Why is the price of capital ‘r’ ? (From Cost function)

Question

according to the Cost formula in microeconomics class,

Total Fixed Cost is represented as “rK” (K as in unchanging, fixed K)

now my economics teacher tells me this ‘r’ is the interest rate at which you rent capital, which is pretty vague and ambiguous to me. If r represents interest “rate” and not the “interest-added cost of capital”, then how can this r be the “price” of capital?

So for example, if the amount of capital is 5 (K=5) I understand that you assume you rent it, or borrow the money from a bank to get the capital, incurring interest. What I don’t understand is the formula. If r is the interest rate (aka rental rate)

isn’t the cost formula supposed to be “K * {P(K) + P(K)*r}”, instead of “rK”? (P(K) here represents the price of capital per unit)

so if a unit of capital costs $10, and the interest rate (r) is 3%,

in the end you will be paying $ 5 * (10 + 10 * 0.03) in total , not 5 * 0.03 right?

I mean, r is just the percentge, not the ‘amount’ that you have to pay for the capital. So I don’t really understand why the textbooks and the teachers tell me that the “price” of capital is r when it seems like it’s supposed to be ‘P(K) * (1+r)’ instead of just ‘r’ .

I hope I can get a satisfying answer here. Would appreciate detailed explanations!

Answer 1

Firstly, in your example the value of $r$ (as used by economists in this context) would be $1.03$, not $0.03$. Economists call this the "interest rate", but you might prefer to think of it as the "rate of return on capital".

Secondly, what we define as constituting one unit of capital is pretty arbitrary. Is a computer one unit of capital or ten units of capital? It doesn't make any difference, just like it doesn't matter whether you measure distance in meters or feet, so long as everyone knows what units you are using. Since this choice is arbitrary, economists usually make the simplest possible choice and measure units of capital in units of money (e.g., dollars), meaning that one unit of capital has a price of one dollar by definition: $P(K)=1$.

If we return to your formula with $r=1.03$ and $P(K)=1$:

$$K[P(K)+0.03P(K)]=P(K)K(1+0.03)=Kr,$$

we get exactly the cost you saw in your course.

Answer 2

Just like the wage $w$ is the (rental) price of buying one unit (say, one person-hour) of labor you can think of $r$ as the (rental) price of renting one unit (say, one machine-hour of a standardized machine) of capital, such that using $K$ units of capital costs you $rK$. In textbooks you can therefore see examples stating that e.g. the price of capital is $r$ = \$$10$.

On the other hand, if the interest rate is $i=3\%=0.03$, then a producer of machines in a competitive rental market would rent out a machine which costs \$$P$ to produce for \$$iP$ per year (under the usual simplifying assumptions), which translates to \$$iP/8760$ per hour. If you now measure capital conveniently in \$$P/8760$ units instead of in machine-hour units, then using $K$ of these (new!) units of capital costs you $iK$.

The $r$ in the term $rK$ can therefore be interpreted both as a (hourly rental) price of capital or as a (yearly) interest rate, depending on the units in which you choose to measure capital. Since the choice of units is arbitrary, the former variant is typically used for exercises while the latter variant is used for explanations, since it establishes the macroeconomically important connection to prevailing interest rates.

Remark: Using $r=1.03$ if the interest rate is $3\%$, as the previous answer suggests, only makes sense if you assume that a machine fully depreciates within one year and has to be replaced on a yearly basis, which is not what is typically assumed.