Micro Lecture 6 Applied Welfare MT 2023 Simon Cowan Worcester College and Department of Economics Recap Last week This week • Competitive equilibrium and welfare • Measurement and recoverability of welfare • Pareto efficiency; Social Welfare Functions • Special case: quasi-linear utility • Two fundamental theorems • Elements of Cost-Benefit Analysis • Externalities and public goods • Social choice: ordinal approach • Preference aggregation rules • Welfare analysis and correction for market failures • Arrow’s impossibility result • Getting out of impossibility Measuring individual welfare • We want to evaluate the effects of changes in the consumer’s environment (change in prices, income, taxes, etc.) on her wellbeing. • Preference-based approach to consumer demand will be of critical importance (choices are not enough). Well-being = utility. • Cardinal approach: we want to be able to say how much so ordinal framework of preference relations won’t be enough. • In the background: rational consumer with well-behaved preferences so we get a well-behaved demand. Measuring aggregate welfare • “Normative individualism”: Normative statements about society based on statements about individual welfare. • In tradition of methodological individualism of Max Weber. • Who is to be included in the set of individuals π = 1, … , π? • All citizens of a country? All human beings? Animals? • Future generations? βΉ time discounting • How should their well-being be aggregated? • What social welfare function should we use? • How much weight should we put in the SWF on the various subgroups? • Welfare analysis is very hard with heterogeneous agents • Often we will work with a representative agent Plan for this lecture 1. How to measure the welfare effect of a price change? • Money-metric measures of welfare 2. What is the best way to tax? • Welfare in a quasilinear environment • The deadweight loss of commodity taxes 3. Introduction to Cost-Benefit Analysis Question #1 How should we measure changes in (indirect) utility or welfare when prices change? Consumer welfare if utility is observed • Objective: measure the effect of a price change from ππ to ππ on the consumer’s welfare. • The consumer has fixed income π > 0, unaffected by prices. • If we know the utility function we can compute changes in indirect utility (to obtain indirect utility put the demand functions into the utility function): π’ π ππ , π − π’ π ππ , π = π£ ππ , π − π£ ππ , π Example 1 1 1 π’ = ln π₯ + ln π₯ 2 2 2 π π π₯ 1 π, π = , π₯ 2 π, π = π₯1, π₯ 2 2π1 2π2 Indirect utility: 1 π 1 π π£ π, π = ln + ln 2 2π1 2 2π2 1 1 = ln π − ln π1 − ln π2 − ln(2) 2 2 Money-metric measures of welfare π₯2 • Problem: we don’t observe utility, utility is usually an ordinal concept; we need a cardinal measure. • We use money-metric measures of welfare. • Constructed using the expenditure function. • Duality: uses 1-1 mapping between two problems. maxπ π’ π subject to π β π ≤ π βΉ π£ π, π = π’ π∗ π, π = π’ΰ΄€ π∗ π, π π, π’ΰ΄€ = β(π, π’) ΰ΄€ π∗ π’ΰ΄€ π₯1 minπ π β π subject to π’ π ≥ π’ΰ΄€ βΉ π π, π’ΰ΄€ = π β π∗ = π Primal problem: Utility maximization Dual problem: Expenditure minimization π∗ π, π : Marshallian demand Obtain indirect utility function π£ π, π π∗ = β(π, π’ΰ΄€ ) : Hicksian demand Obtain expenditure function π π, π’ΰ΄€ Compensating Variation • The Compensating Variation (CV) is the amount of money that must be taken from the consumer, given that a price change happens, to ensure they reach the old utility: π New Prices, π − πΆπ = π Old Prices, π • If the new prices generate higher utility CV is positive: it’s a measure of the welfare gain • Suppose all prices are multiplied by π > 0. The old utility is obtained if income is multiplied by π, so π − πΆπ = ππ πΆπ = 1 − π π Equivalent Variation • The Equivalent Variation (EV) is the change in income at the old prices that would give the same utility as with the new prices π New Prices, π = π(Old Prices,π + πΈπ) • When all prices are multiplied by π > 0: 1−π πΈπ = π λ • Multiplying all prices by π has the same effect as dividing income by π, so π + πΈπ = π/π • Note that EV and CV aren’t equal: in this case πΈπ − πΆπ = when π ≠ 1 1−π 2 π π ≥ 0 with strict inequality