提问

问题如下：

In a simple economy with no foreign sector, the following equations apply:

• Consumption function: C = 2,500 + 0.80 × (Y – T)

• Investment function: I = 500 + 0.30 × Y – 25 × r

• Government spending: G = 1,000

• Tax function: T = –250 + 0.30 × Y

where

Y = aggregate income

r = real interest rate in percent

If the real interest rate is 3% and government spending increases to 2,000, the increase in aggregate income will be closest to:

选项：

A.5,000.00

B.7,143.00

C.5,845.00

解释：

B is correct. With no foreign sector, the GDP identity is: Y = C + I + G.

With substitution from the equations above:

Y = 2,500 + 0.80 × (Y – T) + 500 + 0.30 × Y – 25 × r + 1,000

= 2,500 + 0.80 × (Y + 250 – 0.30 × Y) + 500 + 0.30 × Y – 25 × r + 1,000

Y = 4,200 + 0.86 × Y – 25 × r

Y = 30,000 – 178.6 × r

At 3 percent, Y = 30,000 – 178.6 × 3 = 29,464

Alternatively

Y – 0.86Y = 4,200 – 25 × r

0.14Y = 4,200 – 25 × r

If government spending increased by 1,000 to 2,000, then

Y = 5,200 + 0.86 × Y – 25 × r

Y = 37,142 – 178.6 × r

which at 3 percent would be Y = 37,142 – 178.6 × 3 = 36,607

Representing an increase of 36,607 – 29,464 = 7,143

Alternatively, at 5,200:

Y – 0.86Y = 5,200 – 25 × r

0.14Y = 5,200 – 25 × r

The answer can also be calculated as:

ΔG/(1 – c) = 1,000/(1 – 0.86) = 7,143

where c is the change in consumption per unit change in Y (in this problem 0.80) minus the change in taxes per unit change in Y (–0.24) plus the change in investment per unit change in Y (0.30).

A is incorrect. It is calculated as 1,000/(1 – 0.80) = 5,000

C is incorrect. It substitutes 2,000 for Y in the right side of the following:

Y = 4,200 + 0.86 × Y – 25 × r

Y = 5,845

中文解析：

B正确。如果没有外国部门，GDP等式为:Y = C + I + G。

用上述方程代入:

Y = 2500 + 0.80 × (Y - T) + 500 + 0.30 × Y - 25 × r + 1000

= 2500 + 0.80 × (Y + 250 - 0.30 × Y) + 500 + 0.30 × Y - 25 × r + 1000

Y = 4200 + 0.86 × Y - 25 × r

Y = 30000 - 178.6 × r

在3%时，Y = 30,000 - 178.6 × 3 = 29,464

另外

Y - 0.86Y = 4200 - 25 × r

y = 4200 - 25 × r

如果政府支出增加1000到2000，那么

Y = 5200 + 0.86 × Y - 25 × r

Y = 37,142 - 178.6 × r

3%时Y = 37142 - 178.6 × 3 = 36607

增加了36,607 - 29,464 = 7,143

另外，在5 200:

Y = 5200 - 25 × r

0.14Y = 5200 - 25 × r

答案也可以计算为:

ΔG/(1 - c) = 1000 /(1 - 0.86) = 7143

其中c是每单位Y变化的消费变化量(本题为0.80)减去每单位Y变化的税收变化量(-0.24)加上每单位Y变化的投资变化量(0.30)

A不正确。计算结果为1000 /(1 - 0.80)= 5000

C选项不正确。它用2000代替下面右边的Y:

Y = 4200 + 0.86 × Y - 25 × r

Y = 5,845