# What does “r” represent in the total cost function?

I know that the total cost function of a firm in the short run is:

TC = wL + rK,

where rK is essentially a constant. I understand the variables w, L and K, but I still don't get what r is. I mean, what do I input as "r" when I'm trying to draw up a total cost function?

Thank you.

• If w is the wage that you pay per unit of labour hired, then what would r be for capital? – Maarten Punt Feb 12 '20 at 8:07
• I mean to say, is it rent? Or like if I purchase the capital good is it just the price? – Saad Feb 12 '20 at 8:21
• Are you purchasing the worker (slavery) or are you renting his labor? The cost function is for a given time period, so you are doing the same with the capital good. – Giskard Feb 12 '20 at 8:56
• Thank you, but is it possible for you to give me an example to make it slightly easier for me to understand? – Saad Feb 12 '20 at 9:13

r is the user cost of capital.

User cost refers to the expenses borne by the owner or renter of a capital asset resulting from the use of the asset for a given period of time.

A standard metaphor is that of a machine being worn down over time as it is being used. During the time of use it creates a flow of goods through a production process. The capital gets it value from this flow of goods it can create. Simplify this metaphor to a machine that during three months can produce three pieces of bread - one each month - each sold for 1 dollar. The total value of the machine 3 dollars (ignoring discounting) so if you were to buy the machine you would pay 3 dollars. However if you just want to used it for a month it will only produce 1 piece of bread the value of 1 dollar so one month of machine is just worth 1 dollar.

Things are offcourse very much more complicated but you can google how the user cost of capital is derived.

The $$r$$ in the total cost function is the interest rate, which at the same time constitutes the price of capital. You can think of it as an explicit cost, e.g. if you rent the capital or pay interest to the bank, or as an implicit (opportunity) cost, e.g. if you own the capital and by using it yourself you give up the opportunity to rent it out.

• Thank you, that does make sense. But how do I go about calculating this for one unit of capital that I purchase? Is it just the amount I pay for it? – Saad Feb 12 '20 at 15:55
• Also, is this the same value that is calculated by the equation for rental price of capital? – Saad Feb 12 '20 at 16:36
• The language is a bit confusing. The $r$ is often just called the "price of capital", as I also did in my answer, whereas what is meant is not a payment for purchasing one unit of the capital, but for renting one unit of capital for one unit of time. So yes, it's the "rental price of capital". – VARulle Feb 13 '20 at 8:15