Time

Allocation and time

• The allocation must be established not only for the present but also throughout the future.
• Production processes involve a flow of consumption and production of materials over time.
• Solving the economic problem means studying the compatibility of processes not only in the present but also in the future. It is necessary to establish or estimate which processes will be available in the future, and the compatibility of their joint operation.
• If we are dealing with an optimal economy, where an objective is maximized, the objective can encompass the present and also the future.
• Allocation does not study the past; it always looks forward in time because the past cannot be altered. This is the reason for prospective theory , which the neoclassical economists championed.
• This does not prevent us from studying what past has caused a particular present.
• For example, Robinson has to allocate in such a way that he can not only eat today, but also eat tomorrow and the day after tomorrow.
• Therefore, the study of time is essential, and its influence not only on allocation and production processes but also on prices.
• Allocation, like prices, is defined in time.

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Price and time

• The temporal nature of prices affects all three aspects: the values (or Lagrange multipliers), the weights (or allocation coefficients), and the exchange rates. All three prices have a temporal dimension that cannot be ignored.
• In those societies where only some of these aspects are defined, those that exist will have that temporal dimension.
• For example, on Robinson's island, where there are only values (Lagrange multipliers) but no weights or exchange rates, values have that temporal dimension; they are defined for the present and for the various points in the future that Robinson takes into account. For Robinson, not only do things that exist in the present have value, but also those that may exist in the future.
• In capitalist systems:
• Prices (in all three aspects) will vary depending on the point in time to which they refer, depending on whether they are set for present goods or for those that (are expected) to exist at different points in the future.
• At any given moment in time, the three prices tend to be proportional, through processes similar to those we have outlined.
• Futures markets are those where things are traded over time.
• See the grain futures market, https://www.bolsadecereales.com/mercados

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Values, weightings, exchange rates, and time

• In an optimal economy, one that maximizes an objective, the constraints will determine that what is available of each item at any given moment (for example, because it has been produced since the previous moment) is what can be consumed at that moment. And this applies to every moment throughout the time frame in which the allocation is established.
• The Lagrange values or multipliers will therefore be dated, defined for each instant in time at which the constraints are considered. The value of something at a given instant will be the rate at which the objective we are maximizing increases with the introduction of a small quantity of that thing at that instant.
• If we are dealing with a profit-maximizing economy, the weights or allocation coefficients will have to be defined over the time period covered by the allocation.
• And if we are dealing with a market economy, exchange rates also have that temporal dimension.
• For example, in capitalist systems it is common for exchanges to take place between goods located at different points in time.
• This is how you can buy a car and pay for it in installments. You receive the car now in exchange for paying a series of fixed sums of money over time.

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Three aspects of prices in capitalist systems

• In capitalist systems, as we saw, the three types of prices can be different, but there are forces that cause them to tend to be proportional.
• This is the case for every moment in time in which the allocation is estimated, from the present onwards.
• The three prices will tend to be proportional to each other at any given moment in time, although all three will change over time.
• Let's remember:
• Exchange rates .
• These are the prices that arise in exchanges.
• Weightings or allocation coefficients.
• These are the prices that a company uses to determine its profit target.
• Lagrange values or multipliers.
• They are what increases the objective that is maximized by introducing a small amount of matter.

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Values or

Lagrange multipliers

Exchange rates

Weights or �allocation coefficients

Value and time

• We have seen that in capitalist systems, value was the profit that the commodity could produce.
• We have also noted how exchange rates tended to adjust to values, and how weightings also tended to be proportional.
• This is true whether the merchandise is in the present or located in a specific future.
• Therefore, if a commodity is worth more in the present than in a given future, it is because the benefits that its use allows now are greater than those produced by its use in that future.
• In a growing economy, a certain quantity of something (wheat) today will generally yield more profit than that same quantity a year from now. That's why wheat today is (generally) worth more than wheat to be delivered a year from now.
• We reasoned assuming that the exchange over time involves no risk, no taxes, and no other complications. In reality, the value of wheat one year from now incorporates that risk and those complications.

Optional. Savings and time. Example

Optional. Savings and time. Example

• We will use a very simple system as an illustration, the evolution over time of a biological population, where the population weighted in a few numbers is maximized.
• We are therefore referring to an optimal economy where the objective that is maximized is the weighted population.
• If the weightings referred to a final selling price, we would be maximizing profit; we would be dealing with capitalism. Therefore, we are not talking about socialism.
• We use this simple example because with it we can easily describe its evolution over time.
• The population in the examples follows the reproduction and survival rates of the Spanish population in 1963 (women only).
• We have chosen this data relating to a human population simply because we have it in a very detailed form.
• If it is unsettling to think about a human population, we can assume that we are dealing with an agricultural operation that is dedicated to raising an animal population with a view to selling it at prices determined by the weightings.
• The embedded spreadsheets allow you to define other rates, for another human or biological population, even varying over time. For our purposes, it is not necessary for these rates to remain constant.

