Efficiency: Waste

1. Prerequisites Almost essential Welfare and Efficiency Efficiency: Waste MICROECONOMICS Principles and Analysis Frank Cowell

2. Agenda • Build on the efficiency presentation • Focus on relation between competition and efficiency • Start from the “standard” efficiency rules • MRS same for all households • MRT same for all firms • MRS=MRT for all pairs of goods • What happens if we depart from them? • How to quantify departures from them?

3. Overview… Efficiency: Waste Background How to evaluate inefficient states Basic model Model with production Applications

4. The approach • Use standard general equilibrium analysis to… • Model price distortion • Define reference set of prices • Use consumer welfare analysis to… • Model utility loss • Use standard analysis of household budgets to… • Model change in profits and rents

5. A reference point • Address the question: how much waste? • Need a reference point • where there is zero waste • quantify departures from this point • Any efficient point would do • But it is usual to take a CE allocation • gives us a set of prices • we’re not assuming it is the “default” state • just a convenient benchmark • Can characterise inefficiency as price distortion

6. ~ = p1 ~ = p2 ~ = p3 consumer prices firms' prices ~ = pn A model of price distortion • Assume there is a competitive equilibrium • If so, then everyone pays the same prices • But now we have a distortion Distortion • What are the implications for MRS and MRT? p1 [1+d] p2 p3 … = … pn

7. Price distortion: MRS and MRT For every household marginal rate of substitution = price ratio pj MRSijh= — pi • Consumption: • Production: • for commodities 2,3,…,n pj MRT1j = — p1 • But for commodity 1… [1+ d] pj MRT2j = — p2 pj MRT3j = — p3 … … … Illustration… pj MRTnj = — pn

8. Price distortion: efficiency loss • Production possibilities • An efficient allocation x2 • Some other inefficient allocation • At x* producers and consumers face same prices • At x producers and consumers face different prices • x Producers • x* • Price "wedge" forced by the distortion How to measure importance of this wedge … p* Consumers x1 0

9. Waste measurement: a method • To measure loss we use a reference point • Take this as competitive equilibrium… • …which defines a set of reference prices • Quantify the effect of a notional price change: • Dpi := pi – pi* • This is [actual price of i] – [reference price of i] • Evaluate the equivalent variation for household h : • EVh = Ch(p*,u h) – Ch(p,u h) – [y*h – yh] • This is D(consumer costs) – D(income) • Aggregate over agents to get a measure of loss, L • We do this for two cases…

10. Overview… Efficiency: Waste Background Taking producer prices as constant… Basic model Model with production Applications

11. If producer prices constant… • Production possibilities C(p, u) x2 • Reference allocation and prices • Actual allocation and prices DP • Cost of u at prices p • Cost of u at prices p* • Change in valuation of output • Measure cost in terms of good 2 • x C(p*, u) • Losses to consumers are • C(p*, u)  C(p, u) • x* • L is difference between • C(p*, u)  C(p, u) and DP p p* u 0 x1

12. Model with fixed producer prices • Waste L involves both demand and supply responses • Simplify by taking case where production prices constant • Then waste is given by: • Use Shephard’s Lemma • xih = Hhi(p,uh) = Cih(p,uh) • Take a Taylor expansion to evaluate L: • L is a sum of areas under compensated demand curve

13. Overview… Efficiency: Waste Background Allow supply-side response… Basic model Model with production Applications

14. Waste measurement: general case • Production possibilities C(p, u) x2 • Reference allocation and prices • Actual allocation and prices DP • Cost of u at prices p • Cost of u at prices p* • Change in valuation of output C(p*, u) • Measure cost in terms of good 2 • x • Losses to consumers are • C(p*, u)  C(p, u) • x* p* • L is difference between • C(p*, u)  C(p, u) and DP p u x1 0

15. Model with producer price response • Adapt the L formula to allow for supply responses • Then waste is given by: • where qi (∙) is net supply function for commodity i • Again use Shephard’s Lemma and a Taylor expansion:

16. Overview… Efficiency: Waste Background Working out the hidden cost of taxation and monopoly… Basic model Model with production Applications

