Approximating the gains from trade in large markets

Transcription

1 Approximating the gains from trade in large markets Ellen Muir Supervisors: Peter Taylor & Simon Loertscher Joint work with Kostya Borovkov The University of Melbourne 7th April 2015

2 Outline Setup market model Define the Walrasian quantity traded and level of welfare Approximating welfare Approximating the scaled quantity traded Example computation

3 Setup Consider a market for a homogeneous, indivisible good There are N buyers who demand one unit of the good There are M sellers with the capacity to produce one unit Buyer valuations: V 1,..., V N are IID RVs with AC DF F Seller costs: C 1,..., C M are IID RVs with AC DF G This is a canonical mechanism design model

4 Ordering of buyers and sellers Sellers are ordered from lowest cost to highest cost For j = 1,..., M let C (j) denote the jth lowest seller cost This forms a supply curve Buyers are ordered from highest valuation to lowest valuation For i = 1,..., N let V [i] denote the ith highest buyer valuation This forms a demand curve

5 Ordering of buyers and sellers C (M) V [1] C (1) V [N] 1 N M Figure 1 : The ordered costs and valuations form demand and supply curves

6 Economic welfare The level of welfare generated by a market is essentially the gains from trade The welfare generated by a market is equal to the sum of trading buyer valuations less the sum of trading seller costs If N T and M T are the sets of trading buyers and sellers respectively, the level of welfare is given by i N T V i j M T C j

7 How can we maximise welfare? C (M) V [1] C (1) V [N] 1 K N M Figure 2 : The shaded region shows the maximum level of welfare

8 The Walrasian quantity and level of welfare Definition The Walrasian quantity, K, is the quantity traded that maximises welfare. That is, K = arg max k=0,1,...,n M k ( ) V[i] C (i). i=1 The associated level of welfare, W, is then W = K ( ) V[i] C (i). i=1

9 The problem at hand We want to approximate K ( ) W = V[i] C (i) i=1 for large N and M.

10 Background on empirical quantile functions Definition Let X 1,..., X n be an IID sample of RVs. Then Q n (t) = n i=1 ( i 1 X (i) 1 n < t i ) n is known as the empirical quantile function of the sample.

11 Background on empirical quantile functions Q n (t) X (n) 1 X (1) 1 n n 1 n 1 t Figure 3 : An example of an empirical quantile function

12 Background on empirical quantile functions Proposition (Csörgő and Révész, 1978) Let U 1,..., U n be an IID sample of U(0, 1) RVs and Q n (t) be the associated uniform quantile function. As n, for t (0, 1), Q n (t) = t + n 1 2 B(t) + θn (t)n 1 log n, where B(t) is a Brownian bridge process and θ n = O(1) a.s. Here we use the standard notation := sup t (0,1).

13 Approximating welfare Returning to the problem at hand, K ( ) K K W = V[i] C (i) = V [i] C (i) W V W C. i=1 i=1 i=1 Approximate W V and W C separately.

14 Approximating W C Let G ( 1) be the quantile function of G Let U C 1,..., U C M be an IID sample of U(0, 1) RVs Let Q C M denote the associated uniform quantile function We have K K ( ) K ( ( )) W C = C [i] = G ( 1) U(i) C 1 i = M M G( 1) Q C M M i=1 i=1 i=1

15 Approximating W C Rewriting the sum on the previous slide as an integral, W C = M K i=1 ( ( )) K 1 i M G( 1) Q C M M = M G ( 1) ( Q C M (u) ) du M 0 Then letting B C denote a Brownian bridge process, W C = M K M 0 where θ C M = O(1) a.s. ( ) G ( 1) u + M 1 2 B C (u) + θm C (u)m 1 log M du,

16 Approximating W V Repeating this procedure for W V we obtain W V = N K N 0 ( ) F ( 1) 1 u + N 1 2 B V (1 u) + θn V (u)n 1 log N du, where θn V = O(1) a.s. Here, F ( 1) is the quantile function of F B V is a Brownian bridge process independent of B C To proceed, we must approximate K/N and K/M.

