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1. Consider the model of liquidity and financial intermediation presented in class. The economy has an infinite horizon and is populated by a large number of individuals who each live for three periods. In period t there are N t individuals born where N t = n N t 1 and n 1 . Individuals receive utility from consuming in middle age and when old. That is, individuals have utility function U ( c 2 , t + 1 , c 3 , t + 2 ) = ln c 2 , t + 1 + ln c 3 , t + 2 where ( 0 , 1 ) . For simplicity you can assume that n = 1 and = 1 . Individuals have one unit of time that is supplied inelastically as labor, but possess different productivities denoted by [ 0 , 1 ] . An individual with productivity produces units of the consumption good when young. Assume that the productivity of the young generation is uniformly distributed over the interval [ 0 , 1 ] in every period. There is a risk-neutral bank takes deposits and commits to a one-period return denoted by r d , so that one unit of the consumption good deposited with the bank yields the gross return r d units of the consumption good in t + 1 . When an individual withdraws from the bank, they pay a fixed cost > 0 , measured in units of the consumption good. This fixed cost has many interpretations including the annual fee on a deposit account or the time required to go to a bank (or bank machine) to withdraw (in fact during COVID-19 you could also consider the risk associated with making an in-person withdrawal as a component of this fixed cost). Individuals have three stores of value available; fiat money ( m ) , deposits ( d ) , and capital ( k ) . There is a supply of fiat money available in period t denoted by M t , where M t = z M t 1 for all t where z 1 . Capital has a two period return X > n 2 and a one-period return x = X 1/2 , however, unmatured capital cannot be sold, traded or borrowed against. For parts (a) - (d) you can assume that the real rate of return on fiat money in a stationary equilibrium is n = 1 , and the return on deposits satisfies r d ( n , x ) , where x = X 1/2 . Further for parts (a) - (e) you can assume that the money supply is constant so that z = 1 and M t = M for all t . Please answer the following questions and assume a stationary equilibrium (i.e. c 2 , t ( ) = c 2 ( ) for all t and c 3 , t ( ) = c 3 ( ) for all t ) . (a) Write down the utility maximization problem of a type worker including all budget constraints. Explain how individuals in the economy will use money, deposits and capital. (b) Write down the Lagrangian and solve for the optimal choices of capital, deposits, real money and consumption for any type individual. If it is easier, you an solve the problem by considering a type w worker and doing the following: (i) Assume the bank does not exist so that individuals do not make deposits and can only use fiat money and capital. Solve for the equilibrium levels of consumption ( c 2 , t + 1 and c 3 , t + 2 ) , capital ( k t ) , and individual real money demand ( t m t.
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![1. Consider the model of liquidity and financial intermediation presented in class. The economy
has an infinite horizon and is populated by a large number of individuals who each live for three
periods. In period t there are N t individuals born where N t = n N t 1 and n 1 . Individuals
receive utility from consuming in middle age and when old. That is, individuals have utility
function U ( c 2 , t + 1 , c 3 , t + 2 ) = ln c 2 , t + 1 + ln c 3 , t + 2 where ( 0 , 1 ) . For simplicity
you can assume that n = 1 and = 1 . Individuals have one unit of time that is supplied
inelastically as labor, but possess different productivities denoted by [ 0 , 1 ] . An individual with
productivity produces units of the consumption good when young. Assume that the productivity
of the young generation is uniformly distributed over the interval [ 0 , 1 ] in every period. There
is a risk-neutral bank takes deposits and commits to a one-period return denoted by r d , so that
one unit of the consumption good deposited with the bank yields the gross return r d units of the
consumption good in t + 1 . When an individual withdraws from the bank, they pay a fixed cost
> 0 , measured in units of the consumption good. This fixed cost has many interpretations
including the annual fee on a deposit account or the time required to go to a bank (or bank
machine) to withdraw (in fact during COVID-19 you could also consider the risk associated with
making an in-person withdrawal as a component of this fixed cost). Individuals have three stores
of value available; fiat money ( m ) , deposits ( d ) , and capital ( k ) . There is a supply of fiat
money available in period t denoted by M t , where M t = z M t 1 for all t where z 1 . Capital has
a two period return X > n 2 and a one-period return x = X 1/2 , however, unmatured capital
cannot be sold, traded or borrowed against. For parts (a) - (d) you can assume that the real rate of
return on fiat money in a stationary equilibrium is n = 1 , and the return on deposits satisfies r d (
n , x ) , where x = X 1/2 . Further for parts (a) - (e) you can assume that the money supply is
constant so that z = 1 and M t = M for all t . Please answer the following questions and assume a
stationary equilibrium (i.e. c 2 , t ( ) = c 2 ( ) for all t and c 3 , t ( ) = c 3 ( ) for all t ) . (a) Write
down the utility maximization problem of a type worker including all budget constraints. Explain
how individuals in the economy will use money, deposits and capital. (b) Write down the
Lagrangian and solve for the optimal choices of capital, deposits, real money and consumption
for any type individual. If it is easier, you an solve the problem by considering a type w worker
and doing the following: (i) Assume the bank does not exist so that individuals do not make
deposits and can only use fiat money and capital. Solve for the equilibrium levels of
consumption ( c 2 , t + 1 and c 3 , t + 2 ) , capital ( k t ) , and individual real money demand ( t m
t ) . Note: these should depend on . (ii) Assume now that there is no money so individuals are
only able to make deposits or hold capital. Solve for the equilibrium levels of consumption ( c 2 ,
t + 1 and c 3 , t + 2 ) , capital ( k t ) , and deposits ( d t ) in terms of r d . Note: these should
depend on w . (iii) Now consider the case with both money and bank deposits. Do individuals
hold money, make deposits, or both? If both, find an expression that determines what individuals
acquire money and what individuals deposit their goods with the bank. Explain the result.](https://image.slidesharecdn.com/1-considerthemodelofliquidityandfinancialintermediationpresen-230220235542-72add158/85/1-Consider-the-model-of-liquidity-and-financial-intermediation-presen-docx-1-320.jpg)
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1. The document discusses various theories related to the quantity theory of money, including the Fisher Identity and the Cambridge Cash Balances equation.
2. It analyzes the components and assumptions of these theories, such as the transactions demand for money being proportional to aggregate transactions.
3. Criticisms of the older quantity theories are presented, including that they assumed full employment and an exogenously determined money supply, whereas Keynes argued resources are often underemployed and the money supply is endogenous.
Advanced macroeconomics, 4th edition. Romer. Chapter12.12.1. T.docx
Advanced macroeconomics, 4th edition. Romer.
Chapter12.
12.1. The stability of fiscal policy. (Blinder and Solow, 1973.) By definition, the budget deficit equals the rate of change of the amount of debt outstanding: δ(t) ≡ D ̇(t). Define d(t) to be the ratio of debt to output: d(t) = D(t)/Y(t). Assume that Y(t) grows at a constant rate g > 0.
(a) Suppose that the deficit-to-output ratio is constant: δ(t)/Y(t) = a, where a > 0.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). Is this system stable?
(b) Suppose that the ratio of the primary deficit to output is constant and equal to a > 0. Thus the total deficit at t, δ(t), is given by δ(t) = aY(t) + r(t)D(t), where r(t) is the interest rate at t. Assume that r is an increasing function of the debt-to-output ratio: r(t) = r(d(t)), where r′(•) > 0, r′′(•) > 0, limd→−∞ r(d) < g, limd→∞ r(d) > g.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). In the case where a is sufficiently small that d ̇ is negative for some values of d, what are the stability properties of the system? What about the case where a is sufficiently large that d ̇ is positive for all values of d ?
12.2. Precautionary saving, non-lump-sum taxation, and Ricardian equivalence.
(Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts Y1 and Y2 in the two periods. Y1 is known, but Y2 is random; assume for simplicity that E[Y2] = Y1. The government taxes income at rate τ1 in period 1 and τ2 in period 2. The individual can borrow and lend at a fixed interest rate, which for simplicity is assumed to be zero. Thus second-period consumption is C2 = (1 − τ1)Y1 − C1 + (1 − τ2)Y2. The individualchoosesC1 tomaximizeexpectedlifetimeutility,U(C1)+E[U(C2)].
(a) Find the first-order condition for C1.
(b) Show that E[C2] = C1 if Y2 is not random or if utility is quadratic.
(c) Show that if U ′′′(•) > 0 and Y2 is random, E[C2] > C1.
(d) Suppose that the government marginally lowers τ1 and raises τ2 by the same amount, so that its expected total revenue, τ1Y1 + τ2E[Y2], is un- changed. Implicitly differentiate the first-order condition in part (a) to find an expression for how C1 responds to this change.