17. Application 1: commodity tax • Commodity taxes distort prices • Take the model where producer prices are given • Let price of good 1 be forced up by a proportional commodity tax t • Use the standard method to evaluate waste • What is the relationship of tax to waste? • Simplified model: • identical consumers • no cross-price effects… • …impact of tax on good 1 does not affect demand for other goods • Use competitive, non-distorted case as reference:

18. A model of a commodity tax p1 • Equilibrium price and quantity • The tax raises consumer price… compensated demand curve • …and reduces demand • Gain to the government • Loss to the consumer • Waste revenue raised = tax x quantity • Waste given by size of triangle • Sum over h to get total waste • Known as deadweight loss of tax L Dp1 p1* x1* x1h Dx1h

19. Tax: computation of waste • An approximation using Consumer’s Surplus • The tax imposed on good 1 forces a price wedge • Dp1 = tp1*> 0 where is p1* is the untaxed price of the good • h’s demand for good 1 is lower with the tax: • x1** rather than x1* • where x1** = x1* + Dx1h and Dx1h < 0 • Revenue raised by government from h: • Th = tp1*x1**= x1**Dp1 > 0 • Absolute size of loss of consumer’s surplus to h is • |DCSh| = ∫ x1hdp1 ≈ x1**Dp1−½Dx1hDp1 • = Th−½ t p1* Dx1h > Th • Use the definition of elasticity • e := p1Dx1h / x1hDp1< 0 • Net loss from tax (for h) is • Lh = |DCSh| − Th = − ½tp1* Dx1h • = − ½teDp1x1** = − ½t e Th • Overall net loss from tax (for h) is • ½ |e| tT • uses the assumption that all consumers are identical

20. p1 p1 compensated demand curve Dp1 p1* x1h Dx1h Dp1 p1 p1 p1* Dp1 Dp1 p1* p1* x1h Dx1h x1h x1h Dx1h Dx1h Size of waste depends upon elasticity • Redraw previous example • e low: relatively small waste • e high: relatively large waste

21. Application 1: assessment • Waste inversely related to elasticity • Low elasticity: waste is small • High elasticity: waste is large • Suggests a policy rule • suppose required tax revenue is given • which commodities should be taxed heavily? • if you just minimise waste – impose higher taxes on commodities with lower elasticities • In practice considerations other than waste-minimisation will also influence tax policy • distributional fairness among households • administrative costs

22. Application 2: monopoly • Monopoly power is supposed to be wasteful… • but why? • We know that monopolist… • charges price above marginal cost • so it is inefficient … • …but how inefficient? • Take simple version of main model • suppose markets for goods 2, …, n are competitive • good 1 is supplied monopolistically

23. Monopoly: computation of waste (1) • Monopoly power in market for good 1 forces a price wedge • Dp1 = p1** −p1* > 0 where • p1** is price charged in market • p1*is marginal cost (MC) • h’s demand for good 1 is lower under this monopoly price: • x1** = x1* + Dx1h, • where Dx1h < 0 • Same argument as before gives: • loss imposed on household h: −½Dp1Dx1h > 0 • loss overall:− ½Dp1Dx1, where x1 is total output of good 1 • using definition of elasticity e, loss equals −½Dp12e x1**/p1** • To evaluate this need to examine monopolist’s action…

24. Monopoly: computation of waste (2) • Monopolist chooses overall output • use first-order condition • MR = MC: • Evaluate MR in terms of price and elasticity: • p1** [ 1 + 1 / e] • FOC is therefore p1** [ 1 + 1 / e] = MC • hence Dp1= p1** − MC = − p1** / e • Substitute into triangle formula to evaluate measurement of loss: • ½ p1**x1** / |e| • Waste from monopoly is greater, the more inelastic is demand • Highly inelastic demand: substantial monopoly power • Elastic demand: approximates competition

25. Summary • Starting point: an “ideal” world • pure private goods • no externalities etc • so CE represents an efficient allocation • Characterise inefficiency in terms of price distortion • in the ideal world MRS = MRT for all h, f and all pairs of goods • Measure waste in terms of income loss • fine for individual • OK just to add up? • Extends to more elaborate models • straightforward in principle • but messy maths • Applications focus on simple practicalities • elasticities measuring consumers’ price response • but simple formulas conceal strong assumptions