Answer

Solution

1. 1

   With linear technology and perfect competition $P = W$. The consumer's real

2. 2

   wealth (measured in units of the good) is

3. 3
   $$\Omega = L + \frac{\bar M}{P},$$
4. 4

   where $L$ is labour supplied. Cobb--Douglas preferences imply

5. 5
   $$C = \tfrac{3}{4}\,\Omega, \qquad \tfrac{M}{P} = \tfrac{1}{4}\,\Omega.$$
6. 6

   Full employment means $L = 50$ and goods-market clearing $C = Y = L = 50$:

7. 7
   $$50 = \tfrac{3}{4}\!\left(50 + \tfrac{100}{P}\right) \;\Longleftrightarrow\; \tfrac{200}{3} - 50 = \tfrac{100}{P}\cdot \tfrac{3}{4}\cdot\tfrac{4}{3}.$$
8. 8

   Equivalently, the money-market condition

9. 9
   $$\frac{100}{P} = \tfrac{1}{4}\!\left(50 + \tfrac{100}{P}\right)$$
10. 10

    gives

11. 11
    $$\frac{400}{P} - \frac{100}{P} = 50 \;\Longrightarrow\; \frac{300}{P} = 50 \;\Longrightarrow\; P = 6.$$

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