Chapter 8: Choosing a Discount Rate

The Ideal Market for Loans Demand for loans summarizes borrowers’ choices Supply of loans summarizes lenders’ choices

The Supply of Savings and Individual Time Preference  Given the market interest rate, an individual chooses how much of her current income to consume immediately and how much to save.  Income which is saved collects interest, leading to increased consumption in the future.  Jill has $100 which she can spend now or save for next year at 5% interest.  If Jill consumes $50 and saves $50 this year she will have $50(1.05) or $52.50 next year.

The Efficiency of Capital and the Demand for Loans  The demand for loans in Figure 8-1 is based primarily on the investment decisions of businesses.  Rational and informed firms will choose to invest whenever the returns to capital investment, corrected for factors such as added risk or inflation, are greater than the returns to saving.

Figure 8-4: A Savings-Investment Equilibrium  Figure 8-4 the bulging line is a production possibilities frontier  The curve labeled U 1 represents the highest possible social indifference curve that can be achieved given people’s tastes and the limited productive resources of society.  The straight line T represents the market price of trading current for future consumption. In this perfectly competitive case slope of this line will be -(1+r*), where r* is the competitive social discount rate.

The Effect of Taxes  In Figure 8-5 both the supply of loans (individual savings) and the demand (returns to investment) are taxed.  At investment level I 1 investors will receive a before-tax return of r 3 and an after-tax return of r 2.  Similarly, savings brings a before- tax return of r 2 but an after-tax return of only r 1.  r 1 is the social rate of discount because it represents the marginal benefit to a dollar of saving to individuals after taxes. However, r 1 is below the optimal social rate of discount r* because of the tax effect.

Welfare Loss from Taxes  In Figure 8-6 the equilibrium occurs at point A‘. At this point the slope of the PPF equals –(1+r 3 ) while the slopes of budget line T and the social indifference curve equal – (1+r 1 ).  Also, at point A’ there is more current consumption and less investment and saving than at point A.  The lower social indifference curve clearly shows the loss of efficiency from this inequality in the marginal rates of return

The Shadow Price of Capital Method Estimating the Present Value of a project using the shadow price of capital method takes 4 steps:  Step One: Estimate the Project’s Effect on Investment According to Harberger, nearly all borrowed funds can be assumed to displace investment. On the other hand, projects funded through taxes are more likely to displace consumption.

The Shadow Price of Capital Method  Step 2: Annualize the Capital Cost  The annualized value of capital cost is found by solving the following present value equation for X where r 3 is the rate of return to capital and n is the number of years in the lifespan of the displaced capital.  This equation can be converted to a more useful form that is similar to the annuity value equation introduced in Chapter 7.

The Shadow Price of Capital Method  Step 3 : Include these annualized capital costs from equation (8-2) as a negative value in the net benefits of the project.  Step 4 : Discount the stream of net benefits at the consumption rate of interest (or social rate of discount).

The Shadow Price of Capital Method: An Example for the Student  Assume that the U.S. government is funding a remedial education and employment program called “No Adult Left Behind.” The program is funded through borrowing. The project requires an initial investment of $20 billion dollars, will bring benefits of $6 billion per year to society, and will face operating costs of $2 billion per year after the initial training period in year zero. Assume that displaced consumption and investment both equal $10 billion dollars.  Your Turn 8-1: If the project lasts 5 years, find the annualized lost consumption from the 10 billion dollars of displaced investment using a return to capital of 7 percent.  Your Turn 8-2 : Including the displaced consumption and the other costs and benefits above, find the present value of the project lasting 5 years.

The Weighted Discount Rate  An easier but less accurate method of dealing with the consumption versus investment issue involves discounting the project using a weighted average of the rate of return on capital and social discount rate on consumption.  (8-4) Weighted r = ar 1 + (1-a)r 3 where a is the fraction of the project’s cost which is financed by displaced consumption and (1-a) is the fraction which displaces investment.  With foreign lending included, the weighted discount rate becomes (8-5) Weighted r with foreign lending = ar 1 + br f + (1-a-b)r 3 where r 1 and r 3 are the social discount rate and return to capital, r f is the interest rate on foreign lending, and b is the fraction of the project’s cost financed by foreign lending.