Optional. Assignment and time. Example

• We will assume that our system is an island where we incorporate one million women from the 30-34 age group at the initial moment.
• Or, if you prefer, a farm where we have acquired one million female animals aged 30-34 at the initial moment.
• We can calculate the population of a certain age group higher than the initial one at a given moment as follows:
• The population incorporated from that age group at that moment, plus
• The population of the lower age group at the previous time multiplied by the survival rate,
• The population of the initial age group at any given moment will be:
• The population incorporated from the initial age group at that moment, plus
• The populations of all age groups at the previous time multiplied by their reproduction rates.
• Therefore, we can calculate the population starting from the first instant and calculating forward in time . The population depends on its present and its past.
• We found that, when reproduction and survival rates do not change and when there are no additions to the population, it tends in the long term to the stationary population structure, which will grow over time with the same growth rate.
• But in our model we can modify those rates or impose migrations; we are not assuming that they remain constant.

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Optional. Value and time. Example

• We will assume that our objective is to maximize the production of the 0-4 age group in year 235 (the spreadsheet allows you to define other weights). Therefore, we weight that age group in that year as 1.
• If you think about a farm , years can be seen as weeks, for example.
• Note that in our system there are values and weights, but no exchange rates, unless we assume that this production in the year 235 will actually be sold.
• The values (Lagrange multipliers) tell us how our target population, the 0-4 age group in the year 235, would increase if we introduced a woman of a certain age group in a certain year.
• We can calculate the value that corresponds to an age group at a given moment in two ways:
• Adding a small amount of population from that group at that moment and checking the rate at which the objective we are maximizing grows; that will be the value.
• Based on the maximum conditions:
• The value of the older age group at the next instant multiplied by the survival rate, plus
• The value of a newborn woman in the next instant multiplied by the reproduction rate, plus
• The weighting of that age group at that moment, if there was one.
• Therefore, starting from the maximum conditions, we can calculate the value by beginning with the last instant and calculating backward in time . The value of something depends on its present and its future.
• We found that, when reproduction and survival rates do not change and when we do not add other weights, the long-term values regress to the structure of reproductive values , which will decrease over time at the same rate of interest, equal to the growth rate of the stationary population.
• But in our model we can add weights or modify those reproduction and survival rates; we are not assuming that they remain constant.

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Dispensable. Savings and time. Example

• We will write the formulas for the evolution of the population and the values.
• We will note:
• L _t the (transpose of the) Leslie matrix at time t,
• Q _t the vector of the age-specific population incorporated into the system at time t,
• P _t the vector of weights that correspond to the age population at time t,
• We will assume that L _t , P _t and Q _t , are known data for all t.
• Denoting as X _t the vector of the population by age at time t, we have that the evolution of the population results

X _t = Q _t + X _t-1 L _t-1

• If we know X _t for a t we can calculate X _t+1 , and in general the future of the population is determined.
• X _t P _{t ,} is maximized for given weights, subject to the constraints regarding the evolution of the population (since the population consists of non-negative numbers).
• Then, denoting as Y _t the vector of the Lagrange values or multipliers at time t, the maximum conditions are

Y _t = P _t + L _t Y _t+1

• Knowing Y _t for a t we have that Y _t-1 , and in general the past of the values, is determined.

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Optional. Growth rate and �interest rate

Optional. Stationary population and reproductive value

• To simplify the analysis, we will study the behavior of a biological population in the case where reproduction and survival rates do not change over time.
• And that survival rates are positive, and two consecutive reproduction rates are positive.
• The stationary population and reproductive value inform us about long-term behavioral properties of the population, under the stated assumptions.
• Whatever the initial population, in the long run the system will show the structure of a stationary population, which will grow over time with the growth rate.
• Whatever the weightings, looking back in time the values will show the structure of reproductive values, which will decrease over time with the interest rate.
• The reproductive value of a woman of a certain age is proportional to the number of daughters she will have in the long term, based on the daughters she will have and not on those she has already had, assuming that reproduction and survival rates do not change.
• One way to calculate reproductive values is to add a woman of a certain age at a certain time to the demographic model and establish how the population increases in the long term.

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Optional. Stationary population over time

• Whatever the initial population, in the long run the system will show the structure of the stationary population, which will grow over time with the growth rate.
• We can calculate the stationary population starting from any initial population and allowing the system to evolve over time. The long-term population structure will be that of the stationary population.
• To calculate the stationary population we can also proceed as follows:
1. We start from any initial population, for example by making all age groups 1 in year 0.
2. We calculate the resulting population evolution, and obtain the population of the year 250.
3. We define the population of all ages in year 0 as the population we have obtained for the year 250.
4. We return to step 2 until the (relative) population converges.
1. Generally, two iterations are enough to get a good approximation.

