Economics 304
Economics 304
Winter 2013
Assignment 1 : Due date, Thursday February 7 in Class
1. Consider the following OLG model with money. Assume that the population of the economy grows at rate n such that Nt = nNt−1 for every period and n > 1 and money is also growing in this economy at the rate γ that is Mt = γMt−1 , where γ > 1.
The endowment that each young generation is born with is assumed to be fixed at y. The initial old each receives an equal portion of the money supply Mt .
Suppose that the utility function of a typical agent is given by the familiar log-linear form u(c1,t , c2,t+1 ) = ln c1,t + β ln c2,t+1
(1)
where 0 < β < 1 is the discount factor
(i) Under a monetary economy, what are the first, and second period budget constraints? …show more content…
Combine these constraints into a lifetime budget constraint. [3 marks]
First period budget constraint is: pt c1,t +
Mt
≤ pt y
Nt
(2)
The second period budget constraint is: pt+1 c2,t+1 ≤
Mt
Nt
(3)
combining we have: c1,t +
pt+1 c2,t+1 ≤ y pt (4)
1 mark for the first period budget constraint and 1 mark for the second period budget constraint and 1 mark for combining them. Some people will write the budget constraint as pt c1,t + Mt ≤ pt y
(5)
The second period budget constraint is: pt+1 c2,t+1 ≤ Mt
(6)
This is fine too, give them the marks. If they write the budget constraint with equality. it’s fine too. (ii) What is the real rate of return on fiat money in this economy? [1 mark]
Using the first period budget constraint, we can derive the aggregate money demand which is given by: Mtd = pt (y − c1t )Nt . Assuming equilibrium in the money market, we have money demand=money supply = Mt . Thus we have Mt = pt (y − c1t )Nt pt+1 = pt Mt+1
Mt
Nt (y − c1,t )
Nt+1 (y − c1,t+1 )
1
(7)
γ
Hence given Nt = nNt−1 and Mt = γMt−1 , we have pt+1 = n . This is the real rate of return on pt pt money is given by pt+1 = n . γ If they don’t get it don’t give part marks. They need to show how they arrive at this ratio of prices
(iii) If γ < n, is the value of money falling or increasing over time. Explain your answer carefully.
[1.5 marks]
If γ < n this implies that money is growing at a slower rate than population. In other words, total amount of goods is growing at a faster rate than money. Money can purchase more and more goods, implying that the real value of money is increasing over time.
All marks for intuition.
(iv) Compute the monetary equilibrium of this economy. That is find the optimal value for c1 and c2 [6 marks]
max u(c1 , c2 ) = ln c1 + β ln c2 c1 ,c2
subject to γ c2 ≤ y c1 + n The Lagrangean is set-up as:
ℓ = ln c1 + β ln c2 + λ y − c1 −
γ c2 n
(8)
where λ is the Lagrange multiplier. It can be interpreted as the shadow price of consumption.
The first-order conditions are:
1
=λ c1 γ β =λ c2 : c2 n γ c2 = 0 λ : y − c1 − n c1 :
(9)
(10)
(11)
Using the first-order conditions for c1 and c2 , we have: βn 1
=λ=
c1 γc2 (12)
Thus c2 = nβ c1 . γ Substituting this optimal condition in the budget constraint, we have: y = c1 +
nβ φ γ c1 = (1 + β)c1 ⇒ c∗ =
1
n
1
1+β
y and c∗ =
2
n γ β
1+β
y
(13)
1 mark for setting up and 3 marks for FOC conditions and 2 for the final answer.
If they are using MRS=price ratio, same scheme
2
2. For this question, you will either have to provide a definition or carefully explain your answer.
In both cases, please keep your answer concise and clear.
(i) if J=10, that is there are 10 goods, what is the probability of a successful exchange in a barter economy and in a monetary economy. [1 marks]
1
In a BE, it is J 21 = 100−10 =
−2
1
In a ME, it is J−1 = 1
9
0.5 mark for each
1
90
(ii) Carefully explain what functions a financial asset provide to society [2 marks] Financial assets provide two important economic functions. The first is risk-sharing. Financial assets allow risks to be spread among different parties instead of one party bearing all the risks.
The second is to transfer funds from entities who have surplus funds to invest to those who need funds to invest in tangible assets.
0.5 mark for each
(iii) What do the terms adverse selection, moral hazard and costly state verification mean? Provide an example within the context of a financial transaction/market [4 marks] Adverse selection and moral hazard are two market failures that arise in the credit market. Financial transactions are intrinsically characterized by asymmetric information. Borrowers generally have private information that is more accurate than the information possessed by lenders since they can more easily assess their own risks compared to the lender. Lenders can try to protect themselves from bad borrowers (“lemons”) by setting the contractual terms in a manner that reflects the average quality of their loan applicants. However, by doing so, they run the risk that high risk borrowers will be encouraged to self-select into their loan applicant pool while at the same time low risk borrowers will be encouraged to self-select out of this pool. The resulting adverse effects on the quality of their loan applicant pool constitutes an example of adverse selection.
