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I am trying to isolate a simple model for GDP accounting in the one person economy also known as the Crusoe economy. Because Crusoe lives alone and has dominion (exclusive control) over all items of economic value I assume he assigns zero value for rent as the return on land and extraction of raw materials. This means income is attributed to wages and profit. Also I assume Crusoe invests only in capital and holds zero inventory in the firm or business sector of his economy. He counts all other output as transfers to consumption in the household sector.

In terms of economic value instead of prices I want to know if wages and profit have a mathematical description in terms of respective contributions to total output?

  1. Identity for Gross Domestic Product

Output = Income = Expenditure

  1. Classical factors of production

land, labor, and capital

  1. Classical factor payments

rent, wages, and profit

  1. Income in the circular flow assuming zero rent

income = wages + profit

  1. Expenditure in the circular flow assuming zero inventory

expenditure = capital + consumption

  1. Robinson Crusoe GDP identity with simplifying assumptions:

output = capital + consumption = wages + profit

Questions:

Is there a non-price equation for wages in terms of output and profit?

Is there a non-price equation for profit in terms of output and wages?

Do these identities hold?

wages = consumption?

profit = capital?

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    $\begingroup$ What does "economic value instead of prices" mean? $\endgroup$ – Michael Greinecker Dec 22 '20 at 17:09
  • $\begingroup$ what do you even mean by non-price? By definition wage is the price for labor, also return on capital - from which profit here consist of is the price for capital. If you dont want prices (and by prices I mean at least relative prices, you dont need prices in some currency) you need to get rid of both wages and profit $\endgroup$ – 1muflon1 Dec 22 '20 at 17:50
  • $\begingroup$ So my question perhaps would be better stated as follows: can income as wages and profit be defined in terms of economic value independent of price? The answer seems to be no. The GDP output does have economic value independent of price. The economic value is used up in consumption in the household sector. Crusoe should think of his income as the flow of economic value to his household and should think of labor as the cost of producing his income. Crusoe does not profit from capital he uses it to generate his actual income as household consumption. $\endgroup$ – SystemTheory Dec 22 '20 at 23:54
  • $\begingroup$ You can convert wages, profits etc into utility but for that you need to specify additional utility function and the relation would no longer be GDP which you said you are interested at the beginning. Also, the wage=consumption and profit=capital does not hold in the system above generally - even if we would apply some utility function to it it would not hold. For that you need to change the system above somehow. $\endgroup$ – 1muflon1 Dec 23 '20 at 0:00
  • $\begingroup$ Thank you I think Hyman Minsky makes arguments that profits equal investment spending under heroic assumptions using an identity called the Kalecki-Levy profit identity. However I think you are right that wages and profit are not meaningful in terms of economic value. If one goes away from a pricing system the GDP identities do not hold anymore and I don't think Crusoe would use a pricing system unless he got bored and was educated as an economist. One might imply his use of prices or some ratio of preference when he decides how to allocate labor to a mix of capital and consumption goods. $\endgroup$ – SystemTheory Dec 23 '20 at 0:05
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I will only answer the last question because without further qualification the first two questions do not seem to make sense (see the my comment). Also to make everything cleaner I will use $w$ for wages, $\pi$ for profits, $Y$ for output, $C$ for consumption and $I$ for capital spending.

Do these identities hold?

$$w= C \\ \pi = I$$

No. There is nothing in your system above that would suggest the above should hold. As you remarked on by definition:

$$\pi +w = Y = C+ I \implies \pi + w =C + I$$

Given the above the profits are actually equal:

$$\pi = C + I -w \text{ or } \pi = Y -w $$

and wages are equal to

$$w = C + I - \pi \text{ or } w = Y - \pi $$

The reason why the two identities ($w=C$ and $\pi =I$) won't hold might be best visualized by some practical example. Let us assume some overtly simplistic production function where $Y = L + K$, here marginal product of labor and capital are both $1$ so wage per unit of labor will be $1$ and return on capital also $1$.

Now, I am not sure if this was your question about 'prices' but note we do not need to necessarily need to use some real world currencies here and this prices could be denoted in apples, bananas etc (as commonly done in theoretical models - especially in international economics - see Feenstra Advanced international trade for some examples) but I will leave the units vague for simplicity. In addition note that GDP as a statistical measure - not as a concept of output - just look at production sold at market so that would in Robinson Crusoe economy be zero because there are no market transactions - but I assume here you are interested in the underlaying concept of output rather than statistical measure.

The above out of the way, let us assume that Robinson Crusoe starts 10 units of labor that he gets every time period, that he can invest in capital goods which cost exactly 1 unit of output and let see some example of $Y$ calculation across 2 different periods.

Period 1: Since there is no capital here output is produced with labor, as discussed above the price of labor is 1 so when 10 units of labor are supplied we get that $w=1(10)=10$, So we have $Y=w+\pi = Y=10+0$, now this by definition has to be equal to all spending but there is no reason why we should assume spending on consumption has to be equal to 10. That is one possibility but any combination of $C+I=10$ would work here. Let us assume that $C=5$ and $I =5$.

Period 2: By our assumption that it costs $1$ to acquire $1K$ now the Crusoe has $10L, 5K$. Given that the profit (under which you decided to put returns on capital) are given by $\pi = (1)5$, and wages are $w=(1)10$ so now output is equal to $Y= 5+10 =15$. As previously consumption and investment has to be equal $15$ together but there are various ways how you can achieve that. $C$ could be 5, which would mean $I=10$ or $C=15$ which would mean $I=0$, as long as $C + I =15$ any value for $C$ or $I$ that satisfies the equality will work.

Hence, generally $w\neq C, \pi \neq I$ save for special cases where $C$ and $I$ are choose by Robinson Crusoe to be exactly equal that amount.

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