A single price change π₯2 Equivalent Variation = change in income required at old prices to reach new utility EV Compensating Variation = change in income required at new prices to maintain old utility CV Positive income effect π1 − π2 Note that EV > CV: this holds for a single price change when the income effect is positive. π1′ − π2 π₯1 Using the expenditure function.. πΆπ = Income − Expenditure New Prices, Old Utility πΈπ = Expenditure Old Prices, New Utility − Income Example: Road closure in Oxford • Oxfordshire County Council is considering the closure to car traffic of a major road in Oxford in an attempt to reduce noise and air pollution. • However, this measure could have a potentially serious impact on local businesses and the city would like to discuss the potential welfare loss. • CV: How much would the city have to pay local businesses to keep them as well off as they were before the road closure? (measure implemented) • EV: What is the most that local businesses would be willing to pay to avoid the road closure? (measure not implemented) How much do you value the Internet? • In 2012, this is the question that the Boston Consulting Group asked to consumers in 20 countries. How was this measured? • Consider (hypothetical) policy change of removing the Internet. • π’0 : utility with the Internet; π’1 : utility without the Internet • The survey asked questions of EV and CV type. • CV type: How much money would you require to stop using the Internet? • EV type: Giving up what good/activity would hurt you as much as losing the internet? @ How much do you value the Internet? • CV type • 3%-6% of consumers’ average annual income; average of $1,430 across 20 countries. • Min of $323 in Turkey and max of $4,453 in France; US average of $2,528. • EV type • Most US consumers would trade for the internet • Fast food (83%), Chocolate (77%), Alcohol (73%), Coffee (69%) • Forgo exercise (43%) • Give up their car (10%), showers (7%), sex (21%) • Percentages quite similar in UK and France with small variation: • 21% of Brits would give up cars, 17% showers, 25% sex. • 23% of French people would give up cars, 5% showers, 16% sex. How reliable are these answers to hypothetical choices? As economists, we try to estimate demand curves based on observed purchasing behaviour. But not always possible… How money-metric measures compare • In the first year you learned about the change in consumer surplus, ΔπΆπ, as a measure of the welfare impact of a price change • If a good is normal (the income effect is positive) then if its price changes: CV < βCS < EV • βCS can be a good approximation if the income effect is small because either the budget share or the income elasticity is small. • βCS is the correct measure when utility is quasi-linear so there is no income effect Question #2 What is the deadweight loss of commodity taxes and how can we minimize it? Quasi-linear utility • Now call the two goods π₯ and π¦. Utility is π π₯, π¦ = π’ π₯ + π¦ • π’(π₯) is strictly concave and increasing (over the relevant range) • Budget constraint (normalizing price of π¦ to 1): ππ₯ + π¦ ≤ π • Lagrangian π’ π₯ + π¦ + π[π − ππ₯ − π¦] • FOCs (assuming π is large enough that π¦ > 0): π’′ π₯ − ππ = 0 1−π =0 π − ππ₯ − π¦ = 0 Quasi-linear utility II • So π = 1 and then π’′ π₯ = π. • Demands are π₯ π , π¦ π, π = π − ππ₯ π , indirect utility is π£ π, π = π’ π₯ π − ππ₯(π) + π Consumer Surplus • Income equals profits of the firm, π = ππ₯ − π π₯ , where π(π₯) is the cost function • Indirect utility = welfare is gross utility minus costs: π’ π₯ π − ππ₯ π + ππ₯ − π π₯ = π’ π₯ π − π(π₯) Why quasilinear utility? • The change in consumer surplus βCS gives us an exact measure of the change in welfare: EV = CV = βCS • We can do partial equilibrium analysis: focus on the good of interest and treat all the other goods as a “composite” good. • We can use a representative consumer, with income equal to average income, because income enters additively in indirect utility. • These nice features come at the cost of some strong assumptions. Welfare in this simple framework • Consumer chooses π₯ where π’′ π₯ = π • The firm chooses π₯ where π = π′(π₯) • The resulting π₯ ∗ maximises consumer + producer