# BEE3028 Economic Issues: homework

This page is an experiment in student generated answers to homework questions. In BEE3028 Economic Issues, we covered a diverse number of topics. The lecturer will pose a question and the students will provide the answer. Feel free to improve on other students' answers and pose new questions. Feel free to clarify the questions as well.

## Bank Runs

### Question 1

The (New York) Bank of the United States had a bank run Dec. 10, 1930. (It is a private bank with a catchy name.) The New York Federal Reserve Banking did not arrange financing to keep it open and it closed its doors forever on Dec. 11. Ultimately it was able to pay depositors upwards of 80 cents on the dollar. This indicates that the bank was solvent at the time. Friedman claims that the demise of this bank was one of the downward turning points towards the Great Depression (and also claims that the depression's cause was a credit crunch that was caused by the FED's inaction). Explain how there could be a bank run if the bank was indeed solvent. Also explain how it could have been solvent if it paid "only" 80 cents on the dollar.

If the Bank is solvent, then a bank run may still occur due to public panic. There maybe no financial reason for the run. The bank can be solvent as long as not too many depositors try to withdraw today. EXPAND THIS EITHER USING THE FULL DIAMOND-DYBVIG MODEL OR THE 2X2 GAME(TK)

### Question 2

A reason for not bailing out banks such as the Bank of the United States (or Northern Rock) is moral hazard. Please explain why this is the case. Explain how deposit insurance may also create a moral hazard problem for banks.

If the bank is bailed out there maybe be a problem of moral hazard as the bank will continue to take risks, safe in the knowledge that they can be bailed out of trouble, and will encourage all banks to be less prudent. Thus reducing stability and profitability in the banking sector, and its effectiveness as a whole in the economy.

Risky behaviour will continue with deposit insurance, as a default on any investments will not result in a loss. There would not be risky strategies being undertaken if they could be seen.

THIS SHOULD BE IMPROVED (TK)

### Question 3

Take the Diamond Dybvig model with ${\displaystyle L=.5}$ and ${\displaystyle R=2}$. Let us say that deposits are insured up to fraction ${\displaystyle f}$. For what values of ${\displaystyle f}$ is there only one equilibrium and what values are there two equilibria? (Early withdrawers are guaranteed to get ${\displaystyle 1*f}$ and late get ${\displaystyle 2*f}$.)

The bank expects the two impatient depositors to withdraw today and the two patient depositors to withdraw tomorrow. Hence, yesterday, the bank sets aside $2000 for today and invests$2000 for tomorrow. Now today, the depositors must decide whether to withdraw today or tomorrow. We assume that the impatient depositors withdraw today. Now We can examine this as a game between the two patient depositors. Each has to decide whether or not to withdraw today. When ${\displaystyle f=0}$ the payoffs as discussed in class.

