Separate replacing capital, adding capacity, and making production more effective.
By the end: Evaluate a conditional investment calculation and explain why growth is not guaranteed by saving alone.
A firm builds capacity when it expects future benefits to justify the costs and can obtain finance and inputs. Uncertainty about sales, technology, regulation, or competitors can delay the decision. Even a profitable-looking project can be blocked by credit constraints or missing infrastructure. Conversely, abundant finance can fund projects that later fail.
Net present value discounts expected future net cash flows and subtracts the initial outlay. Paying 100 now for net receipts of 60 in each of the next two years has a value of about 4.13 at a 10% discount rate. This is a conditional calculation, not a guarantee: the cash flows, risk adjustment, and discount rate require justification. A positive private value also need not mean a positive social value if costs fall on others.
Investment uses resources before its benefits arrive. A machine ordered today may require steel, installation work, and a building that cannot simultaneously serve another project. Money finances those claims, but the opportunity cost is the best feasible alternative use of the resources. A firm with cash still faces that cost. Spending its own funds avoids a loan payment but gives up interest, another project, or a payment to its owners. Conversely, obtaining a loan does not create the skilled installers or electricity connection that the machine needs. A financing constraint and a shortage of productive inputs are different possible obstacles.
A discount rate converts receipts at different dates into comparable amounts at one date. Suppose a firm can earn a certain 10% over one year on a genuinely comparable alternative. Putting 50 currency units there today yields 55 next year. Therefore, a certain receipt of 55 next year is worth 50 today in this comparison. To reverse the growth calculation, divide 55 by 1.10. Two years require two successive growth factors. A receipt of 60 after two years must be divided by 1.10 twice, because funds available now could earn the alternative return during both years.
Consider a separate paper project costing 80 now, with certain net receipts of 44 at the end of each of two years. Assume the alternative return is 10% annually, no further cash flows occur, and the machine has no resale value. The first receipt has present value 40. The second has present value about 36.36, since dividing 44 by 1.21 gives that amount. Together the receipts are worth 76.36 today. After subtracting the 80 outlay, net present value is negative 3.64. Adding the undiscounted receipts gives 88, but the apparent excess of 8 ignores the timing of the alternative return.
The same conclusion can be checked at the end of year two. Investing the initial 80 in the alternative would produce 96.80. The project supplies 44 at the end of year one. If that receipt earns 10% for the remaining year, it becomes 48.40. Adding the second receipt of 44 gives 92.40. The project leaves the firm 4.40 behind the alternative at that date. Discounting that difference for two years gives about 3.64 today. Present value and end-date comparison are two ways of expressing the same assumed tradeoff.
Specify net receipts before doing this arithmetic. Sales revenue is not the same as cash available after materials, maintenance, and other project costs. A resale value belongs at the date of sale. Inflation also requires consistent units. Cash flows expressed at future prices need a discount rate appropriate to those money amounts; flows measured in constant purchasing power need a corresponding real rate. Mixing the two changes the comparison without changing the project. None of these calculations establishes the correct treatment of uncertain receipts or effects on people outside the firm. Those require additional assumptions.
An economy's capital stock grows when new investment exceeds depreciation. If a stock of 100 loses 10 of productive value and receives 15 of new capital, it ends at 105 under these valuations. More equipment per worker can raise output, but additional units may yield diminishing gains if knowledge, skills, organization, or complementary infrastructure do not improve.
Labor productivity is output per hour worked. It can rise through better equipment, skills, methods, or organization, and can change when the mix of industries changes. It is not simply a measure of individual effort. Output per person also depends on hours worked per person, including employment and participation. Better productivity can support higher consumption, more leisure, or some combination, depending on distribution and choices.
A stock is measured at a date, while a flow is measured over an interval. The capital stock is the productive equipment and other covered assets available at a date under a stated valuation. Investment adds assets during a period. Depreciation records the productive value used up during that period. In the reconciliation above, subtracting 10 and adding 15 explains why the stock moves from 100 to 105. This is an accounting identity under consistent boundaries and valuations. It does not say that firms chose the best machines or that customers will buy everything those machines could produce.
Take a paper workshop with equipment worth 240 at the start of a year. During the year, depreciation is 24 and new equipment adds 36. Assume no asset sales, destruction, or revaluation. Closing equipment is 252, because 240 minus 24 leaves 216, and adding 36 gives 252. Gross investment is 36, but net investment is only 12 after replacement of used capital. Net investment is 5% of the opening stock. Calling all 36 an expansion would confuse spending needed to maintain capacity with the increase in productive assets. Even the measured 5% stock increase does not establish a 5% increase in output.
