Untuk fungsi permintaan apa monopoli paling berbahaya?

8

Pertimbangkan perusahaan dengan biaya marjinal nol. Jika memberikan produk secara gratis, maka semua permintaan terpenuhi dan kesejahteraan sosial meningkat dengan jumlah maksimum yang mungkin; menyebut peningkatan ini .W

Tetapi karena perusahaan adalah monopoli, itu mengurangi permintaan dan meningkatkan harga untuk mengoptimalkan pendapatannya. Sekarang kesejahteraan meningkat sosial dengan jumlah yang lebih kecil, mengatakan, .V

Mendefinisikan hilangnya relatif kesejahteraan (kerugian bobot mati) sebagai: . Rasio ini tergantung pada bentuk fungsi permintaan. Jadi pertanyaan saya adalah: apakah rasio ini dibatasi, atau dapatkah itu secara sewenang-wenang besar? Khususnya:W/V

  • Jika dibatasi, lalu untuk fungsi permintaan apa ia dimaksimalkan?W/V
  • Jika tidak terikat, lalu untuk keluarga fungsi permintaan apa ia bisa menjadi besar secara sewenang-wenang?W/V

Inilah yang saya coba sejauh ini. Biarkan menjadi fungsi utilitas marjinal konsumen (yang juga merupakan fungsi permintaan terbalik). Asumsikan bahwa ia terbatas, halus, menurun secara monoton, dan diskalakan ke domain . Biarkan menjadi anti-turunannya. Kemudian:u(x)x[0,1]U(x)

kerugian bobot mati monopoli

  • W=U(1)U(0) , total area di bawah .u
  • V=U(xm)U(0), where xm is the amount produced by the monopoly. This is the area under u except the "deadweight loss" part.
  • xm=argmax(xu(x)) = the quantity which maximizes the producer's revenue (the marked rectangle).
  • xm can usually be calculated using the first-order condition: u(xm)=xmu(xm).

To get some feeling of how W/V behaves, I tried some function families.

Let u(x)=(1x)t1, where t>1 is a parameter. Then:

  • U(x)=(1x)t/t.
  • The first-order condition gives: xm=1/t.
  • W=U(1)U(0)=1/t
  • V=U(xm)U(0)=(1(t1t)t)/t
  • W/V=1/[1(t1t)t]

When t, W/V1/(11/e)1.58, so for this family, W/V is bounded.

But what happens with other families? Here is another example:

Let u(x)=etx, where t>0 is a parameter. Then:

  • U(x)=etx/t.
  • The first-order condition gives: xm=1/t.
  • W=U(1)U(0)=(1et)/t
  • V=U(xm)U(0)=(1e1)/t
  • W/V=(1et)/(1e1)

When t, again W/V1/(11/e)1.58, so here again W/V is bounded.

And a third example, which I had to solve numerically:

Let u(x)=ln(ax), where a>2 is a parameter. Then:

  • U(x)=(ax)log(ax)x.
  • The first-order condition gives: xm=(axm)ln(axm). Using this desmos graph, I found out that xm0.55(a1). Of course this solution is only valid when 0.55(a1)1; otherwise we get xm=1 and there is no deadweight loss.
  • Using the same graph, I found out that W/V is decreasing with a, so its supremum value is when a=2, and it is approximately 1.3.

Is there another family of finite functions for which W/V can grow infinitely?

Erel Segal-Halevi
sumber
Zero marginal cost does not imply zero production cost. Who bears the burden of this cost if the product is given away for free, and in what sense does social welfare is maximized then?
Alecos Papadopoulos
"Let u(x) be the consumers' utility function (which is also the inverse demand function)."
.
Isn´t it the consumers marginal utility function ?
callculus
Without having read most of it, harmful depends on the concept of social welfare, and how we weight those two. If we only look at household surplus, a smaller price-elasticity allows the firms to reap more of the surpluses. Consequently, the demand function D(p) = x, is "worst", if we focus consumer surplus.
FooBar
@AlecosPapadopoulos By W I meant increase in social welfare due only to the trade (maybe I should have called it ΔW). In this sense, the production costs are irrelevant.
Erel Segal-Halevi
@calculus You are right, I corrected this, thanks!
Erel Segal-Halevi

Jawaban:

4

An arbitrarily large ratio should occur with demand curve

P={1Qif Q>12Qif Q1.

The monopolist prices at P=1, but the consumers' surplus if P=0 is infinite, because the area under the demand curve contains 11QdQ=.

Sander Heinsalu
sumber
Thanks! Is there any reference where this issue is discussed? I would expect it to appear in standard textbooks in mircoeconomics, but didn't find it in any book I looked at.
Erel Segal-Halevi
I don't know of any references, sorry.
Sander Heinsalu