 Today Tomorrow Today $750,$750 $1000,$0 Tomorrow $0,$1000 $2000,$2000

For a general ${\displaystyle f}$, we must calculate again each of the payoffs. If both withdraw today, the bank can pay the first 3 depositors the $1000. The last depositor will receive ${\displaystyle f}$. Thus, the expected payoff is ${\displaystyle 750+(f/4)*1000}$. If one withdraws today and the other withdraws tomorrow, the bank will be able to pay all three today, and the depositor withdrawing tomorrow receives ${\displaystyle 2000*f}$. Rewriting the game yields.  Today Tomorrow Today$750+1000*(f/4), $750+1000*(f/4)$1000, $2000*f Tomorrow$2000*f, $1000$2000, $2000 We see that if one patient depositor withdraws today, the second patient depositor only has incentive to withdraw today if ${\displaystyle 750+1000*(f/4)>=2000*f}$. Hence, only if ${\displaystyle f<=3/7}$, then there is a possibility of two equilibria. ### Question 4 How would you modify our classroom experiment to test different deposit insurance schemes? Under what parameters do you think we will get a bank run. ### Answer 4 ### Question 5 Take the Diamond-Dybvig model described in class with 2 impatient depositors and 2 patient depositors. Each depositor invested 1000 in the bank and was offered a contract: withdrawing today pays 1000,withdrawing tomorrow pays 2000 ( R=2). The bank had two possible meansof investing its money: a long-term investment and a short-terminvestment. The long-term investment pays R=2 times the amount invested tomorrow. Early liquidation of the investment today pays L=0.5 times the original amount invested. The short term investment pays the original amount if it is withdrawn today or tomorrow. (i) Assuming that the patient depositors wait until tomorrow to withdraw and the impatient depositors withdraw today, how should the bank divide its assets between the short-term and long-term investment to match demands? (ii) Assuming that all the impatient depositors withdraw their money today, represent the decisions of the patient depositors as 2x2 game (draw it). (iii) Indicate any pure-strategy Nash equilibrium of the game. (iv) For what values of L (0<L<1) are there the same equilibria as in part (iii)? (v) Can the Diamond-Dybvig model explain the run on Northern Rock? Why or why not? ### Answer 5 All but part (iv) was discussed in class. (iv) Remember to give the patient depositor 1000 today, the bank must liquidate an amount x where 1000=L*X. Note when L=.5, X must be 2000. Solving gives us X=1000/L. The remaining amount is (2000-1000/L). This amount is 0 when L<=.5 since it can't be negative. The amount the remaining patient depositor would receive is then R(2000-1000/L). There are the two equilibria in (iii) if this amount is LESS than the amount the patient depositor would get if all withdrew today. That amount is (2000+2000*L)/4. Thus, there are two equilibria if ${\displaystyle 2(2000-{\frac {1000}{L}})\leq {\frac {2000+2000\cdot L}{4}}}$ Simplifying we have that there are two equilibria if ${\displaystyle L^{2}-7L+4\geq 0}$ or L<0.63. For example, take L=.8. If both patient depositors withdraw today, then they would get on average 900. If one decides to wait until tomorrow and the other withdraws today, the one withdrawing tomorrow gets 2(2000-1000/.8)=1500. Thus, both withdrawing today can't be an equilibrium. Alternatively, if L=.6, both withdrawing today yields 800. One waiting for tomorrow would receive 2(2000-1000/.6)=666. Intuitively, if L is higher, there is less early liquidation costs. So the bank would have more money left over to pay the patient depositors what they are owed. A student's attempt (i) the bank should take half (2000) and keep it as reserves and take the other half (2000) and put it in the long term investment (ii)  dep. 2 today tomorrow today 500 , 500 1000 , 0 dep. 1 tomorrow 0 , 1000 2000 , 2000  //// should this be 750 in the today today box ////// YES! THERE ARE 4 DEPOSITORS WITHDRAWING TODAY. 