(e) Show that C1 is unaffected by this change if Y2 is not random or if utility is quadratic.
(f) Show that C1 increases in response to this change if U ′′′(•) > 0 and Y2 is random.
12.3
Consider the Barro tax-smoothing model. Suppose that output, Y, and the real interest rate, r, are constant, and that the level of government debt out- standing at time 0 is zero. Suppose that there will be a temporary war from time 0 to time τ. Thus G(t) equals GH for 0 ≤ t ≤ τ, and equals GL there- after,whereGH >GL.Whatarethepathsoftaxes,T(t),andgovernmentdebt outstanding, D(t)?
12.4
Consider the Barro tax-smoothing model. Supp.
Duration and Convexity
This document discusses quantitative finance topics including bond duration and immunization. It provides an example showing how duration and convexity can be used to approximate changes in bond prices from changes in yields. The document also discusses how to construct a bond portfolio with a target duration and convexity. Finally, it briefly defines interest rate swaps, bond options, interest rate caps, and floors.
2 Consider the following Diamond and Dybvig style model with 3 period.pdf
2 Consider the following Diamond and Dybvig style model with 3 periods, t=0,1,2 where agents
each have a $1 endowment at t=0 and care about consuming at t=1 or t=2. Their endowment can
be invested in two assets; a storage asset that allows the costless transfer of one unit from one
time period to another and an illiquid asset that pays out R=2 in period 2 but only pays out L=21
if it is liquidated early in period 1 . At t=1 a fraction people learn that they only care about
consumption at t=1, while a fraction (1) people learn that they only care about consumption at
t=2, and so the ex ante expected utility of an agent in t=0 is u(C1)+(1)u(C2) where the utility
function is u(C)=ln(C). For all questions state any assumptions you are making in deriving your
answers. (a) If =21 what is the expected utility of an optimal bank contract if the possibility of a
bank run, S, is zero i.e. if it is known with certainty that there won't be a bank run? (b) Suppose
that the probability of bank run is S=101, but all banks still offer the optimal contract as if there
was no possibility of a bank run as in part (a). In the event of a bank run all a bank's assets are
distributed equally between depositors. Now what is the expected utility of an agent in period 0
who deposits his/her endowment in a bank? (c) Suppose that the government implemented a
weak narrow banking regulation that each bank must ensure that reserves and the liquidation of
long term assets cover all potential withdrawals i.e. that 1x+xL>C1 where x denotes the per
agent investment in the illiquid asset. What is the expected utility of an optimal bank contract
under these regulations?.
Slides1
1. The document discusses net present value and the time value of money. It introduces concepts like future value, present value, discount rates, and compounding interest.
2. Examples are provided to illustrate how to calculate the net present value of projects using future and present cash flows. A positive net present value indicates the project should be accepted.
3. The concepts of perpetuities, annuities, and multiperiod cash flows are explained. Consumer preferences and budget constraints over time are also covered.
International journal of engineering and mathematical modelling vol2 no2_2015_1
Management of the portfolios containing low liquidity assets is a tedious problem. The buyer proposes the price that can differ greatly from the paper value estimated by the seller, so the seller can not liquidate his portfolio instantly and waits for a more favorable offer. To minimize losses and move the theory towards practical needs one
can take into account the time lag of the liquidation of an illiquid asset. Working in the Merton’s optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. While a standard Black-Scholes market describes the liquid part of the investment the illiquid asset is sold at an exogenous random moment with prescribed liquidation time distribution. The investor has the logarithmic utility function as a limit case of a HARA-type utility. Different distributions of the liquidation time of the illiquid asset are under consideration - a classical exponential distribution andWeibull distribution that is more practically relevant. Under certain conditions we show the existence
of the viscosity solution in both cases. Applying numerical methods we compare classical Merton’s strategies and the optimal consumption-allocation strategies for portfolios with different liquidation time distributions of an illiquid asset.
27 money supply and money demand
The document discusses money supply and money demand. It explains that money supply is determined by the behavior of households, banks, and the Federal Reserve. Banks can create money through fractional-reserve banking by keeping only a fraction of deposits as reserves and lending out the rest. This allows the initial deposit to create additional money in the economy. The money supply and monetary base are also influenced by the reserve-deposit ratio, currency-deposit ratio, and money multiplier. The Federal Reserve can conduct open market operations and adjust reserve requirements and interest rates to influence the money supply. Money demand theories, including portfolio and transactions theories like the Baumol-Tobin model, are also covered.
Consider an environment of overlapping generation model that is simi.pdf
Consider an environment of overlapping generation model that is similar to the model we have
studied in the class. Suppose time is discrete and time horizon is innite, t = 0; 1; 2; :::The
economy is populated with 2-period lived individuals. In period t, Lt new individuals are born
and Lt1 old individuals are dying if they havent died earlier due to covid-19. Assume population
grows at rate n. The lifetime utility of an individual born at t is given by U (c1t ; c2t+1) where
c1t and c2t+1 denote the consumption when young and when old. Utility of an initial old is
simply u (c2; 0). Suppose U is additively separable and the periodic utility, u(c), is an increasing
and concave function of consumption and satises the inada condition. Individuals discount the
future with the discount rate (cid:12) where 0 < (cid:12)(cid:12)< 1: The initial capital stock K0 owned equally by all
initial old individuals. In each period, rms hire labour and rent capital from individuals to
produce output, sell the output in good market, and pay labour and capital. The CRS production
function is Yt = F Kt ; AtL Dt. The intensive form of production: yt = f (kt) has the following
properties: f0(k) > 0, f00 (k) < 0 and limk!0 f0(k) = +1: Suppose At+1 = (1 + gc) At and there is
a drop in the growth of productivity, gc < g. Markets are perfectly competitive. Real interest rate
in period t is rt & the real wage rate per unit of eective labour is wt . Initial conditions (A0, L1
(number of initial old)), K0 are giv
Below are the list of events
(cid:15) (i) Young individuals are born with labour endowments, which is supplied inelastically to rms.
Young individuals may die due to the getting Covid19. Assume that the utility of dead is zero.
With a probability p1 they survive and get to the second period of their life. [Hint: This
probability was one in the model in the class]
(cid:15) (ii) Old individuals are those young individuals that survive the rst period of life. With a
probability p2; they survive to the end of the second period of life. Again the utility of dead is
assumed to be zero. [Hint: This probability was one in the model in the class. Recall that in the
model in the class, all old people die and exit the economy at the end of the second period. Now
they may die before that]. For simplicity we assume even if they die at the middle of the second
period they will consume capital income and existing wealth.
(cid:15) (iii) Firms hire labour from young individuals and rent capital from old individuals to produce
output. Capital and labour are paid.
(cid:15) (iv) Young individuals divide labour income between rst-period consumption and saving to
period t + 1 which form new capital for period t + 1 production. Here for simplicity we assume
that if a young individual dies her savings are kept in an account and will be given to the rm.
Carefully formulate an individual problem for nding the optimal saving. What are the impact of
p1 and p2 on optimal saving?.
Recommended
Qtm
1. The document discusses various theories related to the quantity theory of money, including the Fisher Identity and the Cambridge Cash Balances equation.
2. It analyzes the components and assumptions of these theories, such as the transactions demand for money being proportional to aggregate transactions.
3. Criticisms of the older quantity theories are presented, including that they assumed full employment and an exogenously determined money supply, whereas Keynes argued resources are often underemployed and the money supply is endogenous.
Advanced macroeconomics, 4th edition. Romer. Chapter12.12.1. T.docx
Advanced macroeconomics, 4th edition. Romer.
Chapter12.
12.1. The stability of fiscal policy. (Blinder and Solow, 1973.) By definition, the budget deficit equals the rate of change of the amount of debt outstanding: δ(t) ≡ D ̇(t). Define d(t) to be the ratio of debt to output: d(t) = D(t)/Y(t). Assume that Y(t) grows at a constant rate g > 0.
(a) Suppose that the deficit-to-output ratio is constant: δ(t)/Y(t) = a, where a > 0.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). Is this system stable?
(b) Suppose that the ratio of the primary deficit to output is constant and equal to a > 0. Thus the total deficit at t, δ(t), is given by δ(t) = aY(t) + r(t)D(t), where r(t) is the interest rate at t. Assume that r is an increasing function of the debt-to-output ratio: r(t) = r(d(t)), where r′(•) > 0, r′′(•) > 0, limd→−∞ r(d) < g, limd→∞ r(d) > g.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). In the case where a is sufficiently small that d ̇ is negative for some values of d, what are the stability properties of the system? What about the case where a is sufficiently large that d ̇ is positive for all values of d ?