17 Some simulations Figure 4 : M = 110 Uniformly distributed costs and valuations with N = 100 and

18 Some simulations Figure 5 : M = 220 Uniformly distributed costs and valuations with N = 200 and

19 Some simulations Figure 6 : M = 550 Uniformly distributed costs and valuations with N = 500 and

20 Some simulations Figure 7 : Uniformly distributed costs and valuations with N = 2000 and M = 2200

21 Approximating K/N Take M = λn and approximate K/N only Consider the functional H(k) = sup{t : k(t) < 0} Let G ( 1) M (t) and F( 1) N (t) denote the empirical quantile functions associated with the seller costs and buyer valuations respectively Then K ( N = H G ( 1) M ) (λ 1 t) F ( 1) N (1 t)

22 Approximating K/N For t (0, 1 λ) define k 0 (t) = G ( 1) (λ 1 t) F ( 1) (1 t) Let t 0 = H(k 0 ) so that, as n, K N d t 0 We wish to determine the distribution of the leading order error term

23 A visual illustration Figure 8 : A simulation of K/N with uniform types, N = 50 and M = 50

24 Asymptotic normality of sample quantiles It is well known that, as n, ( 1) N(G M (λ 1 t) G ( 1) (λ 1 t)) d B G (λ 1 t) λg(g ( 1) (λ 1 t)), ( 1) N(F N (1 t) F ( 1) (1 t)) d B F (t) f(f ( 1) (1 t)) Combining these gives, as n, ( 1) N[(G M (λ 1 t) F ( 1) N (1 t)) (G( 1) (λ 1 t) F ( 1) (1 t))] d B G (λ 1 t) λg(g ( 1) (λ 1 t)) + But we want the convergence of N[H(G ( 1) M (λ 1 t) F ( 1) N B F (t) f(f ( 1) (1 t)) (1 t)) H(G( 1) (λ 1 t) F ( 1) (1 t))]

25 The delta method We need a functional version of the delta method. Proposition Let θ, σ 2 be finite valued constants and suppose {X n } n N satisfies n [Xn θ] d N (0, σ 2 ). Then for any function r with r (θ) 0, n [r(xn ) r(θ)] d N (0, σ[r (θ)] 2 ).

26 The derivative of H Compute the functional derivative of H Do this by modifying an example in Section 8, Chapter I of Borovkov (1998) For any v C(, ), H (k 0, v) = v(t 0)λg(G ( 1) (λ 1 t 0 ))f(f ( 1) (1 t 0 )) λg(g ( 1) (λ 1 t 0 )) + f(f ( 1) (1 t 0 ))

27 The Functional Delta Method By Proposition 1 in Borovkov (1985), ( 1) N[H(G (t) F( 1) (1 t)) H(G( 1) (t) F ( 1) (1 t))] N d λg(g( 1) (λ 1 t 0))f(F ( 1) (1 t 0)) λg(g ( 1) (λ 1 t 0))+f(F ( 1) (1 t 0)) N (0, Z. N t 0(1 λ 1 t 0) λ 2 g 2 (G ( 1) (λ 1 t + t 0(1 t 0) 0)) f 2 (F ( 1) (1 t 0)) ) Thus, we have where Z is normally distributed. K d = t 0 + N ( 1 2 Z + O N 1 log N ), N

28 Uniform costs and valuations with N = M In this case, our expression for K/N simplifies to K d = 1 ( N 2 + N 1 2 N 0, 1 ) + O ( N 1 log N ) Y + O ( N 1 log N ) N Our expression for W C simplifies to W C d = N 1 2 +N 1 2 Y 0 u du + N B C (u) du + O (log N) Our expression for W V simplifies to W V d = N N 1 2 Y (1 u) du + N B V (1 u) du + O (log N)

29 Uniform costs and valuations with N = M Direct computation then gives W = W V W C d N = 4 + ( NN 0, 10 ) + O(log N) 192 The dependence of K on B V the expansion and B C affects higher order terms in

30 Example error simulation Figure 9 : The distribution of (W N/4)/ N vs. the density of Y d = N (0, 10/192) for N = M = 1000 with simulations

31 References Borovkov, A. A. (1998), Mathematical Statistics, number 26 27, Gordon & Breach, chapter 1. Borovkov, K. A. (1985), A note on a limit theorem for differentiable mappings, The Annals of Probability 13(3), Csörgő, M. and Révész, P. (1978), Strong approximations of the quantile process, The Annals of Statistics 6(4),