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Optional. Reproductive value over time

• Whatever the weightings, looking back in time the values will show the structure of reproductive values, which will decrease over time with the interest rate, equal to the population growth rate.
• We can calculate the reproductive value by setting weights at a very advanced time and calculating the rate at which the target increases if we add a small number of people from an age group at an instant.
• We can also approximate the (relative) reproductive value corresponding to an age group at a given moment in time by following these steps:
1. We define some arbitrary final weightings, for example weighting all ages in the year 250 as 1.
2. We calculate the resulting values, and obtain the values for year 0.
3. We define the weights of all ages in the year 250 as the values we obtained for the year 0.
4. We return to step 2 until the (relative) values converge.
1. Generally, two iterations are enough to get a good approximation.

Optional. Interest rate and reproductive value

• We can calculate the reproductive value by establishing weights at a very advanced point in time and calculating the rate at which the target increases if we add a small number of people from a particular age group at a given moment. The target could be the sum of the population in the year 250.
• Let's study the case where a woman in the 30-34 age group is introduced into the system in year 0. Analyzing the evolution of the population, we find that in year 250 she would have 10.2 descendants.
• If a woman in the 30-34 age group is introduced into the system in the year 75, in the year 250 she would have 5.17 descendants, about half.
• Therefore, the one incorporated in year 0 will have about twice the value of the one incorporated in year 75, because it will have about twice as many descendants in year 250, in the long run.
• The reproductive value depends on when a woman enters the system. (If the population grows) a woman at an earlier point in time will have more offspring than a woman of the same age introduced at a later point in time. Therefore, she will have a higher value.
• The annual interest rate linking both groups is (10.2/5.17)^(1/75) – 1 = 0.91%, which also happens to be the annual growth rate of the stationary population. The interest rate, the relationship of value over time, is the population growth rate.

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Optional. Reproductive value, age, and time

• We have seen that a woman in the 30-34 age group in year 0 would have 10.2 offspring in year 250.
• If a woman in the 15-19 age group is introduced into the system in year 0, in year 250 she would have 20.2 descendants.
• In the long term, the 30-34 age group would have about half the number of offspring as the 15-19 age group, and therefore the 30-34 age group has about half the reproductive value of the 15-19 age group.
• A woman's reproductive value depends on her age group. A younger woman will have a higher reproductive value than an older woman because she will have more daughters in the future. A woman who has passed reproductive age will have a reproductive value of 0.
• If a woman from the 0-4 age group is introduced into the system in the year 55, in the year 250 she would have about 10.2 descendants.
• In the long term, it would have approximately the same number of offspring as one from group 30-34 in year 0, so both have the same reproductive value.
• That's why in the graph of reproductive values over time both have the same color.
• The group 30-34 would have fewer daughters on average than the group 0-4, but would have them many years earlier.
• The reproductive value depends on the age group and the moment at which the woman is introduced.

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Optional. Interest rate and �growth rate

• If reproduction and survival rates do not change over time, we have:
• The stationary population, the structure that the population adopts in the long term regardless of what the initial structure is.
• Reproductive values, the structure of values taking us back to the long term, regardless of the final weighting.
• The stationary population will grow while maintaining its structure over time, with the growth rate.
• Reproductive values will decrease while maintaining their structure over time, with the interest rate.
• The interest rate is equal to the growth rate of the economy .
• Let's remember that we are talking about a type of economy where the weighted population is maximized in the distant future; we are not talking about socialism.
• If that population were to be sold in the distant future with exchange rates equal to the weights, we could be facing a capitalist enterprise.
• Therefore, we find that the value decreases with the interest rate because the value of a thing is what that thing will produce (in a system like capitalism), also when it is located in any position in time.
• In a growing capitalist system, something available at an earlier time will yield more profit than the same thing available at a later time, because it will be available sooner. Therefore, the later-available item will be worth the earlier one, discounted by the interest rate, which is equal to the economy's growth rate.
• In addition to what has been said, to value things over time, many other circumstances must be taken into consideration, such as risk or the presence of taxes or luxury consumption, for example. We have reasoned in a simple and stable situation to illustrate the point, but in reality, capitalist systems are neither simple nor stable.

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Optional. Reproductive values and �economic theories

• Reproductive value is not only related to the school of reproduction, as its name indicates, but also:
• It is objective, it depends on the conditions of production, as the classics stated,
• It is prospective, it depends on the future of the thing, as the neoclassical economists stated.
• These are Lagrange multipliers, as the optimization school claimed.
• The only school of thought we've studied with which reproductive value isn't directly linked is the general equilibrium school, because we haven't imposed the existence of exchanges (or we've only imagined them as weightings). But if there were exchanges, it would also be related to this school.