12.2. Precautionary saving, non-lump-sum taxation, and Ricardian equivalence.
(Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts Y1 and Y2 in the two periods. Y1 is known, but Y2 is random; assume for simplicity that E[Y2] = Y1. The government taxes income at rate τ1 in period 1 and τ2 in period 2. The individual can borrow and lend at a fixed interest rate, which for simplicity is assumed to be zero. Thus second-period consumption is C2 = (1 − τ1)Y1 − C1 + (1 − τ2)Y2. The individualchoosesC1 tomaximizeexpectedlifetimeutility,U(C1)+E[U(C2)].
(a) Find the first-order condition for C1.
(b) Show that E[C2] = C1 if Y2 is not random or if utility is quadratic.
(c) Show that if U ′′′(•) > 0 and Y2 is random, E[C2] > C1.
(d) Suppose that the government marginally lowers τ1 and raises τ2 by the same amount, so that its expected total revenue, τ1Y1 + τ2E[Y2], is un- changed. Implicitly differentiate the first-order condition in part (a) to find an expression for how C1 responds to this change.
(e) Show that C1 is unaffected by this change if Y2 is not random or if utility is quadratic.
(f) Show that C1 increases in response to this change if U ′′′(•) > 0 and Y2 is random.
12.3
Consider the Barro tax-smoothing model. Suppose that output, Y, and the real interest rate, r, are constant, and that the level of government debt out- standing at time 0 is zero. Suppose that there will be a temporary war from time 0 to time τ. Thus G(t) equals GH for 0 ≤ t ≤ τ, and equals GL there- after,whereGH >GL.Whatarethepathsoftaxes,T(t),andgovernmentdebt outstanding, D(t)?
12.4
Consider the Barro tax-smoothing model. Supp.
Duration and Convexity
This document discusses quantitative finance topics including bond duration and immunization. It provides an example showing how duration and convexity can be used to approximate changes in bond prices from changes in yields. The document also discusses how to construct a bond portfolio with a target duration and convexity. Finally, it briefly defines interest rate swaps, bond options, interest rate caps, and floors.
2 Consider the following Diamond and Dybvig style model with 3 period.pdf
2 Consider the following Diamond and Dybvig style model with 3 periods, t=0,1,2 where agents
each have a $1 endowment at t=0 and care about consuming at t=1 or t=2. Their endowment can
be invested in two assets; a storage asset that allows the costless transfer of one unit from one
time period to another and an illiquid asset that pays out R=2 in period 2 but only pays out L=21
if it is liquidated early in period 1 . At t=1 a fraction people learn that they only care about
consumption at t=1, while a fraction (1) people learn that they only care about consumption at
t=2, and so the ex ante expected utility of an agent in t=0 is u(C1)+(1)u(C2) where the utility
function is u(C)=ln(C). For all questions state any assumptions you are making in deriving your
answers. (a) If =21 what is the expected utility of an optimal bank contract if the possibility of a
bank run, S, is zero i.e. if it is known with certainty that there won't be a bank run? (b) Suppose
that the probability of bank run is S=101, but all banks still offer the optimal contract as if there
was no possibility of a bank run as in part (a). In the event of a bank run all a bank's assets are
distributed equally between depositors. Now what is the expected utility of an agent in period 0
who deposits his/her endowment in a bank? (c) Suppose that the government implemented a
weak narrow banking regulation that each bank must ensure that reserves and the liquidation of
long term assets cover all potential withdrawals i.e. that 1x+xL>C1 where x denotes the per
agent investment in the illiquid asset. What is the expected utility of an optimal bank contract
under these regulations?.
Slides1
1. The document discusses net present value and the time value of money. It introduces concepts like future value, present value, discount rates, and compounding interest.
2. Examples are provided to illustrate how to calculate the net present value of projects using future and present cash flows. A positive net present value indicates the project should be accepted.
3. The concepts of perpetuities, annuities, and multiperiod cash flows are explained. Consumer preferences and budget constraints over time are also covered.
International journal of engineering and mathematical modelling vol2 no2_2015_1
Management of the portfolios containing low liquidity assets is a tedious problem. The buyer proposes the price that can differ greatly from the paper value estimated by the seller, so the seller can not liquidate his portfolio instantly and waits for a more favorable offer. To minimize losses and move the theory towards practical needs one
can take into account the time lag of the liquidation of an illiquid asset. Working in the Merton’s optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. While a standard Black-Scholes market describes the liquid part of the investment the illiquid asset is sold at an exogenous random moment with prescribed liquidation time distribution. The investor has the logarithmic utility function as a limit case of a HARA-type utility. Different distributions of the liquidation time of the illiquid asset are under consideration - a classical exponential distribution andWeibull distribution that is more practically relevant. Under certain conditions we show the existence
of the viscosity solution in both cases. Applying numerical methods we compare classical Merton’s strategies and the optimal consumption-allocation strategies for portfolios with different liquidation time distributions of an illiquid asset.
27 money supply and money demand
The document discusses money supply and money demand. It explains that money supply is determined by the behavior of households, banks, and the Federal Reserve. Banks can create money through fractional-reserve banking by keeping only a fraction of deposits as reserves and lending out the rest. This allows the initial deposit to create additional money in the economy. The money supply and monetary base are also influenced by the reserve-deposit ratio, currency-deposit ratio, and money multiplier. The Federal Reserve can conduct open market operations and adjust reserve requirements and interest rates to influence the money supply. Money demand theories, including portfolio and transactions theories like the Baumol-Tobin model, are also covered.
Consider an environment of overlapping generation model that is simi.pdf
Consider an environment of overlapping generation model that is similar to the model we have
studied in the class. Suppose time is discrete and time horizon is innite, t = 0; 1; 2; :::The
economy is populated with 2-period lived individuals. In period t, Lt new individuals are born
and Lt1 old individuals are dying if they havent died earlier due to covid-19. Assume population
grows at rate n. The lifetime utility of an individual born at t is given by U (c1t ; c2t+1) where
c1t and c2t+1 denote the consumption when young and when old. Utility of an initial old is
simply u (c2; 0). Suppose U is additively separable and the periodic utility, u(c), is an increasing
and concave function of consumption and satises the inada condition. Individuals discount the
future with the discount rate (cid:12) where 0 < (cid:12)(cid:12)< 1: The initial capital stock K0 owned equally by all
initial old individuals. In each period, rms hire labour and rent capital from individuals to
produce output, sell the output in good market, and pay labour and capital. The CRS production
function is Yt = F Kt ; AtL Dt. The intensive form of production: yt = f (kt) has the following
properties: f0(k) > 0, f00 (k) < 0 and limk!0 f0(k) = +1: Suppose At+1 = (1 + gc) At and there is
a drop in the growth of productivity, gc < g. Markets are perfectly competitive. Real interest rate
in period t is rt & the real wage rate per unit of eective labour is wt . Initial conditions (A0, L1
(number of initial old)), K0 are giv
Below are the list of events
(cid:15) (i) Young individuals are born with labour endowments, which is supplied inelastically to rms.
Young individuals may die due to the getting Covid19. Assume that the utility of dead is zero.
With a probability p1 they survive and get to the second period of their life. [Hint: This
probability was one in the model in the class]
(cid:15) (ii) Old individuals are those young individuals that survive the rst period of life. With a
probability p2; they survive to the end of the second period of life. Again the utility of dead is
assumed to be zero. [Hint: This probability was one in the model in the class. Recall that in the
model in the class, all old people die and exit the economy at the end of the second period. Now
they may die before that]. For simplicity we assume even if they die at the middle of the second
period they will consume capital income and existing wealth.
(cid:15) (iii) Firms hire labour from young individuals and rent capital from old individuals to produce
output. Capital and labour are paid.
(cid:15) (iv) Young individuals divide labour income between rst-period consumption and saving to
period t + 1 which form new capital for period t + 1 production. Here for simplicity we assume
that if a young individual dies her savings are kept in an account and will be given to the rm.
Carefully formulate an individual problem for nding the optimal saving. What are the impact of
p1 and p2 on optimal saving?.
Unit 9
The document summarizes the overlapping generations model. It describes the structure of the model where individuals live for two periods and there are always two generations alive at each point in time. The model examines the dynamic efficiency of the economy and whether social security can eliminate dynamic inefficiency. It outlines the key aspects of the model including household optimization, macroeconomic equilibrium conditions, and how the capital labor ratio changes over time according to a dynamic equation. The model can exhibit multiple steady states and local stability depends on where the dynamic equation intersects the 45 degree line.
