Efficient Combination of Labor and Capital

  1. Isoquants
 Marginal Products

Consider a general production function:

Q = F(K,L).

Marginal products of K and L:




An isoquant is a curve or locus of points (K,L), along which the output level remains constant. It shows different combinations of capital and labor inputs required to produce a given output.

It is a concept analogous to that of an indifference curve in consumption. Accordingly, it has properties similar to those of indifference curves.

(i) Recall that indifference are negatively sloped everywhere. This is because human beings are never satiated.

(ii) Isoquants are positively sloped in some regions, i.e., MP of some inputs may be negative.

The slope of an Isoquant = Marginal rate of technical substitution (MRTS) of K for L, i.e.,



 As ΔL becomes infinitesimally small, the chord eventually approaches the tangent line at the initial point (before the change).

In this sense, MRTS is the slope of an isoquant.

Increased use of K reduces the number of unskilled jobs, but the remaining jobs require more skills.

Firms hire skilled workers. Gradually, boring jobs and repetitive tasks tend to disappear, and are replaced by high-tech jobs that require skills.



  2. Iso-Cost Curves
 Definition An isocost curve is the locus of different combinations of capital and labor inputs that yield the same cost. Thus, along an isocost curve, the total cost is fixed. 
Iso-cost curve

 A given isocost curve is expressed as:

Co = wL + rK.

The isocost curve can be solved for the second input, capital. That is,

K = Co/r - (w/r)L.

The isocost curve is negatively sloped, and its slope is the wage-rent ratio, w/r.

Wage-rent ratio 

Wage-rent ratio is the slope of an Iso-cost curve.

Wages and rents differ between countries.

In large countries wages and rents differ between regions. This is an indication that factors do not move freely.

In this situation, firms tend to build factories in regions where wages and rents are lower.

For example, Toyota America has assembly plants in five cities: Huntsville, AL; Georgetown, KT; Princeton, IN; San Antonio, TX; and Buffalo, WV. 

Similarly, Mitsubish has a TV production plant in Mexico, rather than in the US, because of NAFTA.







  3. Least Cost Technique

 A firm's problem is to choose output Y and labor (L) and capital (K) inputs to maximize profit,

 π = pY - (wL + rK),

subject to a production constraint.

This problem can be analyzed in two stages.

Choose optimal level of output. Assume first that somehow the firm knows its desired output level.

Then revenue pY is fixed. Profit maximization requires that the production cost be minimized.

Accordingly, it is necessary for the firm to choose cost minimizing input combinations to produce a given level of output.

 condition for least cost

The firm chooses the point where the isoquant of the desired outupt and the isocost curves are tangent each other. The two curves must have the same slopes, or

MRTS = w/r



  4. SR vs LR Expansion Path
   EP is the locus of cost minimizing input combinations chosen as the desired output level increases. This is analogous to the income consumption curve in the demand theory.
   An EP is obtained by tracing the tangency points between isocost curves and isoquants as output increases. This path depends on the wage-rent ratio.



Rising wage

Black Death was a plague pandemic that devastated Europe in 1348 AD. It began with the mongol siege of Caffa on the Crimean peninsula in 1345. One ship from Caffa arrives in Constantinople in 1347, and the richest city on the planet lost 90% of its population. Caffan ships docked in Sicily and in Genoa, spreading the disease to the rest of Europle.

During the 4 year period (1348-52), the European population shrank by one-third.

At least 20 million people were killed.

Wage rate increased (15-25% in England). In Florence, the wage rates more than doubled within a few years.

Gender equality: Before Black Death, elder son inherited property. After BD, women, as well as all sons, inherited property. Workers became more mobile. Black Death terminated serfdom (peasants were tied to land) in Europe. Due to labor shortages, firms began to explore ways to develop more capital-intensive processes.


It appears that the seed of industrial revolution in Europe was planted by the Black Death. The sudden rise in the wage rate encouraged Europeans to find various ways to invest in capital inputs (e.g., machinery). But this process was slow.



  1. Long Run production cost

Recall that in the SR, a competitive firm can adjust only the variable input, say labor. But in the long run, a firm can change the plant size or capital input.

Plant size = K 


In agriculture, some farms are larger than others.

Likewise, in many retail businesses, some firms are larger than others. 

plant size 

How does one choose an optimal plant size?

A large plant may be underutilized . For instance, many small hospitals bought magnetic resonance imaging (MRI) scanners, but due to lack of patients, they were underutilized in small towns.

If the plant is too small, it tends to be overutilized, and increasing output is costlier.

 discrete case

 In some industries, the plant size is not a continuous variable. For example, a hospital can add cancer division. It cannot simply add one specialist and claim it can treat cancer patients.

Consider the discrete case where the firm has only three options:

K1 = small, K2 = medium, and K3 = large.

These three plants will have different AC curves.

Which plant should the firm choose?

It depends on the output level.

 LRAC curve Long Run Average Cost curve is simply the minimum of the Short Run Average Cost curves. 

  2. LRAC curve (Continuous case)

 In agriculture, the choice of plant size seems to be simpler than the hospital industry, which faces a question: to add a cancer section or not.

Let us consider the continuous case, where optimal plant size can be finetuned, depending on the output level. For example, an ethanol farmer can finetune his operation by determining the corn acreage. His plant size can be finetuned, depending on the acreage.

In the Continuous Case there may be hundreds or thousands of short run AC curves. Maybe the optimal plant size can be tailored to the desired output level.


 In that case, we can generate LAC.

LAC curve = the envelope of all SAC curves.

Imagine that each SAC curve is an egg shell. Basically, you put all eggs in one bag, and the contour of the bag (envelope) is the Long Run Average Cost (LRAC) curve.

   LRAC is NOT obtained by connecting the minimum points of all SRAC curves.