2 IMPATIENT AND 2 PATIENT. THE BANK HAS 2000 FROM THE SHORT-TERM INVESTMENT AND 2000*.5=1000 FROM THE LONG-TERM INVESTMENT. SO 3000/4=750. (iii) there two nash equilibria, whether withdrawing both the money today or do it tomorrow, but always both have to take the same decision. SHOULD MAYBE SAY WHY NEITHER PATIENT DEPOSITORY WOULD HAVE AN INCENTIVE TO DEVIATE IN THE EQUILIBRIA (TK). (iv) ????? (v) yes. Another student's questions. iii) for this question I couldn't find a pure strategy Nash equilibrium for this one, I thought that this was when there were dominant strategies, but my understanding may be incorrect. Tom, tom is a pareto-Nash equilibrium and, today today is a suboptimal Nash. Is this what the question was asking? A NASH EQUILIBRIUM IS SET OF STRATEGIES WHERE GIVEN THE OTHER PLAYER(S) STRATEGY A PLAYER HAS NO INCENTIVE TO DEVIATE FROM HIS STRATEGY. THE NASH EQUILIBRIUM DOES NOT HAVE TO CONSIST OF DOMINANT STRATEGIES. (TK) iv) I think that I have done this question wrong. I found that this same equilibrium holds, when L is in the range of 1/3<L<0.714. To find the upper boundary I calculated when the pay-off in 0* was equal to (tod, tod) payoff(500+500L) . so R(2000L-1000)=(500+500L) and it came out at 0.714. So any value above this, the equilibrium will no longer hold. The lower boundary was when (2000L)<(500+500L) GOOD TRY. YOUR PAYOFFS ARE NOT CORRECT, BUT AT LEAST THE LOGIC IS GOING IN THE RIGHT DIRECTION. THE (TODAY,TODAY) CAN'T BE AN EQUILIBRIUM IF YOUR PAYOFF GOES UP BY WAITING UNTIL TOMORROW. (TK) QUESTION. With regard to the payoff matrix, each depositor recieves 2000 if they are patient and wait until tomorrow. Is this correct? 2000 is invested in an illiquid asset and 2000 is kept in the vault. You state that money in the vault returns the same amount, i.e is not subject to R. Therefore is the payoff not 1500 each, as the 2000 invested long term returns 4000, but the 2000 in the vault is still worth 2000 tomorrow, leaving 6000 to pay all 4 depositors...6000/4 = 1500. THE (TOMORROW, TOMORROW) PAYOFF REFERS TO THE CASE WHERE BOTH PATIENT DEPOSITORS WITHDRAW TOMORROW AND BOTH IMPATIENT WITHDRAW TODAY. ANOTHER QUESTION In the equation R(2000-1000/L) I understand the 1000/L, but to double check, is the 2000 from the maximum the player can get from waiting, ie 1000 * R which is 2000? NO, THE 2000 IS FROM THE INVESTMENT IN THE LONG-TERM INVESTMENT. ## Gift Giving ### Question 1 Examine the second treatment of the Beer-Quiche game where there is a 2/3 chance of the proposer being strong. Payoffs: Proposer, Responder  Flee Fight Beer (Strong)$1.40, $1.25$0.60, $0.75 Quiche (Strong)$1.00, $1.25$0.20, $0.75 Beer (Weak)$1.00, $0.75$0.20, $1.25 Quiche (Weak)$1.40, $0.75$0.60, $1.25 Can there be a pooling equilibrium where both proposers choose Quiche and the responder flees? Does this seem reasonable to you? ### Answer 1 All in all there can be such a pooling equilibrium. It can happen if the responder thinks that people drink beer are weak and fights them. Because of this, everyone continues to eat quiche and the responder flees. However, this is not a reasonable equilibrium because if someone switches to drinking beer then he intuitively should be strong since there is no benefit to a weak person from trying to deviate from the pooling equilibrium (for a weak person, the payoff from drinking beer and the responder fleeing is lower than the payoff from eating quiche and the responder fleeing). There is an incentive for the strong person to try to get the responder to flee while he drinks beer. (This is actually known as the Cho-Kreps intuition criteria.) Notice that the 2/3 chance of being strong is important since it causes the responder to flee. If the chance of being strong is p then the responder (who learns nothing about the strength of the proposer) then the responder for fighting would get p*.75 +(1-p)*1.25. For fleeing, he would get p*1.25+(1-p)*.75. We see that p*1.25+(1-p)*.75=>p*.75 +(1-p)*1.25 if and only if p=>.5. If p=.4, then it can't be an equilibrium since the responder will start fighting and then the strong proposer will start drinking beer. A Student's thoughts. I think initially the proposer would choose beer if strong and quiche if weak, expected payoff would be 2/3 * 1.4 + 1/3 * 0.6 = 1 2/15 presuming the responder would flee beer and fight quiche. However the proposer would have an incentive to signal beer even if weak, increasing expected payoff to 1 4/15. I get slightly confused here because i think that the responder would realise this strategy and start fighting against the proposer even when he says beer. expected payoff if he flees 100% of the time, 1 1/12. If he fights every time the expected pay off would decrease to 11/12. However the proposers expected payoff would reduce to just 7/15. Which is worse than the initial 1 2/15. Todd's response You need to look at a payoff for the proposer when he is weak or strong rather than together. In that case, you are correct in that you should see if the proposer has an incentive to deviate if weak. In that case, the proposer can increase his payoff from .60 to$1 by drinking beer rather than eating quiche.

