University of LagosAkokaLagos

Department of Electrical Engineering

Power System Economics

Instructor: Abimbola Odubiyi1st Semester 2010

Lesson 1Understanding of Cost

Objectives:•Issues on Cost•Cost as it relates to production•Application to the power sector

Cost:

What we pay for. Proper understanding of the term cost, requires we distinguished Economic Cost from Accounting Cost.

Accounting Cost - basically the price of inputs and outputs found in a Balance Sheet. This relates basically to Expenditure incurred by an organisation in the cause of its activities.

Economic Cost* - is what we forgo or give up in order to obtain a good or service. In other words, the Opportunity Cost, which is what we give up in other to obtain something else. Efficient allocation of resources requires serious consideration of Opportunity Cost.

*In this course, emphasis will be on Economic Cost

Cost can be considered in very short, short , medium and long term.

Very Short and Short run, only the fixed cost is incurred.

The period definition depends on the industry and type of goods or services produced.

Types of CostFixed Cost (FC) Variable Cost (VC) Total Cost (TC) Average Cost (AC)

Incurred whether a good or service is produced or not (i.e. do not vary with the number of goods produced). Example, cost of land/ property, insurance, municipal levies, capital, rent, etc. Fixed cost are considered in the short to medium term, in the long term this is no more Fixed.

It varies in the short term as a result of changes in output of goods or services produced. Example, cost of materials, fuel, utilities (i.e. energy, transport, etc), labour (to some extent).

Sum of Fixed and Variable Costs engaged in the production of a good or service.

TC = FC + VC

Total Cost divided by the Quantity (Q) of good or service produced.

AC = TC/Q

Types of Cost (cont)Marginal Cost (MC) Sunk Cost (SC) Transaction Cost (TRC)

This is the change in Total Cost due to a unit change (up or down) in Quantity (Q) of good or service produced.

Also referred to as Incremental Cost.

MC = ΔTC/ΔQ

This is a cost that cannot be recovered when production of a good or service ceases.

For example decommissioning cost of a power station.

This is the cost incurred in transacting a good or service. It consist of:•search costs (the costs of locating information about opportunities for exchange) •negotiation costs (costs of negotiating the terms of the exchange) •enforcement costs (costs of enforcing the contract)

Time Cost TechnologyVery short run All costs are fixed FixedShort run Some costs are fixed FixedLong run No costs are fixed FixedVery long run No costs are fixed Not fixed

Time – Cost- Technology Dimension

Fixed Cost (FC)

Variable Cost (VC)

Total Cost (TC)Cost (N)

Quantity (Q)

Figure 1.1

Marginal Cost (MC)

Average Cost (AC)

Average Fixed Cost (AFC)

Average Variable Cost (AVC)

Cost (N)

Quantity (Q)

Figure 1.2

It’s important to know the cost of producing a particular unit of output. The cost of producing a particular unit of output is its marginal Cost (MC). From the example and the above diagrams, the MC of producing the first unit includes all the Fixed Costs (FC) and some Variable Costs(VC).

Hence, in the short run where some costs are fixed, MC (i.e. Short Run Marginal Costs SRMC) is different from the long run, where no costs are fixed (i.e. Long Run marginal Costs LRMC).

If AVC is constant, MC is level. If productive capacity becomes constrained, MC increases. For a power station the cost of producing a energy changes as more of it is produced.

MC can increase or decrease as more energy is produced. If we assume continuous changes in cost such that MC can be represented as the first derivative of Total Cost (TC).

MC= dTC/dQ. Where d is an instantaneous change in variable.

Quantity Q

Unit Price (N)

Total Cost (N)

Fixed Cost (N)

Variable Cost (N)

Average Cost (N)

Marginal Cost (N)

Total Revenue (N)

Profit (N)

0 5 9 9 0 0 -9

1 5 10 9 1 10 1 5 -5

2 5 12 9 3 6 2 10 -2

3 5 15 9 6 5 3 15 0

4 5 19 9 10 4.75 4 20 1

5 5 24 9 15 4.8 5 25 1

6 5 30 9 21 5 6 30 0

7 5 45 9 36 6.43 15 35 -10

Example:

Table 1.1

From the above table 1.1 we can determine the following:

•Total Revenue (TR) at each Quantity (Q) level •Marginal Cost (MC) •Profit at every quantity level •Fixed Cost •Average Cost

We will go through this problem step-by-step to answer those 5 questions.

