Individuals cannot directly consume commodity wealth. Gold, oil, iron ore , and the like provide no nutrition or protection from the elements. These commodities have little value in direct consumption. However if trade is possible with another nation -- a nation that realizes that the true measure of wealth is in production of necessary goods and services, then these commodities do have value.

Adam Smith was the first to realize that the Wealth of a Nation was not in the accumulation of commodities nor in the resource reserves that a nation may happen to possess. But rather wealth exists in the productive knowledge of its people. The ability to efficiently transform resources (factor inputs) into desired goods and services represents the true source of a nation's wealth.

Physical and human capital represents the true embodiment of wealth. This wealth is used to generate factor income as a payment for the production of desired goods and services X_i [i = 1,2,...,n-goods] and income to be used to purchase these same goods and services. Thus wealth 'W' may be measured in terms of the future stream of income 'Y_t', discounted at some rate 'ρ', generated by the use of physical and human capital:

X_i = f(L_i,K_i,M_i)      for all i = 1,2,...n-goods

Y_t = Σ_[i=1,n]P_i,0X_i,t

W = Σ_{[t = 1, T(i)]} {Σ_[i=1,n]P_i,0 f(L_i,t,K_i,t,M_i,t)} (1+ρ)^-t

The numeric value of wealth is really less important than what it represents. The members of a given society are interested in living standards such that growth in output will at a minimum, equal or exceed the rate of population growth. Thus they are interested in a stock of human and physical capital sufficient to produce desired growth in income.

Note: over time labor input will grow at some rate 'n' proportional or perhaps equal to the rate of population growth:

L_t = L₀(1+n)^t

Growth in capital is affected by savings rates in a given economy to support gross investment (changes in the capital stock overtime) at some rate 'g' less the rate in depreciation 'd'. This rate of depreciation is a reflection of the fact that, over time, capital does where out. Thus 'g - d' represents the net growth in the capital stock over time.

K_t = K₀(1 + g - d)^t

Increases in living standards require more capital per unit of labor thus making each unit of labor more productive or:

(g - d) > n

such that:

K_t/L_t is increasing over time.

To present several examples of production and its relationship to wealth, we will examine the production of a simple restaurant meal, passenger services on an airplane, and production of housing services.

A sole proprietor operating a small restaurant uses raw materials in the form of food ingredients, capital in the form of a stove or oven, and labor input in the form of his own time. The proprietor is motivated by the fact that he is able to create a meal that is valued by his customers over-and-above the value of the individual inputs. This added value is created by his talents and know-how as a cook or chief combined with the physical capital of the restaurant. Wealth in this case is not only in the materials or factors of production but also in the proprietor's knowledge of preparing a meal. Over time this human capital will generate a stream of income for the proprietor for as long as he operates the restaurant and as long a there is a demand for his product.

An airline makes decisions to purchase an airplane based on anticipated demand for transportation services. The value of these services are based on the benefits customers receive by flying relative to other forms of traveling from one place to another. Like the restaurant the value of the airplane (a unit of physical capital) is in the provision of passenger services. Building this airplane represents an addition to wealth in that it will generate a stream of revenue (income) for the corporation generated over its physical life provided demand for passenger services remain.

Finally, the construction of an apartment building represents an addition to wealth based on the demand for apartments by those seeking housing services. The building will generate a stream of rental income over its life based on occupancy levels and rental rates. If however, the building is largely vacant--demand is lacking, its contribution to (national) wealth is close to zero. It becomes an asset with limited realized value even though its construction represents a combination of valuable materials, labor services, and land area.

The creation of wealth is based on knowledge -- the ability to take raw inputs and convert them into output with value greater than the sum of the individual parts. Additionally, this value is determined by correctly assessing the demand for the output -- how it will satisfy needs and wants. Creation of a restaurant, airplane, or apartment building (physical capital) all represent a contribution to a nation's wealth in that they all generate a future stream of income based on the willingness of the members of that nation to purchase food services, transportation services, or housing services to satisfy specific wants. Creation of a school teacher or engineer (human capital) also represent additions to a nation's wealth in that they also generate services desired by others in a given economy and thus produce a stream of income for the individual based on demand for those services.

X^* = f(L,K,M)

where X* is an aggregate measure of goods and services produced (output) in a given economy.

• L represents the quantity and ability of labor input available to the production process.
• K represents capital, machinery, transportation equipment, and infrastructure.
• M represents the availability of natural resources and materials for production

A positive relationship exists among these inputs and the output such that greater availability of any of these factors will lead to a greater potential for producing output. The functional relationship f(.) represents a certain level of technology and know how that presently exists for conversion of these inputs into output such that any technological improvements can also lead to the production of greater levels of output.

In order to understand the creation of wealth and the engine for economic growth that will provide for increasing standards of living, we must first start with an understanding of several characteristics govern aggregate production relationships.

