degree 1.
- Increasing returns to scale
- f(tx) > tf(x)
- example: fixed costs
- Decreasing returns to scale
- f(tx) < tf(x)
- example: a fixed input

Returns to Scale: Examples

- Bausch and Lomb. A 12-oz bottle of saline solution costs $2.79. A 1-oz bottle of eye drops costs $5.65.
- People’s Express. Very successful, attributed partly to management practices (minimal hierachy, training, profit sharing, performance pay). Grew from 300 to 5000 employees. Active involvement became difficult, more hierarch necessary, output increased less than input.
- Oil shippers: unlimited liability in case of a spill. Therefore a small firm with only one ship is preferable.

Short run vs. Long run

- In the short run some inputs may be fixed.
- In the long run, we generally consider all inputs to be variable.
- Example: Capacity

Profit Maximization

- Basic assumption of Economics. Is it right?
- The firm takes actions that will maximize the total revenues – costs.

Max R(a) – C(a)

F.O.C. R’(a) = C’(a)

MR = MC

Profit Maximization - examples

- The level of output should be chosen so that the cost of producing the last unit of output is equal to the revenue from that unit.
- In 1962 Continental Airlines filled only 50% of certain flights. It considered dropping some of these flights, but each flight had a (marginal) cost of $2000 and revenue of $3100.

Profit Maximization

- Revenue = the price of what is sold X the amount that is sold
- Cost = the price of the inputs X the amount of input used.
- Technological Constraints
- Market Constraints
- Assume that the firm is a price-taker (competitive firm)

Profit Maximization- price taker

- Max pf(x) –wx
- foc characterizes profit-maximizing behavior.
- “value marginal product = its price”
- What is the requirement on f(x) for a maximum to exist? What is the SOC?

Profit, supply, and demand functions

- demand function

x(p,w) = argmax pf(x) – wx

- Supply function

y(p,w) = f(x(p,w))

- Profit Function

p(p,w) = py(p,w) – wx(p,w)

Profit, supply, and demand functions

- A function is homogeneous of degree n if f(tx)=tnx
- If a function is homogneous of degree 0, then doubling all its inputs doesn’t change the output.
- If a function is homogenous of degree 1, then doubling all inputs doubles the outputs.

Profit, supply, and demand functions

- Consider the demand function x(p,w). What happens if you double all prices?
- Consider the profit function, p(p,w). What happens if you double all prices?
- What about the supply function?
- What does this say about behavior under inflation?

Properties of the Profit Function

p(p,w) = py(p,w) – wx(p,w)

- Increasing in output prices; decreasing in input prices
- Homogeneous of degree 1 in prices.
- Convex in p (!)

These properties follow only from the assumption of profit maximization.

Alternative proof that the profit function is convex

Let y max profits at p, y’ at p’, and y’’ at p’’, where p’’ = tp + (1-t)p’

Then

p(p’’) = p’’y’’ = (tp + (1-t)p’)y’’

= tpy’’ + (1-t)p’y’’ < tpy + (1-t)p’y’

= tp(p) + (1-t)p(p’)

Why is this true?

Profit function

Suppose the price of output is randomly fluctuating. Is it desirable to stabilize this price?

Jensen’s inequality:

p(E[p]) < E[p(p)] for any convex p.

Thus, firms do better when price fluctuates! Because they change their production plan accordingly.

When might this not be true?

Hotelling’s Lemma

- Use the envelope theorem to find

Hotelling’s Lemma

- Use the envelope theorem to find

Example

- Consider a firm with 3 inputs – x, ec,enc with prices w,pc,pnc
- Carbon emissions are equal to ec
- Carbon is taxed at level t
- Thus total price of ec is pc+t
- Now consider technical change that reduces the carbon intensity.
- Let technical change a reduce the carbon intesity from 1 to (1-a)

Example-continued

- After technical change the total price for ec would be pc+(1-a)t
- The profit function is p(w, pc+(1-a)t,pnc)

Cost Minimization

- In order to maximize profit, a firm must be minimizing the cost of producing its output.
- Cost minimization is an alternate characterization of price-taking firms.

Max py – c(w,y) Where

c(w,y) = min wx

s.t. y = f(x)

- Cost minimization is equally valid for other types of firms

Cost Minimization

Min wx

s.t. f(x) = y

Lagrangian

L(l,x) = wx – l(f(x) – y)

F.O.C.

wi-ldf(x*)/dxi = 0

f(x*) = y

Cost Minimization

Divide the ith constraint by the jth

df(x*)/dxi = wi

df(x*)/dxj wj

Economic rate of substitution = TRS

Cost Minimization

Input 2

f(x1,x2) = y

Input 1

isocost line. slope = -w1/w2

How does this differ from the utility maximization problem?

Example

- Graph the demand for factor x1 as a function of its price w1 for the 3 production functions below.

x2

x2

x2

x1

x1

x1

Kuhn-Tucker Theorem

- In order to solve constrained optimization problems when there might be a corner solution, we need to use the Kuhn-Tucker Theorem.

Cost Minimization

df(x*)/dxi = df(x*)/dxj

wi wj

Marginal product per $

One dollar invested in xi increases output by df(x*)/dxi

wi

To minimize cost you must equalize the rates of return on each input.

Cost minimization - examples

- parking lots: tall versus expansive

Properties of the Cost Function

- Increasing or decreasing in w?
- Homogeneous of degree ? in w?
- Convex or concave?

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