Moral hazard arise in credit markets because borrowers may decide to engage in riskier projects once its demand for a loan has been approved. Once their project/loan is approved, borrowers may have a bigger incentive to engage in riskier investment projects with higher returns since they stand to lose the same if the project fails but stand to gain more if the project succeeds. This problem is exacerbated with higher interest rates.
Costly enforcement of contracts arise because it takes resources to monitor borrowers to ensure that the amount borrowed is repaid back at the end of the contract.
For example, someone borrows money from a financial institution. The financial institution in turn has to monitor the borrower and make sure that the latter is paying back her loan.
1.5 marks for adverse selection and 1.5 for moral hazard and 1 for costly enforcement. make sure they write a sensible answer, otherwise don’t give them the marks.
(iv) “Financial intermediaries provide an essential maturity and liquidity transformation service”.
In your own words, carefully explain what this statement means. [3 marks]
Financial intermediaries do not only pool savings but they also engage in maturity and liquidity transformation. The proceeds banks receive from depositors (savings) are used to grant loans to
3
other economic agents (households, firms, government). The deposits represent financial claims that the financial intermediary issues (secondary securities) whereas the loans represent the financial claims that the financial intermediary purchases and owns (primary securities). Most of the secondary securities (savings) are highly liquid and can be withdrawn at any time. They have short maturity. On the other hand, primary securities (loans) are very illiquid and have long …show more content…
maturity.
There is thus a maturity and liquidity mismatch between these primary and secondary securities.
Financial intermediaries by providing this maturity and liquidity transformation provide an important device as they provide savers the ability to withdraw their funds at any point in time while providing borrowers long with much longer duration. Financial intermediaries can provide this maturity and liquidity transformation only if a portion of savers withdraw their funds in the short-run and only if savers are confident that the will be able to withdraw their funds when needed.
What is important in the answer is that they explain maturity and liquidity. First, they have distinguish between the short-term liabilities (savings) that FI have and the long-term assets (loans). Savings are liquid but not long-term loans (1 mark for saying this). Second, they have to say that savings can be withdrawn at any time
(short-maturity) but loans are usually for long-periods of time (long-maturity). Thus there is a maturity mismatch (1.5 marks). Third, FI provide the essential service of maturity and liquidity transformation only if savers do not withdraw everything at the same time and they have confidence in the FI (0,.5 marks)
4
Winter 2013
Assignment 1 : Due date, Thursday February 7 in Class
1. Consider the following OLG model with money. Assume that the population of the economy grows at rate n such that Nt = nNt−1 for every period and n > 1 and money is also growing in this economy at the rate γ that is Mt = γMt−1 , where γ > 1.
The endowment that each young generation is born with is assumed to be fixed at y. The initial old each receives an equal portion of the money supply Mt .
Suppose that the utility function of a typical agent is given by the familiar log-linear form u(c1,t , c2,t+1 ) = ln c1,t + β ln c2,t+1
(1)
where 0 < β < 1 is the discount factor
(i) Under a monetary economy, what are the first, and second period budget constraints? …show more content…
Combine these constraints into a lifetime budget constraint. [3 marks]
First period budget constraint is: pt c1,t +
Mt
≤ pt y
Nt
(2)
The second period budget constraint is: pt+1 c2,t+1 ≤
Mt
Nt
(3)
combining we have: c1,t +
pt+1 c2,t+1 ≤ y pt (4)
1 mark for the first period budget constraint and 1 mark for the second period budget constraint and 1 mark for combining them. Some people will write the budget constraint as pt c1,t + Mt ≤ pt y
(5)
The second period budget constraint is: pt+1 c2,t+1 ≤ Mt
(6)
This is fine too, give them the marks. If they write the budget constraint with equality. it’s fine too. (ii) What is the real rate of return on fiat money in this economy? [1 mark]
Using the first period budget constraint, we can derive the aggregate money demand which is given by: Mtd = pt (y − c1t )Nt . Assuming equilibrium in the money market, we have money demand=money supply = Mt . Thus we have Mt = pt (y − c1t )Nt pt+1 = pt Mt+1
Mt
Nt (y − c1,t )
Nt+1 (y − c1,t+1 )
1
(7)
γ
Hence given Nt = nNt−1 and Mt = γMt−1 , we have pt+1 = n . This is the real rate of return on pt pt money is given by pt+1 = n . γ If they don’t get it don’t give part marks. They need to show how they arrive at this ratio of prices
(iii) If γ < n, is the value of money falling or increasing over time. Explain your answer carefully.
[1.5 marks]
If γ < n this implies that money is growing at a slower rate than population. In other words, total amount of goods is growing at a faster rate than money. Money can purchase more and more goods, implying that the real value of money is increasing over time.
All marks for intuition.
