Answer to Question #170101 in Economics for Rabbi Afriyie

Question

Find (a) the reduced form. (b) the numerical value of 𝑌𝑒

, and (c) the effect on the multiplier if a 

proportional income tax (t) is incorporated into the model.

𝑌 = 𝐶 + 𝐼,

𝐶 = 𝐶𝑂 + 𝑏𝑌d

, 𝑇 = 𝑇𝑂 + 𝑡𝑌, 𝑌

𝑑 = 𝑌 − 𝑇

where 𝐼 = 𝐼𝑂 = 30, 𝐶𝑂 = 85, 𝑏 = 0.75, 𝑡 = 0.2, and 𝑇𝑂 = 20.

Question 2

Find the equilibrium price and quantity for the three substitute goods below.

𝑄𝑑1 = 23 − 5𝑃1 + 𝑃2 + 𝑃3

𝑄𝑠1 = −8 + 6𝑃1

𝑄𝑑2 = 15 + 𝑃1 − 3𝑃2 + 2𝑃3

𝑄𝑠2 = −11 + 3𝑃2

𝑄𝑑3 = 19 + 𝑃1 + 2𝑃2 − 4𝑃3

𝑄𝑠3 = −5 + 3𝑃3

Answer 1

Question 1

(a) The reduced form is:

Y = C + I = 85 + 0.75(Y - 20 - 0.2Y) + 30 = 100 + 0.6Y.

(b) the numerical value of 𝑌𝑒 is:

0.4Y = 100,

Y = 250.

(c) The effect on the multiplier if a

proportional income tax (t) is incorporated into the model is:

$m = \frac{1} {1 - b(1 - t)} = 1/0.4 = 2.5.$

Question 2

In equilibrium Qd = Qs, so:

-11P1 + P2 + P3 = -31,

𝑃1 − 6𝑃2 + 2𝑃3 = −26,

𝑃1 + 2𝑃2 − 7𝑃3 = −24.

If we substract the 2nd equation from the 3rd, then:

8P2 - 9P3 = 2,

P2 = 0.25 + 9/8P3.

If we add equation 1 and equation 2 multiplied with 11, then we receive:

-64P2 + 23P3 = -317,

P2 = 317/64 + 23/64P3,

0.25 + 9/8P3 = 317/64 + 23/64P3,

49P3 = 61,

P3 = 61/49 = 1.24,

P2 = 0.25 + 9/8×1.24 = 1.65,

P1 = -24 - 2×1.65 + 7×1.24 = -18.62.