surplus • Note: area under demand curve is total utility. π1 π ′ (π₯) Consumer π∗ Surplus Producer Surplus π’′ (π₯) π₯∗ π₯ Effect of commodity tax π π ′ (π₯) CS π‘ Tax revenue π’′ (π₯) DWL PS π₯π‘ π₯∗ • Government taxes the good • Cannot use lump-sum taxation • Tax raises revenue for the government (which it redistributes back) • Tax creates a wedge between producer and consumer price, preventing profitable trades from happening and creating π₯ deadweight loss. Effect of commodity tax π Zooming in: locally everything is linear if π’′ π₯ and c′(π₯) are differentiable πππ‘ π ′ (π₯) (π₯ ∗ − π₯ π‘ ) × π‘ π·ππΏ = 2 π∗ π’′ (π₯) π·ππΏ π‘ πππ‘ π₯π‘ π₯∗ π₯ DWL falls if demand becomes less elastic π1 Less elastic demand πππ‘ π ′ (π₯) (π₯ ∗ − π₯ π‘ ) × π‘ π·ππΏ = 2 π’0′ (π₯) π·ππΏ π∗ π‘ π’1′ (π₯) πππ‘ π₯π‘ π₯∗ π₯ DWL falls if supply becomes less elastic π1 Less elastic supply πππ‘ π ′ (π₯) π∗ (π₯ ∗ − π₯ π‘ ) × π‘ π·ππΏ = 2 π’′ (π₯) π‘ πππ‘ π₯π‘ π₯∗ π₯ Optimal commodity taxation • Suppose we want to raise minimize the sum of the deadweight losses while obtaining tax revenue of πΊ min σπ π·ππΏπ subject to σπ π‘π π₯ π ≥ πΊ π‘1 ,…,π‘π • Assume that all supply elasticities are infinite • Ramsey taxation: tax goods with high demand elasticities less • “Optimal” commodity taxes are, typically, regressive… Frank Ramsey Recap First part This part • Applied welfare: money-metric • Introduction to Cost-Benefit Analysis measures of welfare • How do we measure the welfare effects of policy interventions in practice? • Welfare analysis with • What type of benefits and costs should enter quasilinear utility in the analysis? • Deadweight loss of commodity • By what methods can we estimate them? taxes • How should we weight them over time? What social discount rate should we use? Evaluating big projects… Aerial view of the Olympic Park in 2012. Source: Wikipedia Touted as a bargain… only $14.8 billion… cheap relative to Sochi @ $51 billion. Worth it? National Geographic article Cost-Benefit Analysis • Your mission if you accept it: You are hired by the Cabinet Office to perform a Cost-Benefit Analysis (CBA) of a big project: hosting a major international sporting event. Should the government pursue it? • The project will yield benefits and costs over time. You need to evaluate its net present value i.e., stream of net benefits discounted for the future ∞ π΅π‘ − πΆπ‘ πππ = ΰ· 1+π π‘ π‘=0 • If πππ > 0, then you will declare that the project should be undertaken. • Now you “just” need to find π΅π‘ , πΆπ‘ , and π… Good luck! βΊ Step 1: What should you value? First, you need to come up with a list of costs and benefits. You need to go broad: • Tangible Costs: • Operations • Direct investment (e.g., facilities) • Indirect investment (e.g., new transport, infrastructure) • Intangible Costs: • Congestion and disruption • Tangible Benefits: • Incomes generated by related investment, consumption, tourism. • Intangible Benefits: • Improved local environment • Health benefits from use of new sports facilities • Improved wider social goals (national pride, social inclusion) Step 2: How should you value? • With well-functioning markets, it’s easy. • Relative prices contain all the information we need. • Can obtain directly using revealed preference methods. • How can you value intangibles? What should you do in the case of market failure? • Can infer indirectly using proxies (also revealed preference methods) • Use surveys (stated preference methods) Revealed preference methods 1. Estimate demand directly from market data: • If observe demand at various prices and can make a ceteris paribus assumption, this can be done. • It will become easier and easier with “Big Data”. • “Using Big Data to Estimate Consumer Surplus: the Case of Uber” by Steven Levitt and coauthors (link here) 2. Estimate demand indirectly using proxies: Travel Cost Method: • Employs access costs i.e. transportation and time costs to access facilities as a “price” for the good. • Can build econometric model of relationship between number of visits and travel costs and other determinants of demand Hedonic Pricing: • Use property prices to value regional intangible interventions (e.g., neighbourhood amenities) • Can build