Correcting Market Distortions: Shadow Prices and Discount Rates Chapter 6 Observed market prices sometimes reflect true cost to society. In some circumstances they dont because there are distortions which prevent market prices from conveying true economic values. When this occurs have to correct observed price to calculate the shadow price.

Types of distortions include taxes, subsidies & other forms of govt intervention. In competitive markets D represents marginal benefits to society and supply curve social costs. Social costs are equal to private costs. Likewise private benefits equal social benefits. A Market with a Per Unit Tax Suppose have a market for good but price observed

for the good includes a per unit tax, here price consumers pay is not the price the firms keeps. T is the tax Pc = Pf + T Pc price gross of tax Pf price net of tax

Project Demand with a Per Unit Tax Suppose theres a project that requires the good as an input. Demand for the good increases leads to new equilibrium at point C Output increases from Xe to Xf price firms retain increases

from Pf to Pf Price consumers pay increases from Pc to Pc Non-project demand for the firm falls from X e to Xc Note that the Government requirement of X G comes from two sources: Xf - Xe units of new supply Xe - Xc units of displaced demand

If market werent distorted by the tax, there would not be a problem because consumers marginal benefit would equal the firms marginal costs, this not the case here because of tax (the competitive output should be at Xf ) The tax has driven a wedge between consumers and firms valuation of this input. The tax creates a problem for someone trying to value the input because the market outcomes are distorted by the tax. What the shadow price does is try to take the distorted prices and

correct them for the distortion to get a valuation/price that is distorted. In this example the shadow price takes a weighted average of the opportunity costs of the two sources of the govts input requirement. For example, Suppose the govt needs XG units of X to complete the project, can calculate PG the shadow price as either: or Where Pf price net of tax and Pc is the price gross of tax (Pc = Pf +T)

An alternative expression of the shadow price in the previous example uses elasticities ,where is the elasticity of supply and is the elasticity of demand The shadow price PG will depend critically on elasticities; elasticities will determine how big increases are in new demand as well as how big is displaced demand. Recall that the elasticity determines the slope of the demand and supply curves. A more elastic demand(supply) curve will be flatter

A more inelastic demand(supply) curve will be steeper D1 is flatter than D2 D1 is more elastic than D2 Note that in general the shadow price will fall between gross of tax and net of tax price. However, there are some special cases where

the shadow price takes on specific values. These extreme cases occur when the demand is prefectly elastic and inelastic and supply is perfectly elastic and inelastic Extreme Cases Distortionary Subsidies Analysis is basically the same as a distortionary

tax Choosing and Computing a Discount Rate Recall the NPV =, where r is the discount rate and B and C represent benefits and costs, respectively. The NPV will depend on r as well as benefits and costs. a smaller discount rate will lead to larger values of the NPV, large values of the discount rate lead to smaller values of the NPV

a discount rate of 0 means that society weights the future equally to the present, thought to be altruistic discount rate Marginal rate of time preference Consider whether someone wants a $1 today versus tomorrow Whether someone picks to have the $1 today or tomorrow reflects their time preference, or how they trade off between these alternatives

For example, suppose you have the choice of $1000 today or $1200 one year from today, if you pick $1000 today then your rate of time preference is 20%; you would have a stronger preference for having something today. Can formalize the idea of time preference and choosing between today and tomorrow with the following model. Suppose individuals choose between

consumption today and tomorrow, denoted and subject to a lifetime budget constraint. Assume that individuals have preferences over consumption today and tomorrow The individuals problem can be written as where is the interest rate and T is the present value of income over the individuals lifetime (periods 1 and 2 in this example). Well discuss the solution to this problem in

graphical terms, Absolute value of slope of the indifference curve measures the rate at which individuals are indifferent between substituting current consumption for future consumption, i.e., the MRS between consumption this year and consumption next year, where and is the marginal rate of time preference. An equilibrium for this problem is where the rate at which people are willing to trade consumption today

and tomorrow equals the price of moving consumption allocations, i.e., the interest rate An equilibrium, will occur when the indifference curve is tangent to the budget line, i.e., where If you can freely borrow then you can shift consumption to the future until the MRTP falls to the interest rate you must pay If then save and reduce consumption today If then borrow and increase consumption today

In a prefect capital market Investment demand - Looks at firms making investment decisions - Assumes perfect capital markets - A firm has a variety of investment projects to

select from which have different rates of return associated with them. supply of funds for investment is provided by individual saving if rate of interest > rate of time preference then save

represented by Aggregate savings schedule Market equilibrium occurs where supply of savings schedule equals the demand for investment funds, where rate of return

equals the rate of time preference; the equilibrium point is the market interest rate The previous equilibrium is based on the assumption of prefect capital markets. Generally, the real world is not comprised of perfect capital markets since there are distortions, e.g., taxes, risk, govt borrowing,

which all drives wedges between market and social outcomes, and, consequently, society can end up with under investment. Market Equilibrium with Distortions On previous slide and represent investment demand and supply of funds without taxes Introduction of taxes (both corporate and personal) shifts back the investment demand and supply of

funds curves, denoted by and With taxes the market clearing interest rate would be The marginal return on investment before taxes would be , the opportunity cost of forgone investment The marginal rate of return on savings after taxes would be Suppose the government undertakes a new project/program that it funds by borrowing. This would shift out the demand for funds, shifts out to

Private sector investment falls by crowding out effect Arnold Harberger using this framework suggests the following estimate of the social discount rate: Some empirical evidence suggests that savings is not very sensitive to interest rates, which implies that the savings schedule would be relatively inelastic (i.e., vertical), so that and

and , which implies that Another approximation to social discount rate would be Some argue in favour of as an approximation to social discount rate because social discount rate should be rate at which individuals should be willing to postpone a small amount of consumption for future consumption.

