(1) Derive the IS equation, the LM equation, and the AD equation for this economy. (5%)

(2) Suppose that the price level is fixed at P = 5 in the short run. Compute the short-run equilibrium values of output, the real interest rate, consumption, and investment, and illustrate the short-run equilibrium in an IS-LM-FE diagram. Then determine the long-run equilibrium values of output, the real interest rate, consumption, investment, and the price level, and illustrate the long-run equilibrium in the same diagram. Explain briefly how the economy moves from the short-run equilibrium to the long-run equilibrium. (10%)

(3) Suppose the government aims to restore the economy to its long-run equilibrium, where $\bar{Y} = 182.5$, while the price level remains fixed at P = 5. For each of the following policy scenarios, compute the required policy value separately and illustrate your answers in the IS-LM-FE diagram. (15%)a. With M = 600 and T = 40 unchanged, what level of government purchases (G) is required?b. With M = 600 and G = 30 unchanged, what level of taxes (T) is required?c. With T = 40 and G = 30 unchanged, what level of nominal money supply (M) is required?

(1) First consider the case where $\sigma = 0$. Derive the Euler equation (i.e., the optimality condition representing the trade-off between $c1$ and $c2$). Explain the economic intuition of this condition in words. (8%)

(2) Assume that the net interest rate r satisfies $\beta(1 + r) = 1$. What is the optimal saving $a_2$? Show your mathematical derivation. (6%)

(3) Suppose that $\beta(1 + r) = 1$ still holds. Now consider a small but positive value for $\sigma$. Under what condition on u do we get positive optimal savings $a_2 > 0$? (6%)

(1) Define \( \lambda \) as the Lagrangian multiplier of the financial constraint. Formulate the Lagrangian equation and derive the intertemporal Euler equation. Express the condition in terms of \( c_1, c_2 \) and \( \lambda \). Please ensure that your representation always implies \( \lambda \geq 0 \). (8%) (Hint: You can define other Lagrangian multipliers if necessary)

(2) Using your result from question 1, eliminate \( \lambda \) and rewrite the condition as a weak inequality. Why does the constrained household's optimal condition hold with strict inequality? Explain economic intuition in words. (8%) (Note: You will not receive points if you merely explain the mathematical equation in words)

(3)Given \( y_1 \) and \( \ell \), derive the condition on \( y_2 \) under which households are unconstrained or constrained. You must specify the threshold of \( y_2 \) (denoted as \( \bar{y}_2 \)) that separates constrained and unconstrained households as a function of \( (\beta, r, y_1, \ell) \) in the condition. Show your mathematical derivation. (8%)

(4) Determine whether the following statement is true or false. If true, please prove it; if false, please disprove or provide a counter-example. (6%)"If borrowing requires no interest payment (i.e., $r = 0$), the household always borrows to the limit with any combination of $(y1, y2, \ell) \subset \mathbb{R}_+^3$."

(1) Please yield the Lagrangian expression and the first conditions for a maximum. \( (10\%) \)

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