econometric model πππππππ‘π¦ πππππ = π(π»ππ’π π πΆβπππππ‘ππππ π‘πππ π , ππππβπππ’πβππππ , πΌππ‘πππ£πππ‘πππ) Market failure and shadow prices Market failure: monopoly • Buy inputs from a monopolist and pay price π > ππΆ • What is the appropriate price: π or ππΆ? • ππΆ is resource cost of the input. • π – ππΆ is a transfer payment (to monopolist), not a real resource cost. Shadow prices: labour • What is the cost of employing a worker for an hour? • If that worker is otherwise employed the shadow price = market wage = value of marginal product in alternative job • If unemployed, the shadow price should be the value of leisure foregone, which could be zero General principle: identify opportunity cost – the value of the resource in its best alternative use. Stated preference methods • Use surveys to elicit WTP/WTA for a specific change. • If possible, use incentivized choices to elicit valuation. • Can use a mechanism that will lead people to tell the truth • e.g. WTA for deactivating Facebook for a month • You can read The Welfare Effects of Social Media by Allcott et al. (2020) • Often not possible to incentivize due to the nature of the question. • Contingent valuation methods often suffer “hypothetical bias” • e.g. WTP to save 1,000,000 birds. • We also use surveys to obtain non-monetary measures • QALY’s: express the value of quality of life and length of life in a single number Other concerns • General equilibrium effects / displacement • Might be particularly important to take into account if • There are large distributional effects. • Social value of income is different for winners and losers (depends on SWF!) • There are serious market frictions. Social discount rate π • π is the social discount rate: society’s value of future consumption relative to present. • How do we measure this? • We need to think about intertemporal utility. Welfare is the sum of all discounted utility flows from consumption π over time π‘ = 0, 1, etc.: ∞ 1 π π0 , π1 , … , ππ , … = ΰ· π’ ππ‘ π‘ (1 + π) π‘=0 • π is the pure rate of time preference – the rate at which future utility is discounted The Ramsey framework • With two periods •π= ππ’′′ π − ′ π’ π 1 π π1 , π2 = π’ π1 + π’ π2 1+π π1−π − 1 π’ π = for π ≠ 1; 1−π or π’ π = ln π is the elasticity of the marginal utility of consumption • Identical to relative risk aversion in expected utility theory (Week 6) • π = 1 when π’ π = ln(π) 37 The Ramsey equation: Derivation • π2 is a function of π1 along an iso-welfare curve. Differentiate π(π1 , π2 π1 ) = constant w.r.t. π1 : 1 ππ2 ′ ′ π’ π1 + π’ π2 =0 1+π ππ1 π ππ2 1 + π π’′(π1 ) π2 ⇒ =− =− 1+π ′ ππ1 π’ π2 π1 • The slope of the production possibility frontier that allows reductions in π1 to be transformed into increases in π2 is −(1 + π) where π is the real discount rate for consumption. 38 Derivation II • Equate the slopes 1+π = 1+π π2 π1 π π2 π1 • Take logs, and use = 1 + π, where π is the growth rate ln 1 + π = ln 1 + π + π ln 1 + π • For small x, ln(1 + π₯) ≈ π₯. The Ramsey formula is π = π + ππ 39 Social discount rate π π = π + ππ Pure rate of time preference Impatience Probability of extinction Intergenerational discrimination (what’s the ethical basis?) Elasticity of marginal utility of consumption Determines how fast marginal utility falls as income rises Preference for equality (across space and time!) Rate of consumption growth Less weight put on future generations if they are much richer. Higher π means future consumption is worth less now Social discount rate matters! Stern Review on Climate Change UK Government Green Book π = 0.1% + 1 × 1.3% = 1.4% π = 1.5% + 1 × 2% = 3.5% where 0.1% is the estimated probability of extinction Used in valuing most infrastructure projects William Nordhaus π = 1.5% × 2 × 2% = 5.5% DICE climate change model In summary What we have learned… • Applied welfare is hard in theory and in practice! • We can do this but we often need to make substantive assumptions e.g. no income effects. • CBA: not a neutral exercise; a lot of political choices involved. Next two lectures • We will use the simple quasilinear framework we developed to study public goods and negative externalities. • We will think about how to correct those market failures.
