市场学essay/report/assignment写作
ARE/ECN 115A Spring 2018
Problem Set 3: Market Structure and Asymmetric Information in Credit Markets
Due: Thursday, May 24, 2018 at 1:40 pm
Instructions: An excel template to this problem set can be found in the PS3 excel template on Canvas. Please fill out this Excel file and copy each of the three figures described below into the worksheet that has a title matching the figure number. We will ask you to upload one Excel file at the end of the problem set. For the open-ended questions, we recommend typing your response in this Word document and then copying your answer into the Canvas quiz.
Credit Market Equilibrium under Multiple Activity Choices
In problems 1 — 3, the lender faces a single borrower who has a choice between two activities. In problem 1, the lender offers limited liability loans under symmetric information. In problem 2, the lender offers limited liability loans under asymmetric information. Finally, in problem 3, the lender offers unlimited liability loans under asymmetric information.
The following assumptions describe the borrower and lender throughout questions 1 3.
Borrower: Heng is an entrepreneur who is deciding between two investment projects. Both projects are risky and require an investment of $200. He does not have any money, so he needs a loan in order to undertake one of the projects, which have the following characteristics:
- Project 1 consists of founding an economic consulting firm Excellence with Heng, which is relatively safe: with 70% probability it succeeds and generates $600 of revenues, and with 30% probability it fails and generates only $300 of revenues.
- Project 2 is to open Be a slugger with Heng , a baseball clinic for children. Although Heng is an outstanding baseball player, this project is much riskier: with 30% probability, it succeeds and generates $1100 of revenues, and with 70 % probability it fails and generates only $210 of revenues.
Lender: Qi is a banker who may offer Heng a loan. Qis opportunity cost of money is 10%. In other words, she would earn a 10% interest rate if she invested the money in a bank instead of lending it to Heng.
- Limited Liability and Symmetric Information. We begin by assuming that Qi offers a limited liability loan contract and faces symmetric information (i.e., asymmetric information is not a problem). Under the limited liability contract, if Hengs project succeeds, he must repay 100% of the debt obligation (principal plus interest); however, if his project fails, he only has to repay 65% of the total debt obligation. For example, if the interest rate is 30%, he would have to repay 0.65(1+0.3)200 if his project fails. Symmetric information means that Qi can specify which project Heng must select, and she can enforce this selection. A credit contract thus specifies two terms: the Project and the interest rate.
(a) Derive expressions for ($) and (&), the expected values of Hengs incomes under the
two projects. Report your answers in intercept-slope form. For example, for project 1, report
($)=, where A and B are numbers that you calculate.
()=.([+])+.(.[+])=
()=.([+])+.(.[+])=
(b) Derive expressions for ($) and (&), the expected values of Qis profits on a loan to
Heng when Heng does Project 1 and 2 respectively, as functions of the interest rate, i. Report
your answers in intercept-slope form. For example, for a loan that finances project 1, report
($)=+, where A and B are numbers that you calculate.
()=.=[+](.)?+.=.[+](.)?
=+
()=.=[+](.)?+.=.[+](.)?
=+
(c) Using the Excel template available on Canvas, graph ($), (&), ($) and (&) as
functions of the interest, i. Title this graph Figure 1 : Credit Market under Limited Liability.
-500.
-400.
-300.
-200.
-100.
0.
100.
200.
300.
400.
500.
0 0.5 1 1.5 2 2.5 3
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Figure 1: Credit Market under Limited Liability
E(y1) E(y2) E(1) E(2)
(d) If the credit market is characterized by perfect competition:
Under perfect competition, the equilibrium contract will make Hengs expected income as high as possible while allowing Qis to earn at least zero expected profit. If the contract specifies
project 1, then Qi earns zero expected profit when =
=..^ Hengs expected income under project 1 and this interest rate would be: ()=B
C=.^
If the contract instead specifies project 2, then Qi earns zero expected profit when =
= .. Hengs expected income under project 2 and this interest rate would be ()=
BC=. We can now answer the questions below.
i. What is the equilibrium interest rate charged by Qi?
=
=.
ii. Which project does Qi make Heng do?
Heng does project 1
iii. How much expected profit does Qi earn from the equilibrium contract?
Qi earns zero expected profit
iv. How much expected income does Heng earn from the equilibrium contract?
Heng earns 290 in expected income
(e) If the credit market is characterized by monopoly:
Under monopoly, the equilibrium contract will make Qis expected profit as high as possible while allowing Heng to earn at least zero expected income.