• First, is the law of diminishing marginal productivity. Holding other factors of production constant (i.e., capital and/or materials), increasing quantities of a single input lead to less and less additional output. This property is just an acknowledgment that it is impossible to produce an infinite level of output when some factors of production (machines or land) fixed in quantity.
• Second, is that all factors are essential in production and to some degree one factor may be substituted for another factor of production. Increasing the amount of capital or machinery can replace some labor but not all of the labor in a production process. Increasing amounts of labor (greater care being taken in production to avoid waste) can reduce the need for some material inputs.
• Third is that, for production in the aggregate, there are constant returns to scale. This property refers of the ability to double output simply by doubling the quantity of all the available inputs. A simple way to understand this is based on the idea that any production process may be replicated. Thus if a certain quantity of grain is being produced on one acre of land with 5 units of labor input and 3 pieces of capital, then by replicating this production process the quantity of grain produced may be doubled.

One specific mathematical relationship that possesses these three properties is the Cobb-Douglas production function. This particular representation is one of several possibilities and may be written as follows:

X_i = A_tL^αK^βM^γ

where L,K,M represent the factor inputs listed above, A represents a measure of technology at time period 't', and the exponents represent production parameters (actually output elasticities) for each factor input. The fact that it is multiplicative in the inputs reflects the notion that one factor may be substituted for another. Diminishing marginal productivity requires that the exponents α, β, and γ each take on values less than one. Each input being essential and making a positive contribution to output implies that these exponents be strictly greater than zero. Finally, constant returns to scale implies that α, β, and γ sum to one.

max π = R-C

max π = P_xX - [wL + rK + zM]
s.t

X = f(L,K,M).

The objective function can be rewritten by solving for the variable 'X' as follows:

X = [(π + FC)/P] + (w/P)L

By adding the production function to the above diagram, (click on the next button twice) we find that the input- output combinations as defined by points 'a', 'b', and 'c' are all within the limits of available technology. Point 'd' however, is unattainable -- a level of output of X₂ is impossible with a level of labor input of L₁.

At point 'b', we find that we achieve the greatest level of profits possible with this existing level of technology. At this point the production function is just tangent to iso-profit line 'profit₁. This point is known as a producer optimum.

Mathematically, we can solve for this producer optimum by substitution of the constraint into the objective function,

max π = P_x[f(L,K,M)] - wL -rK - zM

and solving with respect to any of the factor inputs we can derive the following:

dπ/dL = P_xMP_L - w = 0,

dπ/dK = P_xMP_K - r = 0,

dπ/dM = P_xMP_M - z = 0

These results may be interpreted in a variety of ways.

• First we could write (using labor) P_xMP_L= w, where the left-hand side represents the marginal revenue product of labor -- that is the contribution of one more unit of labor input to the firm's revenue. The right-hand side 'w' represents the prevailing nominal wage rate or the cost of hiring that additional unit of labor. Profit maximization implies that the contribution to revenue (using monetary units of measure) is just equal to the contribution to costs in the hiring of one more unit of a factor input.
• Second, we can rearrange the terms and write MP_L = w/P_x. This expression states that profit maximization implies that the contribution of each additional unit of labor input to output must be paid a real wage (a measure of purchasing power) equivalent to that level of contribution.
• Third, we could write P_x = w/MP_L or P_x = MC which states that profit maximization occurs when the revenue from selling one more unit (marginal revenue) of a particular good 'Px' is just equal to the cost of producing one more unit (marginal cost) of that good.

NGDP = PX* = wL + rK,

wL = 0.70PX*, and rK = 0.30PX*

Using the Cobb- Douglas production function with 'L' and 'K' representing labor input and non-labor input respectively and given constant returns to scale (,1-α = β), we can write:

X* = AL^αK^1-α,

MP_L = α(X*/L) and

MP_K = (1-α)(X*/K)

Note that the variable 'M' representing materials has been dropped under the assumption that labor and capital are combined in the extraction of these raw materials and the above factor payment percentages represents compensation for these types of activities.

With profit maximizing behavior,

P_xMP_L - w = 0

or

MP_L = w/P_x

and similarly

MP_K = r/P_x

α(X*/L) = w/P

or

α = wL / PX*

and

(1-α)(X*/K) = r/P

or

(1-α) = rK / PX*

which implies that

α = 0.70 and (1-α) = 0.30

and

X* = AL^0.70K^0.30

dX* = [ΔX*/ΔA]dA + MP_LdL + MP_KdK

dX* = [L^αK ^(1-α)]dA + [αX*/L]dL + [βX*/K]dK

and dividing through by 'X*', we can write:

%ΔX* = %ΔA + α[%ΔL] + (1-α)[%ΔK]

Expressed differently, the rate of economic growth may be expressed as

%ΔRGDP = %ΔX* = %ΔA(t) + 0.70[%ΔL] + 0.30[%ΔK]

Economic growth is thus the sum of the rate of growth in technology in addition to a weighted average of the rate of population growth and the rate in which capital accumulates. An interesting implication of this is that, holding other factors constant, a population growth rate of 1% leads to a less than one percent growth rate in output--a decline in the Standard of Living. In order to maintain or improve these Living Standards, there must be an accumulation of capital and/or technological progress.

• Cobb-Douglas Production Function
• Constant Returns to Scale
• Diminishing Marginal Productivity
• Economic Growth
• Factors of Production
• Iso-profit function
• Production
• Profit Maximization
• Returns to Scale
• Standard of Living
• Wealth