As with shadow prices, the marginal rate of time preference and the rate of return on capital can be distorted. The distortions can include taxes, inflation and risk (default or bankruptcy) Like shadow prices, we can take observed interested rates and correct them for the various distortions. Computing

proxies for a rate of return on low risk private sector investments before taxes but after correcting for inflation Suggests that we can take an observed interest and correct/adjust it to get an estimate of Want to use a low risk corporate bond, so it would have a lower default risk and adjust it for taxes and inflation Three steps in computation, assume that corporate bond rate is 6.86%, corporate tax rate is 35% and

inflation rate is 3.92%: Computing : An Example 1. Figure out before return 2. Adjust for inflation 3. Adjust for bias in CPI Computing proxies for a rate of time preference after correcting for inflation and taxes

Suggests that we can take an observed interest and correct/ adjust it to get an estimate of Want to use a government bond, and a higher level of government, e.g., Federal first, provincial second, and lastly local, so it would have a lower default risk and adjust it for taxes and inflation Three steps in computation assume that interest on government bond is 6.77%, personal tax rate is 30% and inflation rate is 3.92%

Computing : An Example 1. Figure out after tax return 2. Adjust for inflation 3. Adjust for bias in CPI 0.0474 Criticisms tends to produce large discount rate estimates;

computations are based on using corporate bond, which may have a risk premium (e.g. firm may go bankrupt, investors want a higher return to cover this) produces discount rate that are too low; individuals may not properly account for the long run effects of infrastructure programs on future generations Weighted Social Opportunity Cost of Capital

(WSOC) An alternative approach for computing the social discount rate. Takes the perspective the discount rate should reflect social opportunity cost of the resources required for a project, with weights based based on the relative contributions of the different sources of resources The weighted social opportunity cost of capital

can be computed as, where a is the proportion of the projects resources that displace private investment, b is the proportion of resources that are financed by borrowing from foreigners, (1-a-b) is the proportion of resources displacing domestic consumption, and is the government's real long-term borrowing rate Since

We already know how to compute and , but not ; However, is relatively straightforward to compute. Recall that is the governments real long term borrowing rate, so all we need to do is adjust a nominal return government bond for inflation to obtain Computing Only two steps are need to compute . (Figures

continue from previous example) 1. Adjust for Inflation 2. Adjust for Bias in CPI 0.0268+0.01=0.0368 Note: there is no adjustment for taxes because the government doesnt pay taxes to itself. are relatively easy to compute based on available interest rate data The weights, i.e., a, b and (1-a-b) are harder to determine

In a Canadian context, Jenkins suggested using the following values: a=0.75 and b=0.20,which suggest that WSOC=0.75(0.0738)+0.2(0.0368)+0.05(0.0173)= 0.0636 or about 6.4% On the other hand, Burgess suggests that for Canada a is likely to be between 0.26 and 0.32, b is between 0.55 and 0.64 and (1-a-b) is likely to be between 0.1 and 0.13. Picking the

smaller value of a and the bigger value of b produces a smaller value of WSOC; e.g., WSOC=0.26(0.0738)+0.64(0.0368)+0.05(0.017 3)=0.0436 or 4.4% As another example, Suppose have a project that is financed exclusively with taxes, then b=0. The weight should represent the proportion of taxes that reduce investment and 1-a-b should represent the proportion of taxes that reduce consumption.

One can obtain an estimate of a with the ratio of gross fixed investment to real GDP. Recently, this ratio was computed as 16.8%, so that WSOC=0.168(0.0738)+0.0(0.0368)+0.832(0.0173)=0 .0268 or 2.7% Rules of Thumb: United States What do policy makers use in practice? In the United States the Office of Budget Management used a real discount rate of 10 percent during the 1970s,

but had lowered this estimate to about 7 percent by 1992. Recently, the Congressional Budget Office and the General Accounting Office have used the approach to get a discount rate of about 2 percent. Municipalities in the United States tend to use discount rates of 3 percent with sensitivity analysis between 0 and 7 percent. Rules of Thumb: Canada The Federal Treasury Board Secretariat has recommended

from about 1976 to the late-1990s, a discount rate of 10 percent, with a sensitivity analysis at 5 and 15 percent. But they recommend much lower discount rates (0 to 3 percent) for health or environmental cost benefit analysis. More recently, the Treasury Board Secretariat (recommends) a discount rate of about 8 percent, with a sensitivity analysis of 3 and 13 percent. The Treasury Board Secretariat also estimates the social rate of time preference of about 3 percent.