115年 - 115 國立臺北大學_碩士班一般入學考試試題_經濟學系：總體經濟學#147107

(1)Let \( k_t = K_t / L_t \) denote capital per worker. Derive the first-order difference equation governing the evolution of capital per worker, that is, \( k_{t+1} = g(k_t) \), for this economy. (5%)

(2) Continuing from part 1, let \( y{t}, c{t} \), and \( g_{t} \) denote output per worker, consumption per worker, and government purchases per worker, respectively. In the long run, as \( t \to \infty \), derive the steady-state values of capital per worker \( (k^{}) \), output per worker \( (y^{}) \), consumption per worker \( (c^{}) \), and government purchases per worker \( (g^{}) \), expressed in terms of the exogenous parameters. (5%)

(3) Show mathematically that, under the Golden Rule, the exogenous saving rate must equal one of the other exogenous parameters in this system. Identify this parameter and explain the economic interpretation of the result. (Hint: The Golden Rule maximizes steady-state consumption per worker, which depends on capital accumulation driven by saving out of disposable income.) (10%)

(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 $(y1, y2, \ell) \subset \mathbb{R}_+^3$."