If the contract specifies project 1 , Heng earns zero expected income when =
=.. At this interest rate, Qi earns ()=+=+BC=.
If the contract specifies project 2, Heng earns zero expected income when =
=.. At this interest rate, Qi earns ()=+=+(/)=.
We can now answer the questions below.
i. What is the equilibrium interest rate charged by Qi?
=
=.
ii. Which project does Qi make Heng do?
Heng does project 1
iii. How much expected profit does Qi earn from the equilibrium contract?
Qi earns 290 in expected profit
iv. How much expected income does Heng earn from the equilibrium contract?
Heng earns 0 expected income
- Limited Liability and Asymmetric Information. Now lets examine the impact of asymmetric information on the credit market equilibrium. Specifically, in this question we assume that Qi is not able to observe or enforce the project that the borrower chooses. As a result, the loan contract can only specify the interest rate (not the project). Everything else remains as in problem 1. a. What type of asymmetric information problem does Qi face? (MULTIPLE CHOICE: Moral Hazard; Adverse Selection)
Qi faces a Moral Hazard problem.
b. Qi now has to consider how her choice of the interest rate affects Hengs choice of
project. For what interest rates will Heng prefer Project 1 to Project 2?
Heng will prefer Project 1 to Project 2 as long as ()(). Using our equations from above, this implies : .=/
c. For what interest rates will Heng prefer Project 2 to Project 1?
Heng will prefer Project 2 to Project 1 (and still want to borrow) as long as: ()>() and (). From part c, we see that ()>() as long as >.. And Heng will prefer to do Project 2 instead of not borrowing as long () , which requires , or =.. To summarize, Heng will prefer to borrow and finance Project 2 if:^
.<.
d. For what interest rates will Heng prefer not to borrow?
Heng will not borrow if (&)< 0 , which occurs when:
>.
e. If the credit market is characterized by monopoly:
i. What is the equilibrium interest rate?
According to part 2b, the highest interest rate Qi can charge Heng so that Heng chooses project 1 is 0.179. At that interest rate, Qis expected profit is:
()=+=+(.)=.
From part 2d, the highest interest rate she can charge so that Heng chooses project 2 (instead of not borrowing) is 2.16. At that interest rate, Qis expected profit is::
()=+=+(/)=
So Qi will charge =.
ii. What project does Heng choose?
Heng will choose project 2.
iii. How much expected profit does Qi earn?
Qi will earn 257 in profit
iv. How much expected income does Heng earn?
Heng will earn zero expected income.
f. If the credit market is characterized by perfect competition:
i. What is the equilibrium interest rate?
Under perfect competition, the equilibrium loan will give Heng the highest possible expected income while allowing Qi to earn at least zero expected profits.
The interest rate that earns Qi zero expected profit under project 1 is 41/179 = 0.23. But we saw above that at any interest rate above 0.179, Heng prefers project 2 to project 1. So, if Qi charged an interest rate of 0.23, Heng would choose project 2 and Qi would earn negative profits.
So, instead, Qi will end up charging the interest rate such that she earns zero expected profit when Heng chooses project 2. This interest rate is such that:
()=+=
=
=.
ii. What project does Heng choose?
Heng will choose project 2.
iii. How much expected profit does Qi earn?
Qi will earn zero profit
iv. How much expected income does Heng earn?
Heng will earn: ()==(/)=
- Unlimited Liability and Asymmetric Information. Now lets see what happens if Qi instead offers unlimited liability contracts. Under unlimited liability, Heng must fully repay the loan even when his project fails. We continue to assume that Qi faces asymmetric information, so she can only specify the interest rate. a. Derive new expressions for ($) and (&), the expected values of Hengs income under the two projects. Report your answers in intercept-slope form.
()=.([+])+.([+])=
()=.([+])+.([+])=
b. Derive new expressions ($) and (&), the expected values of Qis profits from loans
that finance Projects 1 and 2 respectively, as functions of the interest rate, i. Report your
answers in intercept-slope form.
()=.=[+](.)?+.=[+](.)?
=
()=.=[+](.)?+.=[+](.)?