The rest of your comments I believe are just showing what can't be an equilibrium. There is one important thing to remember which may be confusing you. The proposer cannot COMMIT to a strategy. He may know that in the long run it may hurt him to drink beer when weak, but he cannot help himself if in the short run it will increase his payoff. Just think that each time the proposer and responder are different matches (like in the experiment).

### Question 2

Some gazelles can be seen initially running slowly and jumping high when threatened by a predator such as a lion or cheetah. Can you explain this behavior?

If gazelles run slowly and jump they are effectively standing and fighting their corner, it may be enough to warn off certain weak lions and cheetahs. If they had run for their lives they might have tired themselves out, and the weaker lions may have prevailed. THINK SIGNALLING (TK)

### Question 3

In the KR search model of gift giving, we showed that when the (c,v,p,α)=(1,8,2,.3), the receiver buying the good for himself gives higher social surplus than the giver buying the good for the receiver. Let us now say that the receiver can return the good to the store for a full refund, but it still costs the receiver 1 to travel to the store. Which strategy yields higher welfare now?

## Other questions from other students

1) In the first lecture, slide 9 of 11. There are a couple of questions i was just wondering if my notes were correct on the answers: -How much should the bank sell?- the bank needs another 2000, so must sell L x 2000 to sell illiquid assets to fund todays withdrawals. (in this case i assume we don't know L, so is this just a general case?)

YES -Those withdrawing tomorrow would receive R X 1000 each..... Surely the bank will possibly not have enough money to pay these customers having sold 2000Ls worth yesterday, so what happens?

THEY GET WHAT IS LEFT OVER.

-On average, how much does those withdrawing today receive?--- would this not be 1000R as well as this is what they were guaranteed in the contract agreement. Or is it 10000-(5000+2000L)divided by the remaining 3 customers?

NOT SURE WHAT YOU MEAN HERE.

-At what value of L is the bank unable to meet demands today for those 7 depositors?-----Would this be at a value of 3/2?

WELL, THE BANK HAS 5000 IN SHORT-TERM INVESTMENTS AND NEEDS TO RAISE ANOTHER 2000. A BANK WOULD NEED TO LIQUIDATE 2000/L OF THE LONG-TERM ASSET. WHEN THIS IS LARGER THAN THE 5000 IN THE LONG-TERM ASSET, THE BANK CAN'T EVEN MEET SHORT-TERM DEMAND.

2) In the same lecture and the next slide, the nash equilibrium. How are the payoffs determined. Are the payoffs in terms on interest recieved on their deposits or absoluet vaules in relation to utility from some fucntiopn i did not write down? How is it determined that 2 individuals withdrawing today gets 3/4? Or why does depositor 2 get nothing if he withdaraws tomorrow and the other guy get 1? Does this mean he gets 1 * initial deposit? This box is for two patient depositors withdrawing today.... what would a table look like if they were withdrawing tomorrow?

PAYOFFS ARE JUST THE EXPECTED PAYOFF RECEIVED. THEY TAKE INTO ACCOUNT THE IMPATIENT DEPOSITORS. FOR INSTANCE, IF ALL 4 DEPOSITORS WITHDRAW TODAY, THEN THE BANK MUST LIQUIDATE ALL OF ITS LONG-TERM ASSETS. THE BANK WOULD THEN HAVE 3 TO GIVE TO 4 DEPOSITORS. ON AVERAGE, A DEPOSITOR WOULD RECEIVE 3/4.

3) Lecture 2, slide 11 One suggested solution to bank runs is to make R less risky. R is the interest payed to the customer who withdraws tomorrow. How would one make this less risky? By making it smaller so their is a greater chance that the bank can pay it or .....?

THE R HERE WAS MEANT TO BE THE INTEREST ON THE LONG-TERM ASSET. A LESS RISKY R WOULD BE LESS RISKY LOANS. -- Paying early withdrawals less than 1 and late withdrawals less than R to increase reserves. It says that is not the best contract. Is this referring to the best contract in efficiency terms or fairness to the depositor?

EFFICIENCY.

Would keeping more in reserves merely enable the bank to bail itself out of problems if more depositors withdraw early?

IT WOULD HELP.