Other Issues in the Discount Rate

Inflation and the Discount Rate  Concept: The real interest rate is the rate of interest corrected for inflation. It is generally defined as the nominal rate of interest minus the inflation rate. If the real interest rate is r, the nominal interest rate is i, and the inflation rate is p, the approximate formula for the real interest rate is r = i - p.  If you are using real costs and benefits, a real interest rate is correct. If your costs and benefits are nominal, your discount rate should also be nominal.  If you are estimating the present value of future net benefits, either adjust them upward for future inflation or use a real interest rate when calculating present value.

Inflation Bias in the Official C.P.I  The most commonly used measure of inflation in the U.S. is the Consumer Price Index for all urban workers (CPI-U). This index measures changes in the cost of a fixed set of goods over time.  A price index that measures the cost of a fixed set of goods is likely to exaggerate the costs of inflation for a number of reasons.  Substitution bias refers to the tendency of consumers to substitute away from goods whose prices rise more than the average.  Same store bias : The CPI ignores the gradual movement of consumption to discount stores,  New Product Bias: New goods tend to experience steep price decreases in early years,  Quality bias: Some goods increase greatly in quality over time, particularly technological products such as consumer electronics and automobiles.  In the 1990s the Boskin Commission released a detailed study of these biases and concluded that the consumer price index was biased upward by about 1.1 percentage points per year.

“Chain Weighted” Price Indexes  Chain-weighted price indexes are based on annual spending data rather than a fixed market basket and therefore do not suffer from substitution, same store, or new product bias.  U.S. chain weighted indexes include the chain weighted CPI published by the Bureau of Labor Statistics and the chain weighted GDP price index published by the Bureau of Economic analysis, and the GDP price deflator, which is similar to the GDP price index.

Alternative Price Index Values  Note that the CPI-U is higher than the chain weighted indexes in nearly every case.

The Project’s Time Frame  Interest rates vary with the length of time of the loan, and vary over time with variations in demand and supply conditions.  Therefore, the discount rate should be based on an interest rate for low risk assets such as government bonds that have a time frame similar to that of the investment project.

Intergenerational Equity and Discounting for Long Term Projects

High Discount Rates and Distant Future Present Values  Policies related to long term problems such as nuclear waste or climate change have effects lasting many generations into the future.  Even 100 years is enough for a high discount rate to reduce present value to near zero.

2 Methods of Including Future Generations in Present Value  The utilitarian social welfare function, which weighs current and future utilities equally, may allow some discounting based on growing average incomes and declining marginal utility of income.  The declining discount rate model which is commonly (but less clearly) referred to as a hyperbolic discounting model, lowers the effective rate of discount in future years.

 The utilitarian social welfare function for different generations is: (8-7) SWF U = U (B 1 -C 1 ) + U (B 2 -C 2 ) + U (B 3 -C 3 ), where U refers to utility.  Note that the utilities of the 3 generations are equally weighted and no discounting occurs for utility.  If future generations are richer and the marginal utility of income falls as income rises, some discounting based on the falling marginal utility of income is acceptable according to utilitarianism.

The Ramsey Formula and Discounting  Given the utilitarian assumption that all generations’ utilities will be equally weighted, the discount rate for income or dollar net benefits across generations takes the following form:  (8-9)r g = ηρ g.  where r g is the rate of discount across generations, η = marginal utility of consumption, and ρ g is the annual growth rate of consumption.

The Range of Discount Rates using the Ramsey Formula  Representative estimates of the marginal utility of consumption (η) include 1.6 and 0.7.  The annual growth rate of consumption (ρ g ) is somewhere between 1.5 and 2 percent per year, the mean real growth rate in developed countries over the 20th century. Therefore estimates for ηρ g range from 1.05 to 3.2 based on these representative figures.  These rates are lower than most short term discount rates based on the shadow price of capital or other methods.

Declining Discount Rates  Declining Discount rate models come in many forms, and have multiple justifications. For example, one could define the present value of $1 in year t as  (8-10) PV ddr = 1/(1+ α t), where α is the discount rate and t is time.  Table 8-3 compares the declining discount rate and standard present value using a 5% rate.  Table 8-3 compares values for the declining discount rate and standard present value discounting formulas using a 5 percent rate of interest. This example demonstrates that declining discount rate models are accurately named,

Conclusion  The goal of this chapter was to provide a more detailed view of the theoretical challenges in determining an ideal discount rate for public projects.  There is no generally accepted discount rate for public projects, although economists have moved closer to a consensus over the past two decades for relatively short run projects.  The issue of intergenerational equity in very long run policy analysis remains a controversial issue, however.