Total Revenue (TR) at each Quantity (Q) level

Here we are trying to answer the following question for the company: "If we sell X units, what will our revenue be?

If the company does not sell a single unit, it will not collect any revenue. At quantity (Q) =0, Total Revenue (TR) is 0. We mark this down in our chart.

If we sell one unit, our total revenue will be the revenue we make from that sale, which is simply the price. Thus TR at quantity 1 is N5, since our price is N5. With 2 units, revenue will be from selling each unit. Since we get N5 for each unit, our TR = N10. We continue this process for all the units in the table.

Marginal Cost (MC)

MC are the costs a company incurs in producing one additional unit of a good.

In this question, we want to know what the additional costs to the firm are when it produces 2 goods instead of 1 or 5 goods instead of 4.

From Total Costs (TC), we can calculate the MC from producing 2 goods instead of 1. It is simply:

MC(2nd good) = TC(2 goods) - TC(1 good)

Here the total costs from producing 2 goods is N12 and the total costs from producing only one good is N10. Thus the marginal cost of the second good is N2.

Profit (P) at every quantity level

The standard calculation for profit is simply: Total Revenue (TR) - Total Costs (TC)

If we want to know how much profit we will receive if we sell 3 units, we simply use the formula: Profit(3 units) = Total Revenue (3 units) - Total Costs (3 units)

Fixed Cost (FC)

FC are the costs that do not vary with the number of goods produced. In the short-run factors like land and rent are fixed costs, whereas raw materials and fuel used in production are not. FC are costs the company has to pay before it even produces a single unit. Here we can collect that information by looking at the total costs when quantity is 0. Here that is N9.

Average Cost (AC)

AC is the Total Cost (TC) divided by the Quantity (Q) produced.

AC = TC/Q = FC/Q +VC/Q = AFC +AVC

Average Cost (AC) for producing 1 unit is 10/1 = N10, for 2 units will be 12/2 =N6

Economies of Scale

For many productive activities:•Average Cost (AC) is high for low output•Average Cost (AC) is lower in range of output•Average Cost (AC) is high for high level of output

The above confirms from table 1.1 for small quantities, fixed costs are high; for large quantities, variable increases as production approaches full capacity.

When Average Cost (AC) is falling, Marginal Cost (MC) is below Average Cost (AC). This is because if a unit of production costs less than AC of all previous production, then AC falls.

If unit of production costs more than average, then AC rises. Then MC is greater than AC.

If AC is constant, cost of producing the next unit is equal to AC of producing the previous units of production, so MC is equal to AC and the cost curves intersect.

Economies of Scope

When a single product firm (e.g. a power station) experiences falling AC with increases in output, and MC is below AC, then Economies of Scale results.

If the cost of producing two products by one firm is less than cost of producing the same two products by two separate firms, the production process of the former firm exhibits Economies of Scope. For example is Company A with two power stations of different capacities can operate at a lower cost combined compared to Companies B & C with one power station each, then Company A exhibit Economies of Scope.

Application to the Power Sector

Cost Symbol UnitsFixed FC N/MWh

Variable VC N/MWh

Average ACk = FC + (cfxVC)

ACE = (FC/cf) + VC

cf : Capacity factor

N/MWh

N/MWh

Table 1.3

ACk is the Average Cost of using the capacity

ACE is the Average Cost of energy produced.

Capacity factor (cf) is the fraction of time the station capacity is used.

The size or Capacity of a generator is measured by the maximum flow of power it can produce, therefore it is measured in MW.

Applying the scenario in Figure 1.1 to a power station

Gas Turbine

Coal Station

FC

VC

Cost (N)

Energy (MWh)

Type of Generator

Overnight Capacity Cost ($/KW)

Fixed Cost ($/MWh)

Variable Cost ($/MWh)

Advanced Nuclear 1729 23.88 4.16Coal 1021 14.10 11.77Wind 919 13.85 0Advanced CCGT 533 7.36 20.78Combustion Turbine (Diesel)

315 4.75 34.40

Table 1.4 Typical Fixed and Variable Costs of Generation [Source: US Dept of Energy (2001a)]

Overnight Capacity Cost ($/kW): This is the present-value cost of building the power station. For example the cost of building a 1000MW coal plant is $1,500 million. Hence its Overnight Capacity cost is $1,500/kW.