4) Credit coordination in relation to Long Term Capital managemnet. is this referring to the bailout organized by the NY Fed consisting of the 14 or so banks in return for a share in the fund and the promise of a supervisory board?

YES, I AM SURE IF YOU GOOGLE "LONG TERM CAPITAL" YOU WOULD SEE A DESCRIPTION OR READ THE BOOK "WHEN GENUIS FAILED" BY ROGER LOWENSTEIN.

5)Is it possible that one question may be "explain the advantages and disadvantages of preventative measures and solutions to bank runs." in which case we should extend reading on these, or is this not the sort of question we will be asked?

NO HINTS.

6) Next slide, "better contracts." You suggest that banks can afford to pay early withdrawals more, thus ensuring customers against the need to do so. I am not sure that i get how this is feasible, since if they pay interest on what the bank themselves have not yet had time to invest and make a profit on how would the bank be able to afford to pay the customer the extra money? with R=1.5, in both periods would the full insurance contract not just be 1000*R in both periods., Rather than (2000-X)*R=X, where does the 2000 come from? and what does the X stand for? Moreover where do the potential payoffs of 1000 or 1200, or 1200 for sure come from. I understand why the risk averse would go for the certainty option but i am not sure where the figurers came from even with R=1.5 if we have no value for X?

PAYING 1000*R IN BOTH PERIODS WOULD BE GREAT, BUT NOT FEASIBLE. SINCE THERE IS ONE IMPATIENT DEPOSITOR FOR EACH PATIENT DEPOSITOR. 2000 COMES FROM THE TWO DEPOSITORS: ONE PATIENT, ONE IMPATIENT. NOTE THAT SIMPLY SOLVING THAT EQUATION FOR X, WE GET X=(2000*R)/(1+R).

7) The investor wants R where R*P(x)=1. ----- is R the return? so he wants a return of his investment guaranteed to be at least £1. But considering that P(X)=(3-x)/2 and he has chosen 1, P(X) = 2. and thus for R*P(x) then to equal one then R would have to equal 0.5?? As you can see i am a wee bit confused! I am just not sure what R*P(x) represents. I assume P(X)=(3-x)/2 gives you the probability of achieving your desired payoff so where does this fit together?

Then without insurance, the bank maximizes......P(X)*(X-R) where R=1/P(x) With insurance, Todd only needs R=1. So the bank maximizes P(X)*(X-R) where R=1 as there is no probability because the insurance deposit will cover it 100 % so really R= 1/100% ....... Why does the bank max at P(X)*(X-R)? What is X? Is X the initial deposit? The initial deposit minus the desired return multiplied by the probability of the event occurring?

8) Lecture 3 slide 10, How would evolution stop present giving? Would this be because people realize they never appreciated the full value of the gift and therefore accepted it was not "worth while" giving in the future? slide 12, how is a wedding or hinter gather gift a from of insurance?

THE EVOLUTION ARGUMENTS COME FROM THE FIELD OF EVOLUTIONARY PSYCHOLOGY WHICH BASICALLY SAYS THAT OUR EMOTIONS AND CUSTOMS ARE FORMED BY EVOLUTION. IF SOMETHING IS INEFFICIENT THAN EVOLUTION WILL GET RID OF IT IN THE SAME WAY IT WOULD GET RID OF AN UNNECESSARY 3RD EYE.

THE WEDDING GIFTS ARE A FORM OF INSURANCE IS WHEN ONE HAS A LARGE COST OF A WEDDING THEN THE NUMBER OF GIFTS INCREASES. IN SOME PLACES SUCH AS ISRAEL, MONEY IS GIVEN AND THIS IS EXACTLY LIKE INSURANCE SINCE THE AMOUNT USUALLY PAYS FOR THE WEDDING AND SOMETIMES THERE IS LEFTOVER TO HELP THE COUPLE GET STARTED. HUNTER/GATHERERS SHARE THEIR FOOD WITH GROUPS THAT DON'T MAKE A KILL. THIS AGAIN IS INSURANCE.

9) with the gift giving equation, is v the value the receiver puts on the gift. So for a granny (alpha) would be low as there is a low chance of granny getting a gift correct, in which case there is a high chance of p> V.