H & Q CH 12 Optimization over time.ppt
1) The document introduces multi-period optimization, where consumers maximize utility over multiple time periods rather than just one. Consumers can borrow and lend between periods.
2) Basic assumptions include the availability of one-period bonds for borrowing and lending, with interest rate i. Consumers maximize a multi-period utility function subject to their intertemporal budget constraint.
3) The consumer's rate of time preference η between any two periods should equal the market interest rate ε between those periods for an optimal consumption plan. If η is less than ε, the consumer will borrow, and vice versa if η is greater than ε.
Dynamic Consumption – Savings Framework.pptx
- The document introduces a framework for analyzing intertemporal consumption and savings choices over two time periods (present/period 1 and future/period 2).
- It defines key concepts like income, wealth, budget constraints, and savings. The individual receives income in both periods and can save in period 1 to consume or repay debts in period 2.
- The individual's utility depends on consumption in both periods (c1 and c2). Budget constraints equate the individual's resources (income + savings + interest) to expenditures (consumption + savings carried to next period) in each period.
Unit 10
In a standard two-period overlapping generations model of exchange, money facilitates transactions by allowing individuals to exchange current consumption for future consumption. Without money, individuals cannot reach the socially optimal outcome. The introduction of money as a medium of exchange enables people to save current endowments by exchanging them for money, then use that money to buy future consumption and reach their optimal lifetime utility. At the aggregate steady state equilibrium, the rate of deflation equals the population growth rate, allowing individuals' budget constraints to coincide with their optimal consumption choices.
Homework 51)a) the IS curve ln Yt= ln Y(t+1) – (1Ɵ)rtso th.docx
Homework 5
1)
a) the IS curve: ln Yt= ln Y(t+1) – (1/Ɵ)rt
so the slope is: drt/dyt (is) = -Ɵ/Yt. That means that an increase in Ɵ will result in a steeper curve.
LM curve: Mt/Pt = Yt^(Ɵ/v) (1+rt / rt)^(1/v)
Ln(Mt/Pt) = (Ɵ/v) ln Yt +(1/v)ln(1+rt) – (1/v)ln rt.
0 = (Ɵ/v)(1/Yt)dYt + (1/v)(1/(1+rt)) drt – (1/v)(1/rt)drt.
The slope is: drt/dyt (LM) = (Ɵrt(1+rt))/Yt. That means that an increase in Ɵ will result in a steeper curve.
b) the curve IS is not affected by the value of V. while curve LM shifts upwards, since a decrease in v will result in an increase for the demand for real money.
c) IS is not affected byΓ(.)
optimal money holdings: BΓ’(Mt/Pt) = (it/(1+it)) U’(Ct)
B(Mt/Pt)^(-v) = (it/1+it) Yt^-Ɵ
Mt/Pt= B^(1/v) Yt^(Ɵ/v) (1+rt/rt)^(1/v)
So this means that the LM curve will shift downwards.
2)
a) AC= (PC/)+(αYP/2)i
AC/ = -(PC/^2) + (αYP/2)I = 0
C/^2 = αYi/2
So *=(2C/αYi)^(1/2)
b) average real money holdings:M/P= αY/2
M/P = (αY/2) (2C/αYi)^(1/2)
M/P= (αCY/2i)^(1/2)
Ln(m/p) = (1/2)(lnα+lnY+lnC-ln2-lni)
(1/(M/P))((M/P)/i) = -(1/2)(1/i)
Elasticity of real money with respect to i: ((M/P)/)(i/(M/P)) = -1/2
The elasticity with respect to Y : ((M/P)/Y)(Y/(M/P)) = ½
Average real money holdings increase in Y, and decrease in i.
4)
a)when p is at a level that generates maximum output, LS meets LD.
b) when p is above the level that generates maximum output, will cause unemployment.
7)
a)
b)i)
ii)
iii)
13)
a) the asset has an expected rate of return r. capital gain/loss plus dividends per unit time = rvp. There is no dividends per unit time while searching for the palm tree, and there is b probability per unit time of capital gain of (vc-vp)-c. the difference in the price of the asset is(vc-vp) and –c is what the asset pays, so at the end we have rvp=b(vc-vp-c)
b) there is probability aL that a person will find another person with a coconut and trade with that person and gain u̅. the difference in the price of the asset is (vp-vc). So we end up with
rvp=al(vp-vc+u̅).
c) vp=(rvc/aL)+vc-u̅.
r((rvc/aL )+vc-u̅)= b(vc-(rvc/aL)-vc+u̅-c)
vc(r(r+aL+b))/aL = u̅(r+b)-bc
the value of being in state C: vc= (aL(u̅(r+b)-bc)) / r(r+aL+b)
the value of being in state p: vp= ((u̅(r+b)-bc)/(r+aL+b)) + (aL(u̅(r+b)-bc)/r(r+aL+b)) - u̅
so finally
vc-vp = (bc+u̅aL)/(r+aL+b).
e) vc-vp ≥c
vc-vp = (bc+u̅a(b/a))/(r+a(b/a)+b) = (bc+bu̅)/(r+2b)
(bc+bu̅)/(r+2b) ≥ c
That means that
Bc+bu̅≥c and c(r+2b-b) ≤ bu̅
So finally we have
c≤ bu̅ / (r+b).
f) it is a steady-state equilibrium for no one who finds a tree to climb it for any value of c>0.
Yes there are values of c which there is more than one steady-state equilibrium for 0<c< bu̅/(r+b)
Yes, L = b/a has a higher welfare than L=0. When L=0 people don’t gain any utility since they don’t climb a tree and don’t have a chance to trade with other people and gain a coconut.
0 1 2 3 4 5 -3 -2.2000000000000002 -1.8 -1.8 -2.2000000000000002 -3
0 1 2 3 4 5 7 6.5 5.5 3.5 1
0 1 2 3 4 -2 -2.5 -3.5 -5.5 -8
LD.
Optimal Policy Response to Booms and Busts in Credit and Asset Prices
This document summarizes a model investigating the optimal policy response to booms and busts in credit and asset prices. The model shows how an externality can emerge from decentralized borrowing decisions, potentially leading to overborrowing. The laissez-faire level of borrowing is analyzed and found to differ from the socially optimal level due to agents not internalizing the externality. The analysis suggests that imposing a tax on borrowing could achieve the socially optimal level of borrowing by addressing this externality. However, the magnitude and direction of the optimal tax rate are found to differ from previous findings.
slides_endowment.pdf
This document summarizes key aspects of an intertemporal model of a small open economy's current account. It introduces the model, which involves a two-period economy where households receive endowments and choose consumption and savings. It describes the sequential budget constraints, intertemporal budget constraint, indifference curves, and how households optimize consumption over time by equalizing the marginal rate of substitution to the interest rate. The document also notes some properties of small open economies and defines the interest rate parity condition.
Calisto 2016a 251116
This document summarizes a research paper that examines the optimal investment, consumption, and life insurance selection problem for a wage earner. The problem is modeled using a financial market with one risk-free asset and one risky jump-diffusion asset, along with an insurance market composed of multiple life insurance companies. The goal is to maximize the wage earner's expected utility from consumption during life, wealth at retirement or death, by choosing an optimal investment, consumption, and insurance strategy. The authors use dynamic programming to characterize the optimal solution and prove existence and uniqueness of a solution to the associated nonlinear Hamilton-Jacobi-Bellman equation.
Criteria Mark 1. Professional Presentation 1 · Well organiz.docx
Criteria / Mark
1. Professional Presentation/ 1
· Well organized, neatly presented, word processed, spelling, grammar, and punctuation
· Name and student number as well as lab partner/s name, date and time of experiment
· In-text referencing, reference list at end of report
2. Title and Abstract/ 1
· Title: concise title directly above abstract (even if you have it on the cover page already)
· Abstract: Brief description of the experiment, the major results, associated uncertainties, comparison with theoretical results/expected results.
3. Introduction and Aims/ 1
· A brief paragraph describing key physics and relevant background
· At the end of the introduction, state the aims of experiment
4. Procedure and Apparatus/ 0.5
· Describe the equipment used, you may include labeled sketches, drawing, or photographs of apparatus
· Describe how you conducted the experiment. Remember to use past tense, you are telling the reader what you have done.
5. Risk Assessment/ 0.5
· Use a risk assessment matrix (under lab section of Blackboard) to:
(i) Consider the health and safety risk of this experiment to yourself and others in the laboratory.