A Gas Turbine (GT) has an Overnight Cost closer to $350/kW. Although a GT is three times cheaper to build than a Coal station. However, in many parts of the world, the Coal station is cheaper to run, because its fuel cost per unit of energy (a major component of its Variable Cost) is cheaper than gas.

From tables 1.2 & 1.4 we can determine the Average Cost for the different power station technology.

For example for a CCGT, its ACk = (7.36 +20.78cf) $/MWh.

Home Work

Using data from Table 1.4, select the appropriate generator technology investment based on Average Cost of capacity using the following cf:

0.10.30.50.70.9

Lesson 2

ProfitProfit is the difference between Revenues from sales and the cost of production. The aim of most business organisation is to maximise Profit.

PR = TR – TCTR = Price multiply by Quantity sold (PxQ) : Total Revenue TC: Total CostPR: ProfitFor a particular product j,

PRj = TRj - TCj 2.1

From 2.1

dPRj / dQj = dTRj/dqj – dTCj/dQj 2.2 = MRj – MCj = 0

For profit maximisation, MCj = MRj

By chain rule of differentiation,

MRj = dTRj/dQj = d[P(Qj) . Qj]/dQj 2.3 = (dP/dQj). Qj + P 2.4

Under a competitive market condition, MRj is equal to P (i.e. the market price), because (dP/dQj) = 0.

Therefore a profit maximising firm chooses output such that price is equal to Marginal Cost (MC). Letting P = MRj in 2.2;

Then MCj =P 2.5Thus at 2.5 is when there is Economic Efficient Price.

When there are economic profits and no barriers to entering the market, the following happens:

•Other forms enter•Entering firms increase output in the market•Prices decline (because of outward shift in supply)•Economic profits decline

When there are economic losses and no barriers to exiting the market

•Firms exit•Output declines•Prices rise (because of an inward shift in supply)•Economic losses decline

In summary, the theory of supply is based on:

•A production model of a firm’s Marginal Cost (MC)•The assumption that firms choose input and output levels to maximise profit•Firms enter and exit markets based on maximising profit•Summation of each firm’s supply curve equals and industry supply curve

Industry Efficiency

Technical Efficiency: when the maximum output has been produced with a given set of inputs. This is when the minimum level of inputs have been used to produced a given level of outputs.

Ƞ = Output / Input = (Input-Losses)/Input

Economic Efficiency: when the maximum output has been produced at a given (opportunity) cost, or that a minimum (opportunity) cost has been achieved for a given level of output.

Market Efficiency: when the following occur:•Output is produced by the cheapest suppliers•Output consumed by those willing to pay for it (i.e. no subsidy)•Right amount is produced (i.e. no waste)

Under competition minimum cost is achieved when Price is equal to Marginal Cost (i.e. P = MC), which is the Economic Efficient Price.

P

Q

True demand curve

Competitive supply curve at Marginal Cost (MC)

Consumer surplus

Producer surplusCompetitive Price

Competitive Quantity

Power Supply and DemandInteraction between Demand and Supply is very crucial in the Power Industry than in most industry.

If Demand > Supply; then blackout or rationing

If Supply > Demand; then un-interrupted electricity and its continuous use

Demand for electricity is best describe by a Load-Duration Curve.

Load-Duration Curve – measures the number of hours per year the total Load is at or above any given level of Demand.Duration is measured in hours per year (i.e. 8760h) or percentage.

MW

Duration20% 50% 100%

2,000

4,000

6,000

8,000

10,000

Load

A Load Duration Curve describe the duration or period which a particular level of demand occur in a measured time period (e.g. a year).

From the above, we can deduced that for 20% of the time, demand is about 5,500MW or greater; 50% of the time, demand is about 5,000MW or greater and for 100% or whole duration demand is about 4,000MW or greater.

Price – Elasticity of Demand in the Power Industry

Price Elasticity of Demand ,means a change in Price lead to change in Quantity demanded.

ɛ = -ΔQ%/ΔP% (ɛ is –ve) = -(ΔQ/Q)/(ΔP/P) = -(dQ/dP)(P/Q)

It measures the responsiveness / sensitivity of Demand to Price. Normally referred to as Demand Elasticity. Technically Demand Elasticity is –ve, but in this course we assume it to be +ve.Example:

A 10% change in Price leads to 5% change in Demand. Therefore,

Demand Elasticity ɛ = 0.05/0.10 = 0.5

In the Power industry;

In the Short Run Demand Elasticity is 0. Which means that Demand is Inelastic, due to the fact that consumers do not change their power consumption immediately when there is change in Price of electricity.

