武汉大学2003现代经济理论(含宏微观经济学461)(英文)

武汉大学

2003攻读硕士学位研究生入学考试试题

考试科目:现代经济理论(含宏、微观经济学) 科目代码：４６１

1. (25’) Consider an economic model in which the production function can be written as 10 1≤+=?ββa g L K Y a a p , here g is the government spending per capita, K, L, Y are aggregate capital, population and output respectively. The government spending is financed by an income tax, i.e., Y Lg G τ==. Capital accumulation equation is K sY K δ?=. Suppose the saving rate s ,income tax rate τ,depreciation rate δand population growth rate n are given exogenously, S+τ＜1.

(1)Show that if 1p βα+, then the capital stock per capita will converges

to a steady-state value k * which is positively correlated with income tax rate τ.

(2)Show that if 1=+βα, then the capital stock per capital will increase with a constant growth rate if δτ+n s f .

2. (25’) Consider an economy with identical long-lived individuals whose utility functions can be written as 10 0, ,]lnk lnc [t t 0p p f βγγβdt e t +∫?∞, her c t ,

k t are consumption and capital stock per capita in time t. Suppose the population growth rate is zero. The capital accumulation equation is 1a 0 ,p p k c Ak k a δ??=.

Solve the steady state values of consumption, capital stock and output per capita.

3. (25’) Consider an OLG model in which each individual can live two period. Individuals’ utility function can be written as 12t 1t lnc lnc ++=βt U , here c 1t and c 2t+1 are the consumptions in period t of young and old individuals respectively. Individual’s wealth constraints are

,1t t t w s c =+ )1(112+++=t t t r s c

Here w t , s t and r t+1 are wage, saving and interest rate respectively. Production function can be written as a

t t Ak y =. Suppose that the depreciation rate δ＝1 and population growth rate are zero. Assumption the government levies an output tax with a constant tax rate τ.

(1) Solve the steady state values of capital stock and output per capita.

(2) Show that the relationship between the tax income of the government and tax rate can be described as a Laffer Curve, i.e., as the tax rate increase, the tax income increases when the tax rate is small and decreases when it is large enough. Solve the tax rate with the maximum tax income.

4. (25’) If preferences are locally nonsatiated, and if (x *,y *,p )is an

equilibrium with transfers, then the allocation(x *,y *)is Pareto optimal.

5. (25’) If preference i ≥ is homothetic, assume that x i (p,w i ) is differentiable, then x i (p,w i ) satisfies the uncompensated law of demand property.

6. (25’) Suppose that production set Y is convex. Then every efficient production Y y ∈is a profit-maximizing production for some nonzero price vector 0≥p .

答案部分

武汉大学

2003攻读硕士学位研究生入学考试试题

考试科目:现代经济理论(含宏、微观经济学) 科目代码：４６１

1. Consider an economic model in which the production function can be written as 10 1≤+=?ββa g L K Y a a p , here g is the government spending per capita, K, L, Y are aggregate capital, population and output respectively. The government spending is financed by an income tax, i.e., Y Lg G τ==. Capital accumulation equation is K sY K δ?=. Suppose the saving rate s ,income tax rate τ,depreciation rate δand population growth rate n are given exogenously, S+τ＜1.

(1) Show that if 1p βα+, then the capital stock per capita will converges

to a steady-state value k * which is positively correlated with income tax rate τ.

(2)Show that if 1=+βα,

then the capital stock per capital will increase with a constant growth rate if δτ+n s f .

2. Consider an economy with identical long-lived individuals whose utility functions can be written as 10 0, ,]lnk lnc [t t 0p p f βγγβdt e t +∫?∞, her c t , k t are

consumption and capital stock per capita in time t. Suppose the population growth rate is zero. The capital accumulation equation is 1a 0 ,p p k c Ak k a δ??=.

Solve the steady state values of consumption, capital stock and output per capita.

3. Consider an OLG model in which each individual can live two period. Individuals’ utility function can be written as 12t 1t lnc lnc ++=βt U , here c 1t and c 2t+1 are the consumptions in period t of young and old individuals respectively. Individual’s wealth constraints are

,1t t t w s c =+ )1(112+++=t t t r s c

Here w t , s t and r t+1 are wage, saving and interest rate respectively. Production function can be written as a t t Ak y =. Suppose that the depreciation rate δ＝1 and population growth rate are zero. Assumption the government levies an output tax with a constant tax rate τ.

(1) Solve the steady state values of capital stock and output per capita.

(2) Show that the relationship between the tax income of the government and tax rate can be described as a Laffer Curve, i.e., as the tax rate increase, the tax income increases when the tax rate is small and decreases when it is large enough. Solve the tax rate with the maximum tax income.

4. If preferences are locally nonsatiated, and if (x *,y *,p )is

an equilibrium with transfers, then the allocation(x *,y *)is Pareto optimal.

5. If preference i ≥ is homothetic, assume that x i (p,w i ) is differentiable, then x i (p,w i ) satisfies the uncompensated law of demand property.

6. Suppose that production set Y is convex. Then every efficient production Y y ∈is a profit-maximizing production for some nonzero price vector 0≥p .