If we define the Standard of Living 'SoL' as the ratio of Real GDP and the population of a given country (also known as per-capita Real GDP), then improvement requires that this ratio increase over time:

SoL = RGDP / Population

%Δ RGDP > % Δ population → SoL ↑

In a world of diminishing marginal productivity, this increase can be difficult to achieve. Let us take a look at a basic production relationship. In this case, we will hypothesize an aggregate production function defined as follows:

Y = f(L, K, M)

Where Y represents the output of an economy (RGDP), and 'L', 'K', and 'M' represent the inputs -- the aggregate factors of production: Labor, Capital, and Materials. Looking at production in the short run, holding 'K' and 'M' constant (their contribution is captured in the coefficient of 100) we define the following hypothetical relationship:

Y = f(L) = 100(L)^0.50

We observe that as the amount of labor increases, output increases at a decreasing rate (diminishing marginal productivity). Holding the 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) are fixed in quantity. Also notice that as more labor is used in the short run, the ratio between Y (Real GDP) and L is also falling.

Now suppose that the ratio between the population and the size of the labor force (the labor force participation rate) remains constant:

L = α(Pop) = 0.67(Pop)

We can derive an expression for the Standard of Living (SoL):

SoL = Y/PoP = Φ (Y/L)

Or as more labor is applied to the production of goods and services (holding technology, capital and materials fixed) the SoL declines:

An Increase in Labor Input

In a country with population growth and diminishing marginal productivity what is necessary for improvements to living standards are additions to the capital stock, the level of technology or both. These additions increase the productivity of workers and allow for more output for each and every level of labor input. This can be seen in the diagrams.

The creation of Physical and Human Capital represents production not intended for direct consumption. Rather, the creation and accumulation of capital is intended to increase the level of productivity of a nation and thus allow for an increase in the production of goods and services at future date. Capital is an intermediate good or a good used to produce other goods.

An Increase in Capital K₀ → K₁

The creation and accumulation of capital depends on the ability of a nation to give up current consumption of final goods to make resources available for the accumulation of this capital. Deferring consumption (known as savings,) depends on the ability of that nation to first meet the basic needs of its citizens with existing production technology and resource availability.

3.1.1 Production Possibilities

In every economy, a tradeoff always exists between using resources for the production of intermediate goods and final goods although the consequences of this tradeoff may differ among nations. For example, looking at the table below (with 6 units of labor available), wheat represents a final good -- something that can be directly consumed and plows represent an intermediate good -- something used in the production of future wheat.

Production Possibilities (Final Goods vs. Intermediate Goods)

	L	Ywheat	MPL		L	Kplows	MPL
F	0	0	-	A	0	0	-
E	1	80 bu	80	B	1	50 units	50
D	2	150	70	C	2	90	40
C	3	210	60	D	3	120	30
B	4	260	50	E	4	140	20
A	5	300	40	F	5	150	10

Production Possibilities

As labor is transferred from the production of one type of good to the other in an effort to product more intermediate goods (plows ~ capital), more and more wheat must be sacrificed to produce additional units of capital (plows). These sacrifices might be possible for some countries, but for those nations living at the edge of subsistence (barely able to produce enough calories to support the population), this reallocation might come at the expense of starvation for some. It is this tragic trade-off that has contributed to Economics being labeled at the Dismal Science.