V IS THE VALUE THAT THE RECEIVER PUTS ON A GIFT THAT THEY WANT. THE EXPECTED VALUE OF A GIFT IS ALPHA*V. SINCE GRANNY HAS A LOW ALPHA, WE HAVE P>ALPHA*V.

10) This is copied from the lecture notes: slide 32 lecture 3 "We get a separating equilibria if Vs-Cs>0 and Vw-Cw<0. We get a pooling equilibria if Vs-Cs<0 and Vw-Cw<0 (no one signals)" If weak don't signal would it not be Vw- Cw > 0 as there are no signaling costs? or is the 0 relating to fleeing, i,e Vweak - costs of signaling would be smaller than the payoff from fleeing, and hence the individual does not flee.  ?

11) why in question 3 of wikiversity the payoffs of tod, tod are 750$. it should be 1000*L=1000*0.5=500. why is 750$?

BASICALLY YOU NEED TO TAKE INTO ACCOUNT THE TWO IMPATIENT DEPOSITORS AS WELL. SEE ABOVE.

12) Would it be possible to have a model answer to explain question 1 of the gift giving section. With reference to how the chance of being strong affects the answer?

DONE, SEE ABOVE.

13) can there still be a pooling equilibrium if one of the players has an incentive to deviate?

NO, IT WOULDN'T BE AN EQUILIBRIUM THEN.

14) answer 5 where you say, " the remaining amount is (2000 - (1000/x) " should this nor be 2000 - x , i.e. 2000 - ( 1000/L)

YUP! THANKS FOR POINTING OUT THE TYPO.

15) i understand by paying depositors withdrawing early more than 1, the risk averse will take this ensuring that if they take the gamble of investing long they are unlikely to withdraw early. But where will the money come from

THE BANK LOOKS AT WHAT IS THE BEST CONTRACT GIVEN THAT IT WILL HAVE 2 WITHDRAWING TODAY AND 2 WITHDRAWING TOMORROW. THE BANK THEN DOESN'T SPLIT THE INVESTMENT INTO 2000 AND 2000 AS BEFORE. FOR FULL INSURANCE AND R=1.5 IT WOULD INSTEAD SPLIT IT INTO 2400 FOR THE SHORT-TERM AND 1600 FOR THE LONG TERM. IT CAN THEN PAY OUT 1200 TO BOTH WITHDRAWERS TODAY AND 1200 TO BOTH WITHDRAWING TOMORROW (SINCE IT WOULD HAVE 1600*1.5=2400).

16) I'm wondering if I could ask for your help in clearing up an issue about Deposit Insurance. I think I might be getting confused (or then again I might have the right idea!) as I'm wondering whether there is a difference between deposit insurance and the central bank acting as the "lender of last resort". Doesn't the deposit insurance come from the central bank and fully (sometimes partially) protects your deposits? Or is it the Financial Services Compensation Scheme (FSCS) - in the UK at least - that is in charge of the deposit insurance? If you could clear this issue up then I'd very much appreciate it.

THIS VARIES FROM COUNTRY TO COUNTRY, BUT OFTEN THE DEPOSIT INSURANCE COMES FROM A DIFFERENT GOVERNMENT AGENCY THAN THE CENTRAL BANK. DEPOSIT INSURANCE ONLY IS PAID OUT WHEN THE BANK GOES BUST. THE LENDER OF LAST RESORTS IS A PRELIMINARY STAGE WHERE THE CENTRAL BANK LENDS MONEY AT A HIGHER INTEREST RATE AND IN RARE OCCASIONS TAKING RISKY ASSETS AS COLLATERAL. IN OUR MODEL THIS IS IN SOME SENSE HELPS INCREASE L. OTHER NOTES, THE DEPOSITS AREN'T ALWAYS FULLY GUARANTEED. IT ALSO ISN'T CLEAR WHETHER THE CENTRAL BANK WILL LEND MONEY TO THE LOCAL BANK AND TAKE RISKY ASSETS AS COLLATERAL (AS THEY DID NOW).