We will assume a central marketplace which is surrounded by agricultural land, all of which is of equal fertility. There will be one good, which we call wheat. The landowners hire a single input to production, labour.

*L*units of labour produces

*W(L)*units of wheat. The price of wheat in the marketplace is

*p*while the costs of transporting the wheat to the market is

*$t*per mile per ton. Thus the earnings generated from wheat grown

*m*miles away from the marketplace is

*p-t.m*per ton. Total revenue from the wheat will be

*W(L).(p-t.m)*. Assume that the landowner pays workers a wage of

*w*resulting in a total wage bill of

*w.L*. This means that the landowner's profit from producing wheat

*m*miles from the marketplace will be

*W(L).(p-t. m)-w.L*.

Further, assume that the landowner selects

*L*to maximise profits. This results in a problem we can represent mathematically as

Maximising this objective function with respect tomax_L W(L).(p-t.m)-w.L

*L*gives the first order condition,

this implies(dW(L*)/dL).(p-t.m)-w=0

where(dW(L*)/dL).(p-t.m)=w

*dW(L*)/dL*is the marginal product of labour and

*(dW(L*)/dL).(p-t.m)*is the value of the marginal product of labour. Equation \ref{thunen} tells the firm it wants to select the level of labour such that the value of the marginal product of labour equals the marginal cost of labour, the wage rate.

Put more generally, a profit maximising firm will choose the level of an input so that the value of the marginal product of the input equals the price of that input. Therefore, from the point of view of a firm, the theory indicates how many units of a factor it should demand.

Blaug (1985: 17-8) sums up Thunen's analysis by saying,

[h]is analysis culminates in the perfectly modern statement that net revenue is maximized when each factor is employed to the point at which its marginal value product (Wert des Mehrertrags) is equalized to its marginal factor cost (Mehranfwand). Although the discussion proceeds in verbal terms, illustrated by numerical examples, Thunen correctly points out that the marginal product of a factor is a partial differential coefficient of a multivariable production function. Moreover, apart from clearly recognizing the distinction between fixed and variable factors, and between the average and the marginal returns of a factor, he took great care to define the inputs of capital, labor, and land in strictly homogeneous units, observing that this condition was rarely obtained in practice--this too was literally more than sixty years ahead of his time.

**Refs.:**

- Blaug, Mark (1985). 'The Economics of Johann Von Thunen',
*Research in the History of Economic Thought and Methodology*, 3: 1-25. - Thunen, Johann H. (1826; 1850; 1863).
*Der isolierte Staat in Beziehung auf Landwirtschaft und Nationalokonomie*. Pt. I: Untersuchungen uber den Einfluss, den die Getreidepreise, der Reichtum des Bodens und die Abgaben auf den Ackerbau ausuben. Hamburg: Perthes; Pt. II: Der naturgemasse Arbeitslohn und dessen Verhaltniss zum Zinfuss und zur Landrente. Rostock: Leopold; Pt. III: Grundsatze zur Bestimmung der Bodenrente, der vorteilhaftesten Umtriebszeit und des Werts der Holzbestande von verschiedenem Alter fur Kieferwaldungen. Rostock: Leopold. English translation,*The Isolated State*, Volume 1. Carla Wartenberg, trans. Oxford: Pergamon Press, 1966; Volume 2 in*The Frontier Wage*. B. W. Dempsey, trans. Chicago: Loyola University Press, 1960.