(ii) Determine what measures you would put in place to mitigate/reduce these risks where possible.
· Attach the risk assessment at the end of your report after the references list
6. Data, Critical Analysis and Uncertainties / 2
· Data presented in numbered table format with proper titles and headings ie. “Table 1: Description of Some Tabulated Data”
· Figures to presented with number headings and with proper titles and headings ie. “Figure 1: Graph of some Data”
· You should write text which refers to/describes data in tables and plots
· Show calculations of key parameters and calculations of uncertainties where necessary
· Present data in scientific notation where applicable
· Use appropriate SI units – eg. metres (m), seconds (s), metres per second (m/s).
7. Summary and Discussions /2
· Brief summary the experiment and aims
· Describe the results and put them in context with the expected or theoretical values – do they match? If not, why? You may need to do some research for this part.
· Identify the major source(s) of uncertainty and how they contribute to the final uncertainty
· Statements describing ways to improve the precision and/or accuracy of the final result by improving the method employed as well as considering alternative equipment
· Unnecessary repetition of method or results will not be considered as discussion
8. Answers to specific questions /2
MOD001072
MANAGING THE ECONOMY
Weeks 7-8
Classical model- small open economy
*
Weeks 7-12
The three topics
WKS 7-8
CLASSICAL MODEL‘Long run’ –flexible pricesOpen economy
WKS 9-10
IS-LM [‘Keynesian’] MODEL‘Short run’ – fixed pricesOpen economy
WKS 11-12
INFLATIONARY EXPECTATIONSAdaptive expectationsRational expectations
HOLIDAY BREAK
*
WEEKS 7-8 SUMMARY
CLASSICAL MODEL0. Classical models –basic features, closed v.
Economic Crises Homework Help
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When Fiscal consolidation meets private deleveraging
This document summarizes a model that analyzes the interaction between public and private deleveraging in a small open economy. The model shows that larger and front-loaded fiscal consolidations entail larger output and welfare losses by postponing the end of the private deleveraging process. The model finds that deleveraging-friendly fiscal consolidations that are more gradual in nature minimize these losses.
Resources Economics. Jon M.Conrad. Chapter 1
1. The document discusses renewable and nonrenewable natural resources. It defines renewable resources as those that regenerate on a relevant economic time scale and nonrenewable resources as those that do not regenerate.
2. It explains the difference between resource economics, which studies the allocation of natural resources, and environmental economics, which focuses on conservation.
3. The optimal harvesting of renewable resources over time presents a dynamic optimization problem to maximize economic value over periods. Nonrenewable resources depletion also impacts the accumulation of waste pollution over time.
Froyen21
1) Keynes argued that consumption is determined by disposable income and other factors. Empirical studies from 1929-1941 estimated the consumption function as C = 26.5 + 0.75Yd, supporting Keynes' theory.
2) Kuznets' data from 1869-1938 showed average propensity to consume remained constant despite rising income, contradicting Keynesian predictions.
3) The life cycle and permanent income hypotheses were proposed as alternative explanations for consumption, arguing it depends on lifetime rather than current income. They better explained post-war consumption levels.
1. This question is on the application of the Binomial option
1. This question is on the application of the Binomial option
pricing model.
PKZ stock is currently trading at 100. Over three-months it will either
go up by 6% or down by 5%. Interest rates are zero.
a. [25 marks] Using a two period binomial model to construct a delta-
hedged portfolio, price a six month European call option on PKZ
stock with a strike price of £105.
b. [3 Marks] Using your answer from the first part, together with the
put-call parity, price a put option on the same stock with same
strike and expiry.
COMP0041 SEE NEXT PAGE
2
2. This question is on the Binomial method in the limit δt → 0.
[40 Marks] The binomial model for pricing options leads to the for-
mula
V (S,t) = e−rδt [qV (US,t + δt) + (1 − q) V (DS,t + δt)]
where
U = eσ
√
δt, D = e−σ
√
δt, q =
erδt −D
U −D
.
V (S,t) is the option value, t is the time, S is the spot price, σ is volatil-
ity and r is the risk-free rate.
By carefully expanding U,D,q as Taylor series in δt or
√
δt (as appro-
priate) and then expanding V (US,t + δt) and V (DS,t + δt) as Taylor
series in both their arguments, deduce that to O (δt) ,
∂V
∂t
+
1
2
σ2S2
∂2V
∂S2
+ rS
∂V
∂S
− rV = 0.
COMP0041 SEE NEXT PAGE
3
3. This question is on probability and Monte Carlo
a. Consider theprobabilitydensity function p (x) fora randomvariable
X given by
p (x) =
{
µ exp (−µx) x ≥ 0
0 x < 0
where µ (> 0) is a constant.
i. [15 Marks] Show that for this probability density function
E
[
eθX
]
=
(
1 −
θ
µ
)−1
Hint: You may assume µ > θ in obtaining this result.
ii. [20 Marks] By expanding
(
1 −
θ
µ
)−1
as a Taylor series, show
that
E [xn] =
n!
µn
, n = 0, 1, 2, ....
iii. [15 Marks] Hence calculate the skew and kurtosis for X.
COMP0041 CONTINUED ON NEXT PAGE
4
b. [32 Marks] An Exchange Option gives the holder the right to
exchange one asset for another. The discounted payoff for this
contract V is
V = e−rT max (S1 (T) −S2 (T) , 0) .
The option price is then given by θ = E [V ] where
Si (t) = Si (0) e
(r−12σ
2
i )t+σiφi
√
t
for i = 1, 2, and φi ∼ N (0, 1) with correlation coeffi cient ρ.
Youmayassumethatauniformrandomnumbergenerator isavail-
able. Use a Cholesky factorisation method to show(
φ1
φ2
)
=
(
1 0
ρ
√
1 −ρ2
)(
x1
x2
)
,
where
(
x1
x2
)
is a vector of independent N (0, 1) variables and
has the same distribution as
(
φ1
φ2
)
.
Give a Monte Carlo simulation algorithm that makes use of anti-
thetic variates for the estimation of θ.
COMP0041 SEE NEXT PAGE
5
4. This question is on finite differences
a. [30 Marks] Consider a forward difference operator, ∆, such that
∆V (S) = V (S + h) −V (S) , (4.1)
where h is an infinitessimal. By introducing the operators
D ≡
∂
∂S
; D2 ≡
∂2
∂S2
show that
∆ ≡ ehD −1 (4.2)
where 1 is the identity operator. Hint: start by doing a Taylor
expansion on V (S + h) .
By rearranging (4.2) show that
D =
1
h
(
∆ −
∆2
2
+
∆3
3
−
∆4
4
+ O
(
∆5
))
.
Hence obtain the second order approximation for
∂V
...
1. This question is on the application of the Binomial option
1. This question is on the application of the Binomial option
pricing model.
PKZ stock is currently trading at 100. Over three-months it will either
go up by 6% or down by 5%. Interest rates are zero.
a. [25 marks] Using a two period binomial model to construct a delta-
hedged portfolio, price a six month European call option on PKZ
stock with a strike price of £105.
b. [3 Marks] Using your answer from the first part, together with the
put-call parity, price a put option on the same stock with same
strike and expiry.
COMP0041 SEE NEXT PAGE
2
2. This question is on the Binomial method in the limit δt → 0.
[40 Marks] The binomial model for pricing options leads to the for-
mula
V (S,t) = e−rδt [qV (US,t + δt) + (1 − q) V (DS,t + δt)]
where
U = eσ
√
δt, D = e−σ
√
δt, q =
erδt −D
U −D
.
V (S,t) is the option value, t is the time, S is the spot price, σ is volatil-
ity and r is the risk-free rate.
By carefully expanding U,D,q as Taylor series in δt or
√
δt (as appro-
priate) and then expanding V (US,t + δt) and V (DS,t + δt) as Taylor
series in both their arguments, deduce that to O (δt) ,
∂V
∂t
+
1
2
σ2S2
∂2V
∂S2
+ rS
∂V
∂S
− rV = 0.
COMP0041 SEE NEXT PAGE
3
3. This question is on probability and Monte Carlo
a. Consider theprobabilitydensity function p (x) fora randomvariable
X given by
p (x) =
{
µ exp (−µx) x ≥ 0
0 x < 0
where µ (> 0) is a constant.
i. [15 Marks] Show that for this probability density function
E
[
eθX
]
=
(
1 −
θ
µ
)−1
Hint: You may assume µ > θ in obtaining this result.
ii. [20 Marks] By expanding
(
1 −
θ
µ
)−1
as a Taylor series, show
that
E [xn] =
n!
