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Capital and Net Present Value

19 一月, 2016 - 17:17

Suppose Carol Stein is considering the purchase of a new $95,000 tractor for her farm. Ms. Stein expects to use the tractor for five years and then sell it; she expects that it will sell for $22,000 at the end of the five-year period. She has the $95,000 on hand now; her alternative to purchasing the tractor could be to put $95,000 in a bond account earning 7% annual interest.

Ms. Stein expects that the tractor will bring in additional annual revenue of $50,000 but will cost $30,000 per year to operate, for net revenue of $20,000 annually. For simplicity, we shall suppose that this net revenue accrues at the end of each year.

Should she buy the tractor? We can answer this question by computing the tractor’s net present value (NPV), which is equal to the present value of all the revenues expected from an asset minus the present value of all the costs associated with it. We thus measure the difference between the present value of marginal revenue products and the present value of marginal factor costs. If NPV is greater than zero, purchase of the asset will increase the profitability of the firm. A negative NPV implies that the funds for the asset would yield a higher return if used to purchase an interest-bearing asset. A firm will maximize profits by acquiring additional capital up to the point that the present value of capital’s marginal revenue product equals the present value of marginal factor cost.

If the revenues generated by an asset in period n equal Rn and the costs in period n equal Cn, then the net present value NPV0 of an asset expected to last for n years is:

Table 13.6
NPV0=R0−C0+R1−C11+r+...+Rn−Cn(1+r)n
 

To purchase the tractor, Ms. Stein pays $95,000. She will receive additional revenues of $50,000 per year from increased planting and more efficient harvesting, less the operating cost per year of $30,000, plus the $22,000 she expects to get by selling the tractor at the end of five years. The net present value of the tractor, NPV0 is thus given by:

NPV0=−$95,000+$20,0001.07+$20,0001.072+$20,0001.073+$20,0001.074+$42,0001.075=$2,690

Given the cost of the tractor, the net returns Ms. Stein projects, and an interest rate of 7%, Ms. Stein will increase her profits by purchasing the tractor. The tractor will yield a return whose present value is $2,690 greater than the return that could be obtained by the alternative of putting the $95,000 in a bond account yielding 7%.

Ms. Stein’s acquisition of the tractor is called investment. Economists define investment as an addition to capital stock. Any acquisition of new capital goods therefore qualifies as investment.