Production Theory

Technology and Production

Technology determines how inputs can be transformed intooutputs. Inputs can be generalized to two classes.

Capital K Physical resources, materials, assets, etc.

Labor L Man hours, number of workers, sweat and toil.

Brett Devine Production Theory

Production Functions

A production function expresses the technology by mapping a pair(K , L) to a quantity of output q. Mathematically we have

q = f (K , L)

Production functions have important properties like returns toscale. Isoquants for production functions are like indifferencecurves for utility functions. An isoquant is all the combinations ofK and L that can be combined to produce a specific output q.


Marginal Product and Average Product

The marginal impact of an input on output can be found throughpartial derivatives. If we increase capital by one unit, or if we hireone more person, how many more units will we produce?

Marginal Product of Capital Denoted MPK =f (K ,L)

K

Marginal Product of Labor Denoted MPL =f (K ,L)

L

Sometimes we want to know on average how many units of q weare producing per unit of capital or labor.

Average Product of Capital Denoted APK =f (K ,L)

K

Average Product of Labor Denoted APL =f (K ,L)

L


Common Isoquants

Some common production functions and their isoquants are:

Cobb-Douglass Convex bowed in isoquant.

CES Convex bowed in isoquant, some what specialcurve.

Fixed Proportion L shaped, or kinked isoquants.

Perfect Substitutes Straight, parallel lines for isoquants.


Law of Diminishing Returns and the MRTS

In general many production functions adhere to the Law ofDiminishing returns. If MPK and MPL get smaller as we increaseK and L then they diminish.The slope of the isoquant has an important interpretation, just thethe slope of indifference curves. The slope of the line expresses thetradeoffs between capital and labor that can produce the same q.The ratio of marginal products is called the Marginal Rate ofTechnical Substitution.

MRTSK ,Y =MPL

MPK


Returns to Scale

Returns to scale describes how output changes as we scale theinputs up or down.

Increasing Returns Doubling inputs more than doubles output.

Constant Returns Doubling inputs exactly doubles output.

Decreasing Returns Doubling inputs less than doubles output.

How do you determine the returns to scale?


Checking Returns to Scale

Let q = f (K , L) be our production function. Now let t > 0. Weproceed as follows: multiply all inputs by t, i.e., f (tK , tL). Nowpull the ts out and check the exponent. Suppose q = AK 0.75L0.50

then

A(tK )0.75(tL)0.50 = At0.75K 0.75t0.50L0.50

= t0.75t0.50AK 0.75L0.50

= t0.75+0.50f (K , L)

= t1.25f (K , L)

Since exponent is greater than 1, we have increasing returns toscale.


Profit Maximizing Firms

Firms seek to maximize their profit. If p is the price of the goodthey sell, w the wage rate, and r the rental rate of capital, then

pi = pf (K , L) rK wL

The firm takes p as given and needs to choose K and L tomaximize profit.


Cost

Firms cannot produce for free. They experience several types ofcosts.

Explicit Costs Labor costs: wage rate and Capital costs: rentalrate.

Implicit Costs Economics keeps track of opportunity cost which isimplicit in economic decisions.

Sunk Costs Costs that have already been incurred. You cantrecover them.

Long Run Costs No fixed costs, all costs are variable in long run.


Cost Minimization

Firms that profit maximize also cost minimize.

When maximizing profits, the firm has to payout rK + wL incosts.

Firms find a profit maximizing quantity q to produce andthen need to choose the best (K , L) combo to produce q.

The best (K , L) pair is the cheapest one that still producesq.

The cost function, C (r ,w , q) tells us the minimum possiblecost of producing q units at input prices r and w .


Finding Cost Functions

We find the cost function by solving the Cost MinimizationProblem (CMP). The problem is solved using the Lagrangianfunction. The process is similar to utility maximization with theLagrangian.

min rK + wL

costs

s.t. q = f (K , L)

prod constraint

L = rK + wL+ (q f (K , L))


Necessary Conditions

Take the partial derivatives of the Lagrangian with respect to K , L,and .

L

K= r f (K ,L)

K= 0 (1)

L

L= w f (K ,L)

L= 0 (2)

L

= q f (K , L) = 0 (3)


Necessary Conditions

Manipulating equations 1,2, and 3 from the last slide we can reducethe 3 conditions down to 2 conditions that should look familiar.

MRTSK ,L =MPLMPK

= wr

Tangency Condition

q = f (K , L) Production Constraint Satisfied

Use the above conditions to solve for K and L as functions of(r ,w , q).


Properties of Cost Functions

Cost functions are functions of input prices r ,w and aproduction constraint q.

Cost is increasing in q.

Cost is increasing in (r ,w) together, not necessarilyindividually.

Note: The cost function is frequently called the Total Costfunction and may only be a function of q in some problems.


Expansion Paths

Given r and w as fixed, whenever we give the firm a productionquota q, the cost function will return the minimum cost ofproducing q and will implicity choose the cheapest combination ofK and L. The firms expansion path is the set of cost minimizingtangencies. We can trace out the expansion path be seeing whereall the optimal choices of (K , L) are for all levels of production q.


The Other Cost Functions

There are several important manipulations of the cost functionthat are used for economic analysis.

Marginal Cost This is defined as MC = C(r ,w ,q)q

.

Average Cost This is defined as AC = C (r ,w , q)/q which is justaverage cost per unit produced.


Very Shortrun, Shortrun and Long Run

Decisions made by firms often require careful planning and areaffected by limitations on resources and time. It is important torealize how a firm is capable of reacting to changes in the marketwithin certain time frames.

Very Shortrun Both capital K and labor L are fixed.

Shortrun Only capital K is fixed, labor L is free to be adjusted.

Longrun Neither capital K , nor labor L is fixed, both are freeto move and be adjusted.


Shortrun Average Cost Curves

It can often be easier to analyze costs on a per unit produced basisrather than just the total cost overall. The Shortrun ideamentioned previously can yield us a Shortrun Total Cost functionby fixing k . We define the per-unit produced basis as ShortrunAverage Cost:

SAC =shortrun total costs

total shortrun output=

STC

q

In the Longrun nothing is fixed and so capital and labor areperfectly adjustable.


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Production Theory