µn
, n = 0, 1, 2, ....
iii. [15 Marks] Hence calculate the skew and kurtosis for X.
COMP0041 CONTINUED ON NEXT PAGE
4
b. [32 Marks] An Exchange Option gives the holder the right to
exchange one asset for another. The discounted payoff for this
contract V is
V = e−rT max (S1 (T) −S2 (T) , 0) .
The option price is then given by θ = E [V ] where
Si (t) = Si (0) e
(r−12σ
2
i )t+σiφi
√
t
for i = 1, 2, and φi ∼ N (0, 1) with correlation coeffi cient ρ.
Youmayassumethatauniformrandomnumbergenerator isavail-
able. Use a Cholesky factorisation method to show(
φ1
φ2
)
=
(
1 0
ρ
√
1 −ρ2
)(
x1
x2
)
,
where
(
x1
x2
)
is a vector of independent N (0, 1) variables and
has the same distribution as
(
φ1
φ2
)
.
Give a Monte Carlo simulation algorithm that makes use of anti-
thetic variates for the estimation of θ.
COMP0041 SEE NEXT PAGE
5
4. This question is on finite differences
a. [30 Marks] Consider a forward difference operator, ∆, such that
∆V (S) = V (S + h) −V (S) , (4.1)
where h is an infinitessimal. By introducing the operators
D ≡
∂
∂S
; D2 ≡
∂2
∂S2
show that
∆ ≡ ehD −1 (4.2)
where 1 is the identity operator. Hint: start by doing a Taylor
expansion on V (S + h) .
By rearranging (4.2) show that
D =
1
h
(
∆ −
∆2
2
+
∆3
3
−
∆4
4
+ O
(
∆5
))
.
Hence obtain the second order approximation for
∂V
...
Exam doaa
This document contains questions about system dynamics modeling concepts like stocks, flows, causal loop diagrams, stock and flow diagrams, and reference modes. It asks the reader to identify which variables are stocks and flows in various contexts, draw causal loop and stock and flow diagrams to represent different systems, and build a simple stock and flow model to represent population growth and health sector development over time.
Keynesian Income Determination
This document discusses key concepts in the simple Keynesian theory of income determination, including:
1. Endogenous and exogenous variables - endogenous variables are explained by the theory, while exogenous variables are taken as given. Real output and consumption are initially treated as endogenous.
2. The consumption function - consumption is explained as autonomous consumption plus the marginal propensity to consume times disposable income. This can be shown graphically.
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Macro prudential policy(2)
This document summarizes a research paper that develops a dynamic general equilibrium model to analyze systemic risk in the banking sector. The key aspects of the model are that it includes banks that engage in maturity transformation by issuing non-state contingent debt, and the banks are exposed to risks in capital markets that can affect their solvency. The model shows that individual banks in a competitive system will take on excessive systemic risk due to pecuniary externalities, leading to a higher crisis probability than the socially optimal level. The document then discusses using prompt corrective action (PCA) policies to reduce crisis risk by strengthening bank capital requirements.
Note classical
This lecture note introduces classical macroeconomic theory, which examines the linkages between interest rates, money, output, and inflation. It will first cover classical monetary theory, which assumes money is neutral and its supply only determines prices. It then presents a basic classical model of the real economy with aggregate supply and demand. Equilibrium output and interest rates are determined by the intersection of these curves. Money enters by facilitating transactions but does not affect real variables. Money demand depends on nominal income and interest rates, determining equilibrium money holdings and the price level.
Macrodynamics of Debt-Financed Investment-Led Growth with Interest Rate Rules
This document provides an overview of a macroeconomic model that examines debt-financed investment-led growth. Some key points:
- The model explores whether financial factors can provide stability to an otherwise unstable demand-constrained economy, and whether they can generate growth cycles.
- Investment is determined by a post-Keynesian accelerator function where the sensitivity of investment to capacity utilization depends on financial factors like the risk of default and interest rates.
- Debt dynamics are modeled, where borrowing, repayment, and the debt stock over time are determined. Financial fragility and creditworthiness indicators are also developed.
- Monetary policy follows a Taylor-type rule where the central bank adjusts interest rates in response
1- Demonstrate an in-depth understanding of what a Learning organizat.docx
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4. Examine the context of the evolving global market
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H & Q CH 12 Optimization over time.ppt
1) The document introduces multi-period optimization, where consumers maximize utility over multiple time periods rather than just one. Consumers can borrow and lend between periods.
2) Basic assumptions include the availability of one-period bonds for borrowing and lending, with interest rate i. Consumers maximize a multi-period utility function subject to their intertemporal budget constraint.
3) The consumer's rate of time preference η between any two periods should equal the market interest rate ε between those periods for an optimal consumption plan. If η is less than ε, the consumer will borrow, and vice versa if η is greater than ε.
Dynamic Consumption – Savings Framework.pptx
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Unit 10
In a standard two-period overlapping generations model of exchange, money facilitates transactions by allowing individuals to exchange current consumption for future consumption. Without money, individuals cannot reach the socially optimal outcome. The introduction of money as a medium of exchange enables people to save current endowments by exchanging them for money, then use that money to buy future consumption and reach their optimal lifetime utility. At the aggregate steady state equilibrium, the rate of deflation equals the population growth rate, allowing individuals' budget constraints to coincide with their optimal consumption choices.
Homework 51)a) the IS curve ln Yt= ln Y(t+1) – (1Ɵ)rtso th.docx
Homework 5
1)
a) the IS curve: ln Yt= ln Y(t+1) – (1/Ɵ)rt
so the slope is: drt/dyt (is) = -Ɵ/Yt. That means that an increase in Ɵ will result in a steeper curve.
LM curve: Mt/Pt = Yt^(Ɵ/v) (1+rt / rt)^(1/v)
Ln(Mt/Pt) = (Ɵ/v) ln Yt +(1/v)ln(1+rt) – (1/v)ln rt.
0 = (Ɵ/v)(1/Yt)dYt + (1/v)(1/(1+rt)) drt – (1/v)(1/rt)drt.
The slope is: drt/dyt (LM) = (Ɵrt(1+rt))/Yt. That means that an increase in Ɵ will result in a steeper curve.
b) the curve IS is not affected by the value of V. while curve LM shifts upwards, since a decrease in v will result in an increase for the demand for real money.
c) IS is not affected byΓ(.)
optimal money holdings: BΓ’(Mt/Pt) = (it/(1+it)) U’(Ct)
B(Mt/Pt)^(-v) = (it/1+it) Yt^-Ɵ
Mt/Pt= B^(1/v) Yt^(Ɵ/v) (1+rt/rt)^(1/v)
So this means that the LM curve will shift downwards.
2)
a) AC= (PC/)+(αYP/2)i
AC/ = -(PC/^2) + (αYP/2)I = 0
C/^2 = αYi/2
So *=(2C/αYi)^(1/2)
b) average real money holdings:M/P= αY/2
M/P = (αY/2) (2C/αYi)^(1/2)
M/P= (αCY/2i)^(1/2)
Ln(m/p) = (1/2)(lnα+lnY+lnC-ln2-lni)
(1/(M/P))((M/P)/i) = -(1/2)(1/i)
Elasticity of real money with respect to i: ((M/P)/)(i/(M/P)) = -1/2
The elasticity with respect to Y : ((M/P)/Y)(Y/(M/P)) = ½
Average real money holdings increase in Y, and decrease in i.
4)
a)when p is at a level that generates maximum output, LS meets LD.
b) when p is above the level that generates maximum output, will cause unemployment.
7)
a)
b)i)
ii)
iii)
13)
a) the asset has an expected rate of return r. capital gain/loss plus dividends per unit time = rvp. There is no dividends per unit time while searching for the palm tree, and there is b probability per unit time of capital gain of (vc-vp)-c. the difference in the price of the asset is(vc-vp) and –c is what the asset pays, so at the end we have rvp=b(vc-vp-c)
b) there is probability aL that a person will find another person with a coconut and trade with that person and gain u̅. the difference in the price of the asset is (vp-vc). So we end up with
rvp=al(vp-vc+u̅).
c) vp=(rvc/aL)+vc-u̅.
r((rvc/aL )+vc-u̅)= b(vc-(rvc/aL)-vc+u̅-c)
vc(r(r+aL+b))/aL = u̅(r+b)-bc
the value of being in state C: vc= (aL(u̅(r+b)-bc)) / r(r+aL+b)
the value of being in state p: vp= ((u̅(r+b)-bc)/(r+aL+b)) + (aL(u̅(r+b)-bc)/r(r+aL+b)) - u̅
so finally
vc-vp = (bc+u̅aL)/(r+aL+b).
e) vc-vp ≥c
vc-vp = (bc+u̅a(b/a))/(r+a(b/a)+b) = (bc+bu̅)/(r+2b)
(bc+bu̅)/(r+2b) ≥ c
That means that
Bc+bu̅≥c and c(r+2b-b) ≤ bu̅
So finally we have
c≤ bu̅ / (r+b).
f) it is a steady-state equilibrium for no one who finds a tree to climb it for any value of c>0.
