# Efficiency: Waste - London School of Economics

Prerequisites Prerequisites Almost Almost essential essential Welfare Welfareand andEfficiency Efficiency EFFICIENCY: WASTE MICROECONOMICS Principles and Analysis Frank Cowell Frank Cowell: Efficiency-Waste 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 these rules? How to quantify departures from efficiency?

April 2018 Frank Cowell: Efficiency-Waste 2 Overview Efficiency: Waste Background How to evaluate inefficient states Basic model Model with production Applications April 2018 Frank Cowell: Efficiency-Waste 3 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 April 2018 Frank Cowell: Efficiency-Waste 4 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 were not assuming it is the default state just a convenient benchmark Can characterise inefficiency as price distortion April 2018

Frank Cowell: Efficiency-Waste 5 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 What are the implications for MRS ~ p1 = p1 and MRT? consumer prices April 2018 p2 p3 ~

= p2 ~ = p3 Distortion Distortion [1+d]d]] = ~ pn = pn Frank Cowell: Efficiency-Waste firms' prices 6 Price distortion: MRS and MRT For For every every household household marginal marginal rate rate of of substitution

substitution == price price ratio ratio Consumption: Production: for commodities 2,3,,n But for commodity 1 April 2018 pj MRSij = pi h pj MRT1j = [1+ d]] p1 pj MRT2j = p2 pj MRT3j = p3 pj Illustration

Illustration MRTnj = pn Frank Cowell: Efficiency-Waste 7 Price distortion: efficiency loss Production possibilities An efficient allocation Some other inefficient allocation x2 At x* producers and consumers face same prices At x producers and consumers face different prices x x* Producers Producers Price "wedge" forced by the distortion

p* p* Consumers Consumers 0 April 2018 x1 How Howtotomeasure measure importance importanceofofthis this wedge wedge Frank Cowell: Efficiency-Waste 8 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 April 2018 Frank Cowell: Efficiency-Waste 9 Overview Efficiency: Waste Background Taking producer prices as constant Basic model Model with production Applications

April 2018 Frank Cowell: Efficiency-Waste 10 If producer prices constant C(p, C(p, u) u) x2 DP Production possibilities Reference allocation and prices Actual allocation and prices 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*, C(p*, u) u) Losses to consumers are C(p*, u) C(p, u) x* pp 0 April 2018 L is difference between C(p*, u) C(p, u) and DP p* p* u

x1 Frank Cowell: Efficiency-Waste 11 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 Shephards 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 April 2018 Frank Cowell: Efficiency-Waste 12 Overview Efficiency: Waste Background Allow supply-side response

Basic model Model with production Applications April 2018 Frank Cowell: Efficiency-Waste 13 Waste measurement: general case C(p, C(p, u) u) Production possibilities Reference allocation and prices Actual allocation and prices Cost of u at prices p Cost of u at prices p* Change in valuation of output x2 DP

C(p*, C(p*, u) u) Measure cost in terms of good 2 x Losses to consumers are C(p*, u) C(p, u) x* p p* p* L is difference between C(p*, u) C(p, u) and DP u April 2018 0

x1 Frank Cowell: Efficiency-Waste 14 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 Shephards Lemma and a Taylor expansion: April 2018 Frank Cowell: Efficiency-Waste 15 Overview Efficiency: Waste Background Working out the hidden cost of taxation and monopoly

Basic model Model with production Applications April 2018 Frank Cowell: Efficiency-Waste 16 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 April 2018 Frank Cowell: Efficiency-Waste

17 A model of a commodity tax p1 Equilibrium price and quantity The tax raises consumer price and reduces demand compensated demand curve Gain to the government Loss to the consumer Waste revenue revenueraised raised == tax taxxx quantity quantity Dp1 Waste given by size of triangle Sum over h to get total waste L

L Known as deadweight loss of tax p * 1 x1* x1h Dx1h April 2018 Frank Cowell: Efficiency-Waste 18 Tax: computation of waste An approximation using Consumers Surplus The tax imposed on good 1 forces a price wedge Dp1 = tp1* > 0 where is p1* is the untaxed price of the good hs demand for good 1 is lower with the tax: x1** rather than x1* , where x1** = x1* +d] Dx1h and Dx1h < 0

Revenue raised by government from h: Th = tp1* x1**= x1**Dp1 > 0 Absolute size of loss of consumers surplus to h is |DCSh| = x1h dp1 x1** Dp1 Dx1hDp1 = Th t p1* Dx1h > Th Use the definition of elasticity e := p Dx h / x hDp < 0 1 1 1 1 Net loss from tax (for h) is Lh = |DCSh| Th = tp1* Dx1h = teDp1x1** = t e Th Overall net loss from tax (for h) is |e| tT uses the assumption that all consumers are identical April 2018 Frank Cowell: Efficiency-Waste 19 Size of waste depends upon elasticity p1 p1

Redraw previous example e low: relatively small waste compensated demand curve e high: relatively large waste Dp1 p1* x1h Dx1h Dp1 p1 p1 p1* Dp1 Dp1 p1* p1* x1h Dx1h April 2018

Dx1h x 1h x1h Dx1h Frank Cowell: Efficiency-Waste 20 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 April 2018 Frank Cowell: Efficiency-Waste

21 Application 2: monopoly Monopoly power is supposed to be wasteful but why? We know that monopolist: charges price above marginal cost so equilibrium solution is inefficient But how inefficient? Take simple version of main model suppose markets for goods 2, , n are competitive good 1 is supplied monopolistically April 2018 Frank Cowell: Efficiency-Waste 22 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) hs demand for good 1 is lower under this monopoly price:

x1** = x1* +d] Dx1h, where Dx1h < 0 Same argument as before gives: Dp1Dx1h > 0 loss overall: Dp1Dx1, where x1 is total output of good 1 using definition of elasticity e, loss equals Dp12 e x1**/p1** loss imposed on household h: To evaluate this need to examine monopolists action April 2018 Frank Cowell: Efficiency-Waste 23 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 the loss: p1* * x1* * / |e| Waste from monopoly is greater, the more inelastic is demand highly inelastic demand: substantial monopoly power highly elastic demand: approximates competition April 2018 Frank Cowell: Efficiency-Waste 24 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 April 2018 Frank Cowell: Efficiency-Waste 25

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