To examine output, state a separate production assumption. Suppose this workshop initially makes 1,200 identical items using 600 labor hours, so labor productivity is 2 items per hour. After equipment installation and training, it makes 1,380 items using the same 600 hours. Productivity is now 2.3 items per hour. The gain is 0.3 divided by the starting 2, or 15%. The stock reconciliation and this productivity comparison use different denominators. One measures asset value at a date; the other measures a flow of output relative to hours. Neither ratio can substitute for the other.
A contrasting case keeps the new productivity of 2.3 but reduces available work to 500 hours because orders fall. Output becomes 1,150 items. That is 50 fewer than the original 1,200, a decline of about 4.17%, despite higher productivity. There is no arithmetic contradiction. Output per hour rose while total hours fell enough to offset it. In a larger economy, output per person also depends on how many hours are worked relative to the population. A productivity improvement does not require longer hours, higher employment, or any particular division of additional output between wages and profits.
Do not infer a causal equipment effect merely from the workshop's before-and-after figures. Training, product mix, worker experience, and demand may have changed at the same time. Measuring identical items avoids one quality problem but does not remove these alternative explanations. A causal study would need a credible comparison for what would have happened without the investment. The paper case stipulates its numbers to explain the ratios. It cannot establish how much of an observed economy's growth came from machines, knowledge, management, or institutions.
Receipts of 60 arrive at the end of years one and two; initial cost 100 is paid now. The required annual return is 10%. Currency units; no additional cash flows.
| Capital reconciliation | Amount |
|---|---|
| Opening productive capital | 100 |
| Depreciation | −10 |
| New investment | +15 |
| Closing productive capital | 105 |
The displayed valuation uses a cost of 100 paid now and net receipts of 60 at the end of years one and two. The 10% annual required return means dividing the first receipt by 1.10 and the second by 1.21. Their present values are about 54.55 and 49.59. Use the unrounded amounts when adding them, giving about 104.13, then subtract 100 to obtain about 4.13. The small difference between summing displayed rounded numbers and the final result is rounding, not another cost. The calculation assumes that each receipt is available at its stated year end, rather than spread through the year.
There is no investment slider or playback on this page. The equation and capital reconciliation are static worked examples. Explore them on paper by changing one assumption at a time while leaving dates and units explicit. First keep the 10% return and the first receipt of 60, but reduce the second receipt to 48. Its present value becomes about 39.67. Adding the first present value gives about 94.21, so net present value becomes negative 5.79. Losing 12 in the second year reduces today's value by about 9.92, not by 12, because the missing receipt was two years away.
Next restore both receipts to 60 and use an annual required return of 8%. The first receipt is worth about 55.56 and the second about 51.44 today. Their unrounded sum is approximately 107.00, leaving a positive value of 7.00 after the outlay. At 12%, the present values are about 53.57 and 47.83, leaving only 1.40. These are sensitivity calculations, not competing estimates of the correct rate. The alternative return should match the horizon and relevant risks. Selecting a rate just to make a preferred project pass reverses the purpose of the comparison.
A positive value also does not guarantee enough cash at every date. Suppose, separately, that the original project must repay a financing contract requiring 70 at the end of year one. Its first receipt is only 60. Without reserves or refinancing, it has a cash shortfall of 10 at that date even though the project valuation is positive at the stated discount rate. Solvency, liquidity, and project profitability ask different questions. Liquidity concerns meeting payments when due. The simple net-present-value equation does not contain a loan schedule and therefore cannot diagnose that shortfall by itself.
Finally, expected receipts need not occur. If two equally likely project outcomes yield second-year receipts of 20 and 100, their arithmetic average is 60. That average does not promise a payment of 60 in either outcome. Whether discounting an expected receipt is an adequate decision rule depends on risk, diversification, financing limits, and whose losses matter. A firm unable to survive the low outcome may reject a project that a diversified owner accepts. The displayed positive residual is a conclusion within its cash-flow and discount assumptions, not evidence that a particular real investment will succeed.
Knowledge can be expensive to discover yet cheap for others to reuse. Patents, secrecy, public research, prizes, and collaborative production offer different ways to finance discovery and diffuse it. Stronger exclusion may increase some incentives while slowing access and follow-on innovation. There is no universal optimal protection period implied by the fact that research costs money.
