Correcting Market Distortions: Shadow Prices, Shadow Wages and Discount Rates Chapter 6 • Observed market prices sometimes reflect true cost to society. In some circumstances they don’t 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 gov’t 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 there’s 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 Xe to Xc • Note that the Government requirement of XG comes from two sources: – Xf - Xe – units of new supply – Xe - Xc – units of displaced demand • If market weren’t 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 gov’t’s input requirement. – For example, Suppose the gov’t 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 − Ω = ,where Ω = , is the elasticity of −Ω 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 Sudsidies • Analysis is basically the same as a distortionary tax Shadow Wage Rates • Governments sometimes use discretionary fiscal policy to create jobs. • If labour markets functioned perfectly market wages would reflect true opportunity cost of hiring labour for a project • However, markets usually don’t function perfectly because of distortions so have to calculate the shadow price of labour. i.e. the shadow wage rate. • Can draw labour for public projects from three sources: i. ii. iii. those employed elsewhere in the economy the voluntarily unemployed the involuntarily unemployed • There are implications of i) to iii) • RE: i.) Those employed elsewhere in the economy: – When draw someone who already has a job into a public job, the value of the output they produced is forgone. – Measure by their marginal product of labour i.e. market wage • Recall firm is a profit maximizer or cost minimizer so they expand output as long MPL exceeds the wage rate. – Denote market wage by w1 • RE: ii.) Voluntary unemployed, e.g., retired people or people in school – opportunity cost of these people is the minimum amount required to induce them to work – measure their opportunity cost by the wage the project pays them w2 • RE: iii.) Involuntary unemployed people willing to work at the market wage but unable to find a job – On the one hand, can argue that this opportunity cost is zero – On the other hand, can also argue that market wage is not sufficiently large enough to induce them to work [should work for the market wage rate X] – Denote this wage by w3 – Since don’t really know what this wage is you have to make some assumptions about its values – One assumption that is used is that the value is at least 20% greater than the market wage. • Let 1 , 2 , 3 denote the proportions of the type of labour hired, i.e., i) to iii) • The shadow wage rate for the project can be written as = 1 1 + 2 2 + 3 3 , where 1 + 2 + 3 = 1 • The expression above is one potential estimate for the shadow wage rate. • Another approach is the Hughes formula • The Hughes formula computes the shadow wage rate as = + 1 − ,where is the wage rate paid to workers on the project; is the proportion of the wage paid that represents the opportunity cost of hiring involuntarily unemployed labour; is the probability of hiring a person who is already employed and 1- is the probability of someone who is involuntarily unemployed • You can make an assumption on , but Hughes estimated it as follows = 1 − 0.5 − − − 2 +1, where ∙ is evaluated in radians, is the unemployment rate in a region, is the historical minimum of the unemployment rate in the region, is the historical maximum of the unemployment in the region • can also be set by assuming values, but Pinfold proposed the following estimate = 1 − − UI 1 − , where is the wage rate, UI is unemployment insurance benefits, is the tax on wages and is the tax rate on UI benefits. Choosing and Computing a Discount Rate − =1 1+ , • 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 1 and 2 subject to a lifetime budget constraint. • Assume that individuals have preferences over consumption today and tomorrow • The individual’s problem can be written as 2 max 1 , 2 1 + = , 1+ where is the interest rate and T is the present value of income over the individual’s lifetime (periods 1 and 2 in this example). We’ll 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 = 1 + , 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 1+ =1+ • 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, gov’t 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 0 and 0 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: = + 1 − ∆ ∆ ℎ = 1 − = ∆ + ∆ ∆ + ∆ • 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 ∆ ≈ 0 and 1 − ≈ 0 and a ≈ 1 , 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 0.0686 = 0.1055 1 − 0.35 2. Adjust for inflation 0.1055 − 0.0392 = 0.0638 1 + 0.0392 3. Adjust for bias in CPI 0.0638 + 0.01 = 0.0738 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 1 − 0.3 0.0677 =0.0474 2. Adjust for inflation 0.0474 − 0.0392 = 0.0073 1 + 0.0392 3. Adjust for bias in CPI 0.0073 + 0.01 = 0.0173 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 = + + 1 − − , 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 government’s 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 0.0677−0.0398 = 0.0268 2+0.0398 2. Adjust for Bias in CPI 0.0268+0.01=0.0368 • Note: there is no adjustment for taxes because the government doesn’t 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(0.037)+0.05(0.0173) =0.05696 or about 5.7% • 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(0.037)+0.05(0.0173 )=0.0437 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(0.037)+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.