115 年 國立臺灣大學財務金融研究所甲組《經濟分析》

Suppose the Phillips curve is given by

$u=u^*-\alpha(\pi-E\pi),$

where $u^*=5\%$, $\alpha=0.5$, and expected inflation equals last period’s inflation ($u$: unemployment rate, $\pi$: inflation). If actual inflation turns out to be 2 percentage points higher than expected inflation, what happens to unemployment?

(A) It increases by 1 percentage point.
(B) It decreases by 1 percentage point.
(C) It decreases by 2 percentage points.
(D) It remains unchanged.
(E) Cannot be determined.

Assume a central bank minimizes a loss function over inflation and unemployment. If inflation expectations are adaptive, which outcome is most likely after a surprise monetary expansion?

(A) Permanent reduction in unemployment
(B) Permanent increase in inflation with no unemployment effect
(C) No change in inflation or unemployment
(D) Immediate return to natural unemployment
(E) Temporary reduction in unemployment

Assume the following closed-economy IS–LM model:

$Y=C+I+G$

$C=50+0.75(Y-T)$

$I=180-8r$

$\left(\frac{M}{P}\right)^d=Y-40r$

$G=250,\quad T=200,\quad M=4000,\quad P=4.$

Here $Y$ denotes aggregate output (income), $C$ denotes aggregate consumption, $I$ denotes aggregate investment, $G$ denotes government spending, $T$ denotes lump-sum taxes, $r$ denotes the real interest rate, $M$ denotes the nominal money supply, and $P$ denotes the price level. The term $M/P$ represents the real money supply.

An increase in government spending shifts:

(A) the IS curve to the right
(B) the IS curve to the left
(C) the LM curve to the right
(D) the LM curve to the left
(E) both the IS and LM curves

Two economies are identical except that Economy A has a higher marginal propensity to consume (MPC) than Economy B. Both experience the same monetary expansion. Compared to Economy A, Economy B will experience

(A) A larger increase in output
(B) A smaller increase in output
(C) A larger decrease in interest rates
(D) No change in output
(E) A higher inflation rate

Which of the following changes would most likely reduce the natural rate of unemployment?

(A) Stronger employment protection laws
(B) Higher unemployment benefits
(C) Improved job-matching technology
(D) An increase in minimum wages
(E) A contractionary monetary policy

When a central bank conducts open market sales of government bonds, which outcome is most likely in the short run?

(A) Higher output and lower interest rates
(B) Lower output and higher interest rates
(C) Higher inflation and higher output
(D) Lower inflation and higher output
(E) No change in macroeconomic variables

If Taiwan imposes a quota on imports of Japanese cars due to claims of unfair trade practices, and Japanese demand for Taiwanese exports increases at the same time, then, in the long run ____, everything else held constant:

(A) the Japanese yen will depreciate relative to the New Taiwan dollar
(B) there will be no effect on the Japanese yen relative to the New Taiwan dollar
(C) the Japanese yen will appreciate relative to the New Taiwan dollar
(D) the Japanese yen will either appreciate, depreciate or remain constant against the New Taiwan dollar
(E) None of the above

The Bruce the