New technologies can displace tasks and firms while creating others. Whether affected workers find better jobs depends on skills, geography, demand, institutions, and transition support. Growth also uses energy and materials and can generate environmental damage. Higher GDP is not automatically sustainable or equally shared. A growth explanation must distinguish measurement, proximate inputs, and the institutions and choices that change those inputs.
Innovation changes feasible production methods or products rather than merely adding another copy of existing equipment. Discovery can involve a fixed research cost followed by a low cost of copying a useful result. That cost pattern creates a financing problem. Users can benefit from knowledge without each repeating the discovery expense, but a researcher who cannot obtain revenue may not cover that expense. Private incentives depend on who can charge for use, how quickly competitors learn, and whether complementary goods provide another source of income. The existence of this problem does not determine a single best ownership rule.
Consider a paper process improvement whose research requires 120 currency units of real resources now. Assume its benefit and any payments also occur now, so no discounting is needed. It saves six factories 30 each in production costs, for total resource savings of 180. If discovery is certain and there are no other effects, net resource benefit is 60. Suppose the inventor owns one factory and cannot charge the others. Its private saving is only 30, leaving a private net cost of 90. Rejection can then be consistent with a positive total benefit under these assumptions. The disagreement arises from the boundary of the decision, not an arithmetic error.
Now suppose enforceable licenses let the inventor collect 25 from each factory that adopts, including the separate operating account of its own factory. Six payments total 150. Each factory retains 5 of its 30 saving, so all six adopt under the stated rule. The research account has 30 left after paying the 120 research cost. Factories together retain another 30. Those two amounts total the original net benefit of 60. License receipts are transfers between participants; the real research cost and production savings determine this total. Charging 35 instead would exceed each factory's saving. Without some additional benefit or compulsion, the assumed adoption would no longer follow.
Uncertainty changes the comparison. Suppose successful discovery occurs only half the time and failure gives no savings, while research still costs 120 in either outcome. Expected gross savings become 90, half of 180, so expected net resource benefit is negative 30. Even complete appropriation of the successful benefit would not make this project worthwhile under a simple expected-value criterion. Public funding does not remove the cost of failure. Nor does private rejection prove that research lacks public benefits. One must separately estimate success chances, spillovers, and opportunity costs rather than moving directly from a financing problem to a preferred policy.
Institutions also affect who can bear the delay and risk. A large incumbent with retained earnings, a worker cooperative, and a small entrant dependent on a lender can face different funding constraints for the same technology. Patent scope, procurement rules, research grants, and shared technical standards change both access and incentives. Assessing them requires an explicit comparison with feasible alternatives and evidence about actual behavior. Higher measured output after adoption would still leave questions about displaced workers, environmental effects, and distribution. These are additional outcomes to investigate, not reasons to redefine a production gain as either complete social progress or necessarily worthless.
Output per hour rises from 20 to 22 while total hours stay 1,000. Find output growth. Must every worker's wage rise 10%?
Compute production, then separate distribution.
| Before | After | |
|---|---|---|
| Output per hour | 20 | 22 |
| Hours | 1,000 | 1,000 |
| Total output | 20,000 | 22,000 |
Output rises from 20,000 to 22,000, or 10%. Individual pay need not rise by the same amount. Prices, bargaining, profits, taxes, and differences across workers determine how the gain is distributed.
A project costs 100 now and pays a certain 110 in one year with no other costs. What is its net present value at 5% and 15%?
Discount the receipt before subtracting the initial cost.
| Discount rate | Present value of receipt | Net present value |
|---|---|---|
| 5% | 104.76 | 4.76 |
| 15% | 95.65 | −4.35 |
At 5%, value is 110 divided by 1.05 minus 100, about 4.76. At 15%, it is about negative 4.35. The same cash flow is attractive under the first required return and unattractive under the second, assuming the rates are the appropriate alternatives.
The lessons and answers stand alone. These optional references identify underlying models, evidence, and perspectives. All numerical examples here are invented unless explicitly attributed; none is a forecast or a personal investment recommendation.
Definitions of capitalism, production, historical change, technology, and institutions. The course is a mechanism-based introduction, not a claim that one institution alone explains growth.
Spending feedback, credit constraints, stabilization, and aggregate demand. The interactive rounds are not a calibrated policy multiplier.