### Basic Engineering Ecomomics

##### Anil Pahwa

Most of the engineering decisions require commitment of financial resources. Some of these resources would be spent in the present and more would be needed for the future. In addition, there could be benefits that accrue on an annual basis. For example, consider a utility that decides to install capacitors in distribution system to improve power factor and reduce losses. This would require spending money to purchase capacitors and related hardware and for their installation. Subsequently, the utility will have to spend certain amount of money annually to maintain the capacitors, but reduction in losses will result in an annual benefit. Since expenses are taking place in different years, we cannot simply add all the costs and all the benefits to obtain net expenses for the project. This is so because the value of money is not constant but decreases from one year to the next. Hence we need to examine some of the factors that are commonly used in engineering economic analysis and arrive at simple approaches to determine costs and benefits of projects over a period of time.

##### Interest Rate

Interest rate is the rate that is applied to a financial transaction. The rate can be different depending on the type of transaction. For example if you deposit money in your checking account you get a very small interest, but if you deposit the money in a certificate of deposit (CD) you will get higher interest. However, you won’t be able to use the money that you deposited in a CD until it matures. Similarly, the interest rate that you have to pay to borrow money to buy a car is very different from the interest rate that you would pay to buy a house. The rates are different due to risks, flexibility, and other factors related to the transaction.

##### Inflation

Inflation is the rate at which the price of goods and services increase with time. Do you remember your parents telling you that when they were your age certain item used to be so cheap and that it has become too expensive now. Whether one likes it or not, inflation in fact is an integral part of the modern economy. It varies from one year to another and from one country to another. Nobody seems alarmed if inflation stays at a small value, but everyone gets worried if its value becomes very large. Inflation, however, is a complex phenomenon and its value at a given time depends on spending behavior of people, policies of the government, international events, and many other factors. It is almost impossible to predict inflation for the future years.

**Discount Rate**

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Discount rate is the rate that is used by businesses to discount money with time to make financial decision that span several years. Different businesses use different discount rates based on their specific conditions. This topic is rather complex and thus we will not go into the details of it. Most people consider discount rate to be related to opportunity cost of investing capital. In other words, if an investor considers receiving $100 today as equivalent to receiving $112 one year from today, then the discount rate is 12% for that investor. Discount rate is usually much higher than the interest rate that you would get from the bank since it includes risk, cost of capital, government policies, and other business factors. Generally, discount rate used for engineering economic analysis does not include effect of inflation. However, if inflation rates for future years are available, they can be included in the discount rate. If the discount rate does not include inflation, all the future costs and benefits are represented in real dollars (today’s dollar), but if it does include inflation then all the future monies are represented in inflated or nominal dollars [Khatib]. It turns out that either way the results are the same. Therefore, most people prefer to leave inflation out to keep things simple.

**Time value of money**

As we have seen in the previous sections, the money loses value with time, i.e., the more the investor waits to receive the money the higher the amount he would expect. It is a common practice in financial analysis to convert all the future financial outlays to a present value by defining present worth factor or discount factor, which is a function of discount rate (*r*) and the number of years from the present (*n*):

Thus, a discount rate of 12% and *n* = 1 gives a present worth factor of 0.893, which implies that $100 one year today have a present worth of $89.30. Similarly, n = 5 with the same discount rate gives a present worth factor 0.567, and therefore, $100 five years from today have a present worth of $ 56.70.

##### Annuity

Annuity is a term that defines equal financial outlays that occur at fixed interval of time. For example, if you take a loan of 3 years from the bank to buy a car, you will be paying the bank a fixed amount of money every month for the 3 years. The amount of payment will depend of amount of loan and the interest. Similarly, in engineering applications, the same amount of expense or benefit may take place on a yearly basis, such as annual cost of maintaining an equipment or annual benefits accrued due to implementation of a new technology.

##### Present Worth of Annuity

An annuity of $A over *N* years can be converted to present worth by first finding the present worth of financial outlay for each year and then summing the present worth,

Now add and subtract A from the right hand side of this equation and let Therefore,

The quantity within the parenthesis is a finite geometric series equal to . Upon substitution and simplification we get,

##### Present Worth of Geometric Series

In some situations certain quantities will increase at a certain rate annually. An example of such a quantity is losses in distribution feeders. Since losses are dependent on load and if the load grows at a certain rate, the losses will also grow annually. With *j* as the rate of load growth, we get a geometric series,

for *k *= 2 to *N*

* *

Where is the cost due to losses in year *k*. The present worth of such a series over *N *years with the total cost at the end of the first year equal to A is

or

where .

Now, the quantity within the parenthesis is . Substituting this in the above expression and simplifying gives

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