Screening Curves

When demand is inelastic or when it faces a fixed price so that the load duration curve is fixed, this curve can be used to find the optimal mix of generation technologies

N/MWh

MW

Duration 10

Capacity Factor

GT

Coal

GT

Coal

3. Cost of Capital

Financing the an investment can involve borrowing money from financial institutions or the owner putting in its own money from his/her savings.

Borrowing money from a financial institution (e.g. a bank) requires the borrower pays interest on the borrowed money (i.e. principal). This type of fund is Debt Capital. The cost of this borrowing is the Rate of Interest charged.

However, if the owner of the investment put in its own money, this type of fund is called Equity Capital. The Rate of Return is the rate the owner earn from their investment.

Supply

Demand

Quantity of Financial Capital

Inte

rest

Rat

e

Risk

In investment activities, uncertainty is referred to as Risk. Investors will do all they can to avoid / minimise the Risk (i.e. Risk Aversion).

An investor expects to gain a premium on his investment (i.e. regain his money, plus a return that tallies with the market), to allow for the following:

Inflation (i)Risk taking (r)Expectation of a real return

Investment in Government Bonds/Securities (e.g. 3 month FGN Bond) is associated as risk-free investment. The return measured as nominal Risk-Free Interest Rate Rf.

Rf = i + rf +(i . rf) 3.1 ≈ i + rf 3.2

i : inflation, rf : risk-free rate

If inflation is low (e.g. < 5%) and real risk free rate is low (e.g. <3%), then 3.2 prevails. Hence rf = Rf – i.

For risky projects, investors charge a risk premium (RP) to compensate them for the unknown rate of return.

The real (risky) interest rate is equal to the real risk-free rate plus a real risk premium (RP).

i.e r = r f + RP 3.3

The nominal (risky) interest rate is equal to the nominal risk free rate plus a nominal risk premium, RP*, which also compensates the lender for the riskiness of an unknown inflation rate.

i.e R = R f +RP* 3.4

Discounting

Financial flows (streams of expenditures and income) of money from projects occur after a year onwards for the entire life of the projects. Hence due to inflation, the value of N1 million today, is different in 3 years and very different in 10 years time.

To understand the proper worth of money in future years, its value has to be adjusted to today’s value or the base year value. This phenomenon is term Time value of Money for the following reasons:

•Future incomes are eroded by inflation. The purchasing power of N1m today is higher than N1m in 2 year’s time•The existence of risk, due to future uncertainty.•The need for a return on the money by the investor. An investor needs to be reward for forgoing expenditure today and undertaking the investment for future benefits.

Due to eroding effect of inflation, to properly account for the worth of a Future earnings in today’s money, we apply a Discount Factor to the Future earnings.

At an interest rate r, one Naira today (Present Value) will be worth N1r in one year’s time (Future Value). FV = PV + r . PV = (1+r) . PV 3.5

PV = [1/(1+r)] . FV 3.6

1/(1+r) or (1+r)-1 is the Discount Factor (DF) (i.e. the Present Worth Factor), which is who much a Naira in the future is worth today.

From the above, the Discount Factor in year n = 1/(1+r)n 3.7

Present Value in year n PVn = FVn x [1/1+r)n] 3.8

Therefore assuming a Discount Factor of 10%, a N10,000 earnings in 5 years will have a Present Value of

N10,000 x [1/(1+0.1)5] = N6,209.21.

Similarly N10,000 occurring in 30 years is N573.1 in today’s money.

(1+r)n is the Compound Factor (CF) 3.9 This applies when past earnings are to be expressed in today’s worth.i.e. N1,000 earn last year at interest rate of 10% is worth N1,000 x (1+0.1)1 is N1,100 in today’s money.

Taking current year as the base year, for future years, n is (+ve), for past years n is (-ve), for current year n = 0 (i.e. base year).

The annual Discount period can be broken down into sub periods within the year. For example into monthly and daily.

For Monthly period

DF monthly = [1+(r/12)]-12

For Daily period

DF daily = [1+(r/365)]-365

Using Compound Factor (CF) on FV, on a continuous timescale, CF = [1+(r/m)]m ≈ er

If we are to consider a series of cash flows in future years discounted to the present.

