No theft model of corruption

Question

Consider the model of corruption explored by Shleifer and Vishni’s where there is one government-produced good $X$. There is a demand for that good described by the inverse demand equation $$Q_d = 10 – P.$$ The official government price for the good is $P_g=3$. The government pays the cost of producing the good. A bureaucrat can restrict the supply of $X$. Since there is no risk of detection, the public official has incentives to ask for a bribe to supply the good.

Consider the model of “no theft” where the consumer pays the official government price plus a bribe in order to obtain $X$. Assume that the official marginal revenue for selling the good in this context is given by $$Q_c=\frac{7-P}{2}.$$

a) In the model of “no theft” what is the amount of the bribe that the corrupt official will charge?

+1, first time i am seeing an attempt to mathematically model bribery. Awesome! — gt6989b, Feb 13 '14 at 16:40

Ross Millikan · Answer 1 · 2014-02-13T18:33:42.457

1

Hint: The official wants to maximize his income, so will set the price so the marginal income is zero. The bribe is the excess of the price over $3$.

edited Feb 13 '14 at 18:33

answered Feb 13 '14 at 16:48

Ross Millikan

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Does this make the answer 2? – Dimoftie Cristina Diana Feb 13 '14 at 17:37
I think I'm working it out wrong, I got 7-3/2? – Dimoftie Cristina Diana Feb 13 '14 at 17:53
In my first answer I assumed the official got everything over $3$. Watch for an update – Ross Millikan Feb 13 '14 at 18:07
Qc=(7/2)-(1/2)P or (7/2)-(3/2), so Qc must be 2 as 3.5-1.5=2, the total payed by the comsumer is 5, 3 being the set price and 2 being the bribe. Is this right? – Dimoftie Cristina Diana Feb 13 '14 at 19:01
$P$ is the selling price including the bribe, so it is not fixed at $3$ – Ross Millikan Feb 13 '14 at 19:02
Can you explain what i am doing wrong so i can understand? – Dimoftie Cristina Diana Feb 13 '14 at 19:08
The question is whether $Q_c$ is marginal revenue per unit or total. If $Q_c$ is marginal revenue for all the units, you want to set it to zero, as with a lower price the official can increase his take by charging more and with a higher one he can increase his take by charging less. Then we get $P=7$. – Ross Millikan Feb 13 '14 at 20:06
If $Q_c$ is marginal revenue per unit, for a $1$ unit increase in price he loses $\frac 12$ per unit but gains $1$ unit in sales. I am reading it total, but am not sure. – Ross Millikan Feb 13 '14 at 20:08

No theft model of corruption

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