Yes there are values of c which there is more than one steady-state equilibrium for 0<c< bu̅/(r+b)
Yes, L = b/a has a higher welfare than L=0. When L=0 people don’t gain any utility since they don’t climb a tree and don’t have a chance to trade with other people and gain a coconut.
0 1 2 3 4 5 -3 -2.2000000000000002 -1.8 -1.8 -2.2000000000000002 -3
0 1 2 3 4 5 7 6.5 5.5 3.5 1
0 1 2 3 4 -2 -2.5 -3.5 -5.5 -8
LD.
Optimal Policy Response to Booms and Busts in Credit and Asset Prices
This document summarizes a model investigating the optimal policy response to booms and busts in credit and asset prices. The model shows how an externality can emerge from decentralized borrowing decisions, potentially leading to overborrowing. The laissez-faire level of borrowing is analyzed and found to differ from the socially optimal level due to agents not internalizing the externality. The analysis suggests that imposing a tax on borrowing could achieve the socially optimal level of borrowing by addressing this externality. However, the magnitude and direction of the optimal tax rate are found to differ from previous findings.
slides_endowment.pdf
This document summarizes key aspects of an intertemporal model of a small open economy's current account. It introduces the model, which involves a two-period economy where households receive endowments and choose consumption and savings. It describes the sequential budget constraints, intertemporal budget constraint, indifference curves, and how households optimize consumption over time by equalizing the marginal rate of substitution to the interest rate. The document also notes some properties of small open economies and defines the interest rate parity condition.
Calisto 2016a 251116
This document summarizes a research paper that examines the optimal investment, consumption, and life insurance selection problem for a wage earner. The problem is modeled using a financial market with one risk-free asset and one risky jump-diffusion asset, along with an insurance market composed of multiple life insurance companies. The goal is to maximize the wage earner's expected utility from consumption during life, wealth at retirement or death, by choosing an optimal investment, consumption, and insurance strategy. The authors use dynamic programming to characterize the optimal solution and prove existence and uniqueness of a solution to the associated nonlinear Hamilton-Jacobi-Bellman equation.
Criteria Mark 1. Professional Presentation 1 · Well organiz.docx
Criteria / Mark
1. Professional Presentation/ 1
· Well organized, neatly presented, word processed, spelling, grammar, and punctuation
· Name and student number as well as lab partner/s name, date and time of experiment
· In-text referencing, reference list at end of report
2. Title and Abstract/ 1
· Title: concise title directly above abstract (even if you have it on the cover page already)
· Abstract: Brief description of the experiment, the major results, associated uncertainties, comparison with theoretical results/expected results.
3. Introduction and Aims/ 1
· A brief paragraph describing key physics and relevant background
· At the end of the introduction, state the aims of experiment
4. Procedure and Apparatus/ 0.5
· Describe the equipment used, you may include labeled sketches, drawing, or photographs of apparatus
· Describe how you conducted the experiment. Remember to use past tense, you are telling the reader what you have done.
5. Risk Assessment/ 0.5
· Use a risk assessment matrix (under lab section of Blackboard) to:
(i) Consider the health and safety risk of this experiment to yourself and others in the laboratory.
(ii) Determine what measures you would put in place to mitigate/reduce these risks where possible.
· Attach the risk assessment at the end of your report after the references list
6. Data, Critical Analysis and Uncertainties / 2
· Data presented in numbered table format with proper titles and headings ie. “Table 1: Description of Some Tabulated Data”
· Figures to presented with number headings and with proper titles and headings ie. “Figure 1: Graph of some Data”
· You should write text which refers to/describes data in tables and plots
· Show calculations of key parameters and calculations of uncertainties where necessary
· Present data in scientific notation where applicable
· Use appropriate SI units – eg. metres (m), seconds (s), metres per second (m/s).
7. Summary and Discussions /2
· Brief summary the experiment and aims
· Describe the results and put them in context with the expected or theoretical values – do they match? If not, why? You may need to do some research for this part.
· Identify the major source(s) of uncertainty and how they contribute to the final uncertainty
· Statements describing ways to improve the precision and/or accuracy of the final result by improving the method employed as well as considering alternative equipment
· Unnecessary repetition of method or results will not be considered as discussion
8. Answers to specific questions /2
MOD001072
MANAGING THE ECONOMY
Weeks 7-8
Classical model- small open economy
*
Weeks 7-12
The three topics
WKS 7-8
CLASSICAL MODEL‘Long run’ –flexible pricesOpen economy
WKS 9-10
IS-LM [‘Keynesian’] MODEL‘Short run’ – fixed pricesOpen economy
WKS 11-12
INFLATIONARY EXPECTATIONSAdaptive expectationsRational expectations
HOLIDAY BREAK
*
WEEKS 7-8 SUMMARY
CLASSICAL MODEL0. Classical models –basic features, closed v.
Economic Crises Homework Help
I am Barry A. I am an Economics Crisis Assignment Expert at economicshomeworkhelper.com/. I hold a Ph.D. in Economics. I have been helping students with their homework for the past 5 years. I solve assignments related to Economics Crisis.
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You can also call on +1 678 648 4277 for any assistance with Economics Crisis Assignments.
When Fiscal consolidation meets private deleveraging
This document summarizes a model that analyzes the interaction between public and private deleveraging in a small open economy. The model shows that larger and front-loaded fiscal consolidations entail larger output and welfare losses by postponing the end of the private deleveraging process. The model finds that deleveraging-friendly fiscal consolidations that are more gradual in nature minimize these losses.
Resources Economics. Jon M.Conrad. Chapter 1
1. The document discusses renewable and nonrenewable natural resources. It defines renewable resources as those that regenerate on a relevant economic time scale and nonrenewable resources as those that do not regenerate.
2. It explains the difference between resource economics, which studies the allocation of natural resources, and environmental economics, which focuses on conservation.
3. The optimal harvesting of renewable resources over time presents a dynamic optimization problem to maximize economic value over periods. Nonrenewable resources depletion also impacts the accumulation of waste pollution over time.
Froyen21
1) Keynes argued that consumption is determined by disposable income and other factors. Empirical studies from 1929-1941 estimated the consumption function as C = 26.5 + 0.75Yd, supporting Keynes' theory.
2) Kuznets' data from 1869-1938 showed average propensity to consume remained constant despite rising income, contradicting Keynesian predictions.
3) The life cycle and permanent income hypotheses were proposed as alternative explanations for consumption, arguing it depends on lifetime rather than current income. They better explained post-war consumption levels.
1. This question is on the application of the Binomial option
1. This question is on the application of the Binomial option
pricing model.
PKZ stock is currently trading at 100. Over three-months it will either
go up by 6% or down by 5%. Interest rates are zero.
a. [25 marks] Using a two period binomial model to construct a delta-
hedged portfolio, price a six month European call option on PKZ
stock with a strike price of £105.
b. [3 Marks] Using your answer from the first part, together with the
put-call parity, price a put option on the same stock with same
strike and expiry.
COMP0041 SEE NEXT PAGE
2
2. This question is on the Binomial method in the limit δt → 0.
[40 Marks] The binomial model for pricing options leads to the for-
mula
V (S,t) = e−rδt [qV (US,t + δt) + (1 − q) V (DS,t + δt)]
where
U = eσ
√
δt, D = e−σ
√
δt, q =
erδt −D
U −D
.
V (S,t) is the option value, t is the time, S is the spot price, σ is volatil-
ity and r is the risk-free rate.
By carefully expanding U,D,q as Taylor series in δt or
√
δt (as appro-
priate) and then expanding V (US,t + δt) and V (DS,t + δt) as Taylor
series in both their arguments, deduce that to O (δt) ,
∂V
∂t
+
1
2
σ2S2
∂2V
∂S2
+ rS
∂V
∂S
− rV = 0.