Annuity: This is a case when we expect to earn the same amount A over a period of time in the future from an investment . The Present Value (PV) of such investment is expressed as

PV0 = A∙ (1+r)-1 + A ∙ (1+r) -2 + A ∙ (1+r) -3 + …… + A ∙ (1+r) –n 3.10

n can be number of years

PV0 = A ∙ [ (1+ r) n -1 ] / [ r ∙ (1+r) n ] 3.11

The inverse of the last part of 3.11 = [r(1 + r) n] / [(1+r) n -1] is the Capital Recovery Factor (CRF).

A = PV0 ∙ CRF 3.12

In Investment terms A is known as the Levelized Capital Cost (LCC). Which gives a uniform payment from future years from an investment made in the base year.

Example: What amount must be collected for 30 years at annual interest rate (r) of 10%, if an initial investment of N1000 is made today

Using equation 3.12, with r = 10%, a sum of N106.08 must be collected each year for 30 years.

As the number of periods (e.g. years) increases, Annuities approach a Perpetuity, and PV ≈ A∙ r or A ≈ PV ∙ r

Net Present Value

The electricity industry is capital intensive. Investment are based of the prospects of future profits.

Profit (PR) wish is (i.e. +ve net cash flow) = Total Revenue (TR) – Total Cost (TC)

The Net Present Value (NPV) of Profit over a time period n is expressed as:

NPV0 (PR1, PR2, …, PR n) = PR1(1+r) -1 +PR2(1+r) -2 + …. + PR n(1+r) -n

= (TR1 – TC1) ∙ (1+ r) -1 + (TR2 – TC2) ∙ (1+r) -2 + ….. + (TR n – TC n) ∙ (1+r) -n 3.13

NPV of an investment is the Present Value of future Revenues (TRn) less Present value of Future Cost (TCn)

Since Total Cost (TC) = Fixed Cost (FC) + Variable Cost (VC)

Then NPV0 = -FC0 + (TR – VC) ∙ [ (1+r)n -1] / [r ∙ (1+r)n ] 3.14 = -FC0 + (TR - VC) ∙ (1/CRF) 3.15A firm will only invest in a project with +ve NPV

Internal Rate of Return (IRR)

With IRR, the investor calculates the rate of return that yields NPV = 0, and compares with the firm’s rate of return.

If IRR is > cost of capital, the firm selects the project.

From 3.14 NPV0 = 0 = -FC0 + (TR – VC) ∙ [ (1+IRR)n -1] / [IRR ∙ (1+ IRR)n ]

FC0 / (TR - VC) = (1/IRR) – [1/IRR ∙ (1 + IRR)n ] 3.16

If IRR > cost of capital, then firm should invest.

(Equation 3.16 is difficult to solve analytically, so it is usually solved numerically)

Weighted Average Cost of capital (WACC)

For most capital intensive investment (e.g. building a power station), a firm uses debt and equity to finance a project.

For this type of investment, the cost of capital is a weighted average of the rate of interest on debt and the expected return on equity.

Thus Weighted Cost of Capital (WACC)

= [Debt / (Debt + Equity)] ∙ R debt + [Equity / (Debt + Equity)] ∙ R equity

For example, if 70% of an electric utility’s financing is in corporate bonds and 30% is in corporate equities, R debt = 7% and R equity = 12%, then WACC = 8.5%

Payback Analysis

Payback Rule is used in small investment. For example whether to install a fluorescent light bulbs to reduce cost of electricity.

The decision to be made is to make the investment that pay back the cost of the initial investment after some arbitrarily chosen (payback) period, e.g. 2 years.

For example, if

The cost of fluorescent bulb is N10 greater than the cost of the incandescent bulbThe reduction in electricity is 100 wattsThe bulb is used 10 hours daily,The price of electricity is N0.10/kWh (or N100/MWh)

Then the payback period is 100days.

NPV equivalent payback period = (1/r) -1 / [r ∙ (1 + r)n ]

Risk and Return

An investor seeks is debt finance from the money/financial markets. Financial markets determine the risk premium for risky business.

From the money market we can determine the probability distribution for rates of return r n to an investor over the n periods: r 1, r 2, r 3,…….,f n

For example, if a firm earned rates of return of 5%, 10%, 0% and 5 % during the last 4 years, the mean is the average of these values : 5%. So the Expected Return on investment must have rate of return >= 5%