=
c. Using the Excel template available on Canvas, graph graph ($), (&), ($) and
(&) as functions of the interest rate, i (i.e., put i on the horizontal axis and graph over
the range i = 0 to i = 4 with 0.1 intervals). Title this graph Figure 2: Credit Market
under Unlimited Liability.
d. If the credit market is characterized by monopoly:
i. What is the equilibrium interest rate?
ii. What project does Heng choose?
iii. How much expected profit does Qi earn?
iv. How much expected income does Heng earn?
e. If the credit market is characterized by perfect competition:
i. What is the equilibrium interest rate?
ii. What project does Heng choose?
iii. How much expected profit does Qi earn?
iv. How much expected income does Heng earn?
f. Compare your answers to: A) questions 2e versus 3d and B) questions 2f versus 3e. Does
the ability to require unlimited liability contracts makes a difference to the credit market
equilibrium? If yes, why? If no, why not? Do you see any limitations to offering
unlimited liability contracts?
Part d, e, and f:
Looking at Figure 1, Qi will earn the highest expected profit if she makes Heng do Project 1
at the interest rate that allows Heng to earn zero expected income ()=. From part (a),
that interest rate ==.. So the equilibrium contract is: <Project 1, i = 1.55>
Qis expected profit under monopoly is: ()=BC=.
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-200.
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0.
100.
200.
300.
400.
500.
600.
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Interest Rate
Figure 1: Credit Market under Unlimited liability Contract
E(y1) E(y2) E(1) E(2)
Since Qi is a monopolist, she extracts all the economic surplus from Heng such that his
expected income is 0.
Looking at Figure 1, Heng is always better off (for a given interest rate) doing project 1. The
lowest interest rate that Qi would be willing to offer is the one that allows her to earn zero
expected profits. From part (a), it is easy to see that interest rate is 0.1 (this makes sense,
since 0.1 is her opportunity cost of money). So the equilibrium contract is: <Project 1, i =
0.1>
Under perfect competition, Qis expected profit is: ()=(.)=.
Under perfect competition, Hengs expected income is: ()=(.)=.
Notice that the total surplus (lenders profit plus borrowers income) is the same under the
different market structure scenarios. This is because Heng does Project 1 in both cases.
From societys point of view, this is exactly what we want because Project 1 is the more
profitable project; it generates $290 of expected profit. This is because expected revenues are
510 (= 0.70*600+0.30*300) and the cost of undertaking the project is 220 = 200*(1 + .1) (i.e.,
we have to value the opportunity cost of the $200 investment at the lenders cost of money,
which is 10%). Project 2 on the other hand, is less profitable as it only generates $257 of
expected profits. Under symmetric information, the distribution of the surplus is all-or-
nothing depending on market structure. Under monopoly, Qi gets to earn all of the surplus.
Under perfect competition, Heng gets to keep all of the surplus.
Credit Market Equilibrium under Multiple Borrower Types
Now we turn to a different problem; namely, what happens when lenders face borrowers of different types. In problem 4, the lender is a monopolist who offers limited liability loans under symmetric information. In problem 5, the lender is a monopolist who offers limited liability loans under asymmetric information.
- Limited Liability and Symmetric Information. Diego is a moneylender who lives in the village of Nyabiheke in Rwanda. Half of the farmers in Nyabiheke are SAFE farmers and the other half are RISKY farmers. Both types of farmers need a loan of $250 in order to farm. Farmers will take a loan as long as they can earn at least zero expected income. SAFE farmers have a good harvest in which they earn revenues of $450 with 100% probability. They never have a bad harvest. RISKY farmers have a good harvest in which they earn revenues of $650 with 50% probability. They have a bad harvest in which they earn revenues of $0 with 50% probability. Diego has perfect information about the farmers, i.e. he knows who is a SAFE farmer and who is RISKY. As a result, he can offer different contract terms to SAFE and RISKY types. Diegos opportunity cost is money is 20%. Diego offers limited liability credit contracts in which the farmers must repay the full loan plus interest if harvest is good, but nothing if harvest is bad. (Note: This limited liability contract is a little bit different than the one in questions 1 and 2.)
(a) Let ^ and _ denote the incomes of SAFE and RISKY farmers, respectively. Derive
expressions for (^) and (_), the expected incomes of SAFE and RISKY farmers
respectively. Report your expressions in intercept-slope format as in the questions above.
()=([+])+=
()=.([+])+=
(b) Let ^ and _ denote Diegos profits from a loan to SAFE and RISKY farmers, respectively.
Derive expressions for (^) and (_), the expected values of Diegos profits from loans
to SAFE and RISKY farmers respectively, as functions of the interest rate, i. Report your
expressions in intercept-slope format as in the questions above.
()==[+](.)?+=
()=.=[+](.)?+.=(.)?=
(c) Graph (c), (d) , (^) and (_) as functions of the interest rate, i (i.e., put i on the
horizontal axis and graph over the range i = 0 to i = 3 ). Label this Figure 3.