(iv) Compute the monetary equilibrium of this economy. That is find the optimal value for c1 and c2 [6 marks]
max u(c1 , c2 ) = ln c1 + β ln c2 c1 ,c2
subject to γ c2 ≤ y c1 + n The Lagrangean is set-up as:
ℓ = ln c1 + β ln c2 + λ y − c1 −
γ c2 n
(8)
where λ is the Lagrange multiplier. It can be interpreted as the shadow price of consumption.
The first-order conditions are:
1
=λ c1 γ β =λ c2 : c2 n γ c2 = 0 λ : y − c1 − n c1 :
(9)
(10)
(11)
Using the first-order conditions for c1 and c2 , we have: βn 1
=λ=
c1 γc2 (12)
Thus c2 = nβ c1 . γ Substituting this optimal condition in the budget constraint, we have: y = c1 +
nβ φ γ c1 = (1 + β)c1 ⇒ c∗ =
1
n
1
1+β
y and c∗ =
2
n γ β
1+β
y
(13)
1 mark for setting up and 3 marks for FOC conditions and 2 for the final answer.
If they are using MRS=price ratio, same scheme
2
2. For this question, you will either have to provide a definition or carefully explain your answer.
In both cases, please keep your answer concise and clear.
(i) if J=10, that is there are 10 goods, what is the probability of a successful exchange in a barter economy and in a monetary economy. [1 marks]
1
In a BE, it is J 21 = 100−10 =
−2
1
In a ME, it is J−1 = 1
9
0.5 mark for each
1
90
(ii) Carefully explain what functions a financial asset provide to society [2 marks] Financial assets provide two important economic functions. The first is risk-sharing. Financial assets allow risks to be spread among different parties instead of one party bearing all the risks.
The second is to transfer funds from entities who have surplus funds to invest to those who need funds to invest in tangible assets.
0.5 mark for each
(iii) What do the terms adverse selection, moral hazard and costly state verification mean? Provide an example within the context of a financial transaction/market [4 marks] Adverse selection and moral hazard are two market failures that arise in the credit market. Financial transactions are intrinsically characterized by asymmetric information. Borrowers generally have private information that is more accurate than the information possessed by lenders since they can more easily assess their own risks compared to the lender. Lenders can try to protect themselves from bad borrowers (“lemons”) by setting the contractual terms in a manner that reflects the average quality of their loan applicants. However, by doing so, they run the risk that high risk borrowers will be encouraged to self-select into their loan applicant pool while at the same time low risk borrowers will be encouraged to self-select out of this pool. The resulting adverse effects on the quality of their loan applicant pool constitutes an example of adverse selection.
Moral hazard arise in credit markets because borrowers may decide to engage in riskier projects once its demand for a loan has been approved. Once their project/loan is approved, borrowers may have a bigger incentive to engage in riskier investment projects with higher returns since they stand to lose the same if the project fails but stand to gain more if the project succeeds. This problem is exacerbated with higher interest rates.
Costly enforcement of contracts arise because it takes resources to monitor borrowers to ensure that the amount borrowed is repaid back at the end of the contract.
For example, someone borrows money from a financial institution. The financial institution in turn has to monitor the borrower and make sure that the latter is paying back her loan.
1.5 marks for adverse selection and 1.5 for moral hazard and 1 for costly enforcement. make sure they write a sensible answer, otherwise don’t give them the marks.
(iv) “Financial intermediaries provide an essential maturity and liquidity transformation service”.
In your own words, carefully explain what this statement means. [3 marks]
Financial intermediaries do not only pool savings but they also engage in maturity and liquidity transformation. The proceeds banks receive from depositors (savings) are used to grant loans to
3
other economic agents (households, firms, government). The deposits represent financial claims that the financial intermediary issues (secondary securities) whereas the loans represent the financial claims that the financial intermediary purchases and owns (primary securities). Most of the secondary securities (savings) are highly liquid and can be withdrawn at any time. They have short maturity. On the other hand, primary securities (loans) are very illiquid and have long …show more content…
maturity.
There is thus a maturity and liquidity mismatch between these primary and secondary securities.
Financial intermediaries by providing this maturity and liquidity transformation provide an important device as they provide savers the ability to withdraw their funds at any point in time while providing borrowers long with much longer duration. Financial intermediaries can provide this maturity and liquidity transformation only if a portion of savers withdraw their funds in the short-run and only if savers are confident that the will be able to withdraw their funds when needed.
What is important in the answer is that they explain maturity and liquidity. First, they have distinguish between the short-term liabilities (savings) that FI have and the long-term assets (loans). Savings are liquid but not long-term loans (1 mark for saying this). Second, they have to say that savings can be withdrawn at any time
(short-maturity) but loans are usually for long-periods of time (long-maturity). Thus there is a maturity mismatch (1.5 marks). Third, FI provide the essential service of maturity and liquidity transformation only if savers do not withdraw everything at the same time and they have confidence in the FI (0,.5 marks)
4