COMP0041 SEE NEXT PAGE
3
3. This question is on probability and Monte Carlo
a. Consider theprobabilitydensity function p (x) fora randomvariable
X given by
p (x) =
{
µ exp (−µx) x ≥ 0
0 x < 0
where µ (> 0) is a constant.
i. [15 Marks] Show that for this probability density function
E
[
eθX
]
=
(
1 −
θ
µ
)−1
Hint: You may assume µ > θ in obtaining this result.
ii. [20 Marks] By expanding
(
1 −
θ
µ
)−1
as a Taylor series, show
that
E [xn] =
n!
µn
, n = 0, 1, 2, ....
iii. [15 Marks] Hence calculate the skew and kurtosis for X.
COMP0041 CONTINUED ON NEXT PAGE
4
b. [32 Marks] An Exchange Option gives the holder the right to
exchange one asset for another. The discounted payoff for this
contract V is
V = e−rT max (S1 (T) −S2 (T) , 0) .
The option price is then given by θ = E [V ] where
Si (t) = Si (0) e
(r−12σ
2
i )t+σiφi
√
t
for i = 1, 2, and φi ∼ N (0, 1) with correlation coeffi cient ρ.
Youmayassumethatauniformrandomnumbergenerator isavail-
able. Use a Cholesky factorisation method to show(
φ1
φ2
)
=
(
1 0
ρ
√
1 −ρ2
)(
x1
x2
)
,
where
(
x1
x2
)
is a vector of independent N (0, 1) variables and
has the same distribution as
(
φ1
φ2
)
.
Give a Monte Carlo simulation algorithm that makes use of anti-
thetic variates for the estimation of θ.
COMP0041 SEE NEXT PAGE
5
4. This question is on finite differences
a. [30 Marks] Consider a forward difference operator, ∆, such that
∆V (S) = V (S + h) −V (S) , (4.1)
where h is an infinitessimal. By introducing the operators
D ≡
∂
∂S
; D2 ≡
∂2
∂S2
show that
∆ ≡ ehD −1 (4.2)
where 1 is the identity operator. Hint: start by doing a Taylor
expansion on V (S + h) .
By rearranging (4.2) show that
D =
1
h
(
∆ −
∆2
2
+
∆3
3
−
∆4
4
+ O
(
∆5
))
.
Hence obtain the second order approximation for
∂V
...
1. This question is on the application of the Binomial option
1. This question is on the application of the Binomial option
pricing model.
PKZ stock is currently trading at 100. Over three-months it will either
go up by 6% or down by 5%. Interest rates are zero.
a. [25 marks] Using a two period binomial model to construct a delta-
hedged portfolio, price a six month European call option on PKZ
stock with a strike price of £105.
b. [3 Marks] Using your answer from the first part, together with the
put-call parity, price a put option on the same stock with same
strike and expiry.
COMP0041 SEE NEXT PAGE
2
2. This question is on the Binomial method in the limit δt → 0.
[40 Marks] The binomial model for pricing options leads to the for-
mula
V (S,t) = e−rδt [qV (US,t + δt) + (1 − q) V (DS,t + δt)]
where
U = eσ
√
δt, D = e−σ
√
δt, q =
erδt −D
U −D
.
V (S,t) is the option value, t is the time, S is the spot price, σ is volatil-
ity and r is the risk-free rate.
By carefully expanding U,D,q as Taylor series in δt or
√
δt (as appro-
priate) and then expanding V (US,t + δt) and V (DS,t + δt) as Taylor
series in both their arguments, deduce that to O (δt) ,
∂V
∂t
+
1
2
σ2S2
∂2V
∂S2
+ rS
∂V
∂S
− rV = 0.
COMP0041 SEE NEXT PAGE
3
3. This question is on probability and Monte Carlo
a. Consider theprobabilitydensity function p (x) fora randomvariable
X given by
p (x) =
{
µ exp (−µx) x ≥ 0
0 x < 0
where µ (> 0) is a constant.
i. [15 Marks] Show that for this probability density function
E
[
eθX
]
=
(
1 −
θ
µ
)−1
Hint: You may assume µ > θ in obtaining this result.
ii. [20 Marks] By expanding
(
1 −
θ
µ
)−1
as a Taylor series, show
that
E [xn] =
n!
µn
, n = 0, 1, 2, ....
iii. [15 Marks] Hence calculate the skew and kurtosis for X.
COMP0041 CONTINUED ON NEXT PAGE
4
b. [32 Marks] An Exchange Option gives the holder the right to
exchange one asset for another. The discounted payoff for this
contract V is
V = e−rT max (S1 (T) −S2 (T) , 0) .
The option price is then given by θ = E [V ] where
Si (t) = Si (0) e
(r−12σ
2
i )t+σiφi
√
t
for i = 1, 2, and φi ∼ N (0, 1) with correlation coeffi cient ρ.
Youmayassumethatauniformrandomnumbergenerator isavail-
able. Use a Cholesky factorisation method to show(
φ1
φ2
)
=
(
1 0
ρ
√
1 −ρ2
)(
x1
x2
)
,
where
(
x1
x2
)
is a vector of independent N (0, 1) variables and
has the same distribution as
(
φ1
φ2
)
.
Give a Monte Carlo simulation algorithm that makes use of anti-
thetic variates for the estimation of θ.
COMP0041 SEE NEXT PAGE
5
4. This question is on finite differences
a. [30 Marks] Consider a forward difference operator, ∆, such that
∆V (S) = V (S + h) −V (S) , (4.1)
where h is an infinitessimal. By introducing the operators
D ≡
∂
∂S
; D2 ≡
∂2
∂S2
show that
∆ ≡ ehD −1 (4.2)
where 1 is the identity operator. Hint: start by doing a Taylor
expansion on V (S + h) .
By rearranging (4.2) show that
D =
1
h
(
∆ −
∆2
2
+
∆3
3
−
∆4
4
+ O
(
∆5
))
.
Hence obtain the second order approximation for
∂V
...
Exam doaa
This document contains questions about system dynamics modeling concepts like stocks, flows, causal loop diagrams, stock and flow diagrams, and reference modes. It asks the reader to identify which variables are stocks and flows in various contexts, draw causal loop and stock and flow diagrams to represent different systems, and build a simple stock and flow model to represent population growth and health sector development over time.
Keynesian Income Determination
This document discusses key concepts in the simple Keynesian theory of income determination, including:
1. Endogenous and exogenous variables - endogenous variables are explained by the theory, while exogenous variables are taken as given. Real output and consumption are initially treated as endogenous.
2. The consumption function - consumption is explained as autonomous consumption plus the marginal propensity to consume times disposable income. This can be shown graphically.
3. Induced saving and the marginal propensity to save - as disposable income increases, the remaining portion after consumption is induced saving. Actual U.S. data from 1929-2001 is presented to illustrate these concepts.
Macro prudential policy(2)
This document summarizes a research paper that develops a dynamic general equilibrium model to analyze systemic risk in the banking sector. The key aspects of the model are that it includes banks that engage in maturity transformation by issuing non-state contingent debt, and the banks are exposed to risks in capital markets that can affect their solvency. The model shows that individual banks in a competitive system will take on excessive systemic risk due to pecuniary externalities, leading to a higher crisis probability than the socially optimal level. The document then discusses using prompt corrective action (PCA) policies to reduce crisis risk by strengthening bank capital requirements.
Note classical
This lecture note introduces classical macroeconomic theory, which examines the linkages between interest rates, money, output, and inflation. It will first cover classical monetary theory, which assumes money is neutral and its supply only determines prices. It then presents a basic classical model of the real economy with aggregate supply and demand. Equilibrium output and interest rates are determined by the intersection of these curves. Money enters by facilitating transactions but does not affect real variables. Money demand depends on nominal income and interest rates, determining equilibrium money holdings and the price level.
Macrodynamics of Debt-Financed Investment-Led Growth with Interest Rate Rules
This document provides an overview of a macroeconomic model that examines debt-financed investment-led growth. Some key points:
- The model explores whether financial factors can provide stability to an otherwise unstable demand-constrained economy, and whether they can generate growth cycles.
- Investment is determined by a post-Keynesian accelerator function where the sensitivity of investment to capacity utilization depends on financial factors like the risk of default and interest rates.
- Debt dynamics are modeled, where borrowing, repayment, and the debt stock over time are determined. Financial fragility and creditworthiness indicators are also developed.
- Monetary policy follows a Taylor-type rule where the central bank adjusts interest rates in response
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Optimal Policy Response to Booms and Busts in Credit and Asset Prices
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Criteria Mark 1. Professional Presentation 1 · Well organiz.docx
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When Fiscal consolidation meets private deleveraging
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