The professor who derived the LRAC curve asked his secretary to perform precisely that task.

He found out that such a curve exists, but lies above the envelope, and hence cannot be the LRAC. 



Consider a production function

Y = F(K,L) = AK1/3L2/3.

Along an isoquant, the output level is fixed, i.e., dY = 0.

Totally differentiating the production function, we get

FKdK + FLdL = 0 = dY,

where the subscribt denotes marginal products, Fi = ∂F/∂i.

Thus, the slope of an isoquant is -dK/dL = FL/FK

Marginal products are:

FK ≡ ∂F/∂K = (A/3)K-2/3L2/3 = (1/3)Y/K,

FL ≡ ∂F/∂L = (2A/3)K1/3L-1/3 = (2/3)Y/L. 

 derive MRTS

 MRTS = 2K/L.

Evaluate MRTS at (L,K) = (150 million,$60 trillion).

K/L = 60,000,000,000,000/150,000,000 = 400,000.

Thus, MRTS = 800,000.

This means that a typical American worker uses $800,000 worth of capital input.

In equilibrium, MRTS = w/r.

In the US, the wage-rent ratio is $40,000/.025 = $1, 600,000.

In China, the wage-rent ratio is $10,000/.05 = $200,000.

Due to high wage-rent ratio in the US, crowd scenes of James Bond movies are made in developing countries such as Turkey (Skyfall, 2012) and Mexico (Spectre, 2015)

Industrial Revolution

in the 1820s


It took more than 3 centuries for the Europeans to modify the production process in response to the rising wage. An average shoemaker made two pairs of shoes per day before the industrial revolution. Due to rising productivity, firms were able to pay higher wages.

In 1880, with the aid of new machines, he made 300 pairs a day.

Initially, the new technology created unemployment as machines replaced workers. In time, workers moved to other areas and learned new skills.

The fourth Industrial Revolution

Artificial Intelligence and robots may replace many routine jobs (e.g., taxi drivers may become obsolete; they may be replaced by self-driving taxis.)

Many routine and tedious jobs may disappear.

  1. Substitution between K and L

Virtually all industries use both capital and labor inputs, but the exact combination of capital and labor depends on the local input prices. iPhones are produced in various Foxconn factories in China, because wage rate is much lower in China than in the US.

Labor supply in China has become volatile due to the recent coronavirus pandemic, and its wage rate is rising in China. Accordingly, Apple is moving some of its iPhone production to other countries like India and Vietnam.

However, moving factories out of China is a very slow process, because capital is fixed in the SR.

While capital input or factory size is a fixed input in the SR, it is a variable input in the LR.


In the car industry, automobiles used to be produced by workers on assembly lines. Henry Ford began to mass-produce automobiles (Model T) in 1908, using assembly lines.

Now, industrial robots are beginning to be used extensively to perform repetitive or boring tasks, and fewer workers are employed.

This is an attempt on the part of firms to reduce high labor costs.

Industrial robots are used in countries where the wage rate is high.

Robots are rarely used in low-wage countries. 


  6. Animation Film Industry

 Interest rate r is stationary. Federal funds rate has been stable at 2% per year for the past six decades. In contrast, wage has been steadily increasing. Firms are forced to labor-saving techniques and machinery. As a result, production process becomes more capital-intensive.

Walt Disney films were produced employing hundres of animators drawing most of the film by hand. Each film contains more than 150,000 drawings. (Glenn Hubbard and Anthony Patrick O'Brien)

The first Lion King film in 1994 cost only $50 million but the company earned more than $1 billion in profit. Encouraged by this, the company decided to produce more animation films. The new version was more skilled-labor intensive.

The new Lion King by Jon Favreau cost $260 million, and grossed over $1.6.

 Computers (capital)

Pixar Animation Studios released the film, Toy Story. This was the first animation film produced using computers and no hand-drawn drawings. Snow White cost $1.5 million in 1937, and grossed $7.8 million.

Production cost of Ratatouilli was about $150 million but the movie grossed $624 million. Typical movies need 150 people or more per year. Animation movies became more capital-and skilled labor-intensive.

The animation film industry began to produce films using more capital-intensive method (more skilled animators and software) and hiring fewer extras.)



  7. Rising Wage and Expansion Path
   The expansion path chosen by a firm depends on the wage-rent ratio, w/r.

Input prices may change over time.

Also, factor prices may differ in different countries. 

A rise in w/r

Blue Mosque, Istanbul (e.g., as shown in Skyfall)

 As the wage rate rises, firms tend to choose a more capital-intensive technique.

Movies will tend to use fewer people (less labor-intensive), and crowd scenes appear less frequently. Movies tend to use animation to simulate crowd scenes.

Extras get paid $100 - 200 per day (Screen Actors Guild)

Crowd scenes are outsourced to other countries (e.g., Turkey, China, Mexico).

 Health care bill

Who pays the health care bill?

If employers or firms pay for the health care, the bill effectively raises the wage.

This forces firms to choose a more capital-intensive technique, which shifts the expansion path of every firm. 

  A point on the expansion path shows the capital-labor ratio. The steeper the expansion path, the more capital-intensive technique the firm uses. 
 Low wage countries Firms in low wage countries such as China and India, the production technique tends to be more labor-intensive, whereas firms in European Union and the United States tend to be more capital-intensive. 


Agriculture in Asian countries tend to be labor-intensive.


 Some industries are more capital-intensive than others.


Let Ki and Li denote the amounts of capital and labor used in industry i, and let ki ≡ Ki/Li denote the capital intensity of industry.

Capital intensity simply measures the amount of capital the average worker uses in the production process. In the case of CRS industry, it is also the slope of the expansion path.

In the above diagram, industry 2 is more capital-intensive than industry 1.