1. **Problem Statement:** We want to find the effects of a decrease in government spending $G$ on all endogenous variables $Y, C, I, r, W, P, N^S, N^D$ by differentiating each with respect to $G$ using the given system of equations. 2. **Given Equations and Conditions:** - $Y = C + I + G$ (1) - $C = C(Y_D)$ with $Y_D = Y - T$ - $I = I(r)$ - $\frac{M}{P} = L(Y, r)$ (2) - $Y = F(N, K)$ (3) - $\Phi(N^D) = \frac{W}{P}$ (4) - $\Psi(N^S) = \frac{W}{P}$ (5) - $N^D = N^S$ (6) With: - $C_{Y-T} > 0$, $I_r < 0$, $P > 0$, $L_Y > 0$, $L_r < 0$, $\Phi_{N^D} < 0$, $\Psi_{N^S} > 0$ 3. **Step 1: Solve for $dN$ using eq (6):** From $N^D = N^S$, differentiate: $$dN^D = dN^S$$ Using implicit differentiation on (4) and (5): $$d\left(\frac{W}{P}\right) = \Phi_{N^D} dN^D = \Psi_{N^S} dN^S$$ Since $dN^D = dN^S = dN$, we get: $$\Phi_{N^D} dN = \Psi_{N^S} dN \implies (\Phi_{N^D} - \Psi_{N^S}) dN = 0$$ Because $\Phi_{N^D} < 0$ and $\Psi_{N^S} > 0$, $\Phi_{N^D} - \Psi_{N^S} \neq 0$, so: $$dN = 0$$ 4. **Step 2: Solve for $dY$ using eq (3):** From $Y = F(N, K)$, with $K$ constant: $$dY = F_N dN = F_N \times 0 = 0$$ 5. **Step 3: Solve for $dr$ using eq (1):** Differentiate $Y = C + I + G$: $$dY = dC + dI + dG$$ But $dY = 0$ and $dG < 0$ (decrease in $G$), so: $$0 = dC + dI + dG$$ 6. **Step 4: Solve for $dI$ using investment function:** $I = I(r)$, so: $$dI = I_r dr$$ 7. **Step 5: Solve for $dC$ using consumption function:** $C = C(Y_D)$ with $Y_D = Y - T$, $T$ constant, so: $$dC = C_{Y-T} dY = C_{Y-T} \times 0 = 0$$ 8. **Step 3 continued:** Substitute $dC$ and $dI$ into eq (1) differential: $$0 = 0 + I_r dr + dG \implies I_r dr = -dG$$ Since $I_r < 0$, dividing both sides: $$dr = \frac{-dG}{I_r} > 0$$ Because $dG < 0$ and $I_r < 0$, $dr$ is positive. 9. **Step 6: Solve for $dP$ using eq (2):** From $\frac{M}{P} = L(Y, r)$, differentiate: $$-\frac{M}{P^2} dP = L_Y dY + L_r dr$$ Since $dY=0$: $$-\frac{M}{P^2} dP = L_r dr$$ Rearranged: $$dP = -\frac{P^2}{M} L_r dr$$ Given $L_r < 0$ and $dr > 0$, $dP > 0$. 10. **Step 7: Solve for $dW$ using eq (4) or (5):** From (4): $$\Phi_{N^D} dN^D = d\left(\frac{W}{P}\right) = \frac{dW}{P} - \frac{W}{P^2} dP$$ Since $dN=0$: $$0 = \frac{dW}{P} - \frac{W}{P^2} dP \implies dW = \frac{W}{P} dP$$ Since $dP > 0$, $dW > 0$. **Summary of effects of decrease in $G$:** - $dN = 0$ - $dY = 0$ - $dr > 0$ - $dI = I_r dr < 0$ - $dC = 0$ - $dP > 0$ - $dW > 0$ Thus, a decrease in $G$ raises the interest rate $r$, lowers investment $I$, leaves output $Y$ and consumption $C$ unchanged, and increases price level $P$ and real wage $W$.