(d) Using your equations and graph, answer the following questions:
i. What is the highest interest rate a SAFE farmer would be willing to pay for a loan?
The highest interest rate that SAFE farmer is willing to pay is when ()= ,
i.e. =. or 80%. This is also clear from the graph.
ii. What is the highest interest rate a RISKY farmer would be willing to pay for a loan?
The highest interest rate that a RISKY farmer is willing to pay is when ()= ,
i.e. =. or 160%. This is also clear from the graph when red line ()
intersects the horizontal axis.
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200.
400.
600.
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Figure 3
E(yS) E(yR) E(S) E(R)
iii. What is the lowest interest rate Diego would be willing to charge on a loan to a SAFE
farmer?
The lowest interest rate that Diego would charge a SAFE farmer is when ()=
, i.e. =. or 20%. See graph.
iv. What is the lowest interest rate Diego would be willing to charge on a loan to a
RISKY farmer?
The lowest interest rate that Diego would charge a RISKY farmer is when ()=
, i.e. =. or 140%. See graph.
(e) Assume that Diego is a monopolist.
i. What is the equilibrium interest rate Diego would charge to a SAFE farmer?
The equilibrium interest rate that Diego would charge a SAFE farmer is the
highest interest rate that a SAFE farmer is willing to pay. We know from (c) i. that
when ()= , i.e. =. or 80%.
ii. What is the expected profit that Diego earns on this loan to SAFE farmers?
Diegos expected profit is: (250*0.80)-50=150.
iii. What is the equilibrium interest rate Diego would charge to a RISKY farmer?
The equilibrium interest rate that Diego would charge a RISKY farmer is the
highest interest rate that a RISKY farmer is willing to pay. We know from (c) ii.
that when ()= , i.e. =. or 160%.
iv. What is the expected profit that Diego earns on this loan to RISKY farmers?
Diegos expected profit is: (125*1.6)-175=25.
- Limited Liability and Asymmetric Information. Diego has decided to retire. Yaxi is a lender from a neighboring village who decides to offer loans in Nyabiheke. However, since she is from a different village, he does not know the farmers in Nyabiheke. She only knows that half of the farmers are SAFE and half are RISKY. As a result, he has to charge a single interest rate to everybody who wants a loan. Like Diego, Yaxis opportunity cost is also 20%.
(a) What type of asymmetric information problem does Yaxi face?
ADVERSE SELECTION
(b) What is the maximum interest rate Yaxi can charge so that both types of farmers would want
to borrow?
The maximum interest rate that Yaxi can charge so that both types of farmers would
borrow is =. or 80%. If she charges more than =. or 160%, then the RISKY
type would not borrow and if she charges more than 80% the SAFE types would not
borrow. So the maximum that she can charge to have both types borrow is =. or
80%.
(c) Let be Yaxis profit. Derive an expression for (), the expected value of Yaxis profit
from a loan, as a function of the interest rate, i when the interest rate is less than or equal to
the value you identified in part (b). (Remember: Over this range of the interest rate Yaxi
cannot tell to which type of farmer she has given the loan!).
()=( )=[+](.)?
+( )p.=[+](.)?
+.=(.)?q
=.()+.()
=..
(d) Explain what will happen if Yaxi increases the interest rate above the interest rate you identified in (b)? If Yaxi increases the interest rate above =. , then the SAFE type of borrowers will drop out and only the RISKY type will remain in the market.
(e) What is the maximum interest rate Yaxi can charge so that at least one type of farmer will want a loan? The maximum interest rate that Yaxi can charge is =., and at this rate only the RISKY type will want a loan.
(f) Derive an expression for Yaxis expected profit, (), as a function of the interest rate for values between the interest rates you identified in part (b) and part (e).
In this range, only the risky type borrows so we just use () , which we found above:
()=.=[+](.)?+.=(.)?=
(g) What will happen if Yaxi increases the interest rate above the interest rate you identified in (e)?
Neither type will borrow, so Yaxis expected profit will be zero.
(h) Use the expressions from parts (c) and (f) to graph Yaxis expected profit as a function of the interest rate for interest rates between 0 and 3. Label this Figure 4: Lender Expected Profit under Asymmetric Information
(i) What is the equilibrium interest rate charged by Yaxi?
The best Yaxi can do is to charge the highest interest rate such that both want a loan. From part d above, we saw that was =.
(j) What is Yaxis expected profit?
Yaxi earns 37.5 in expected profits
(k) Which type of types of farmers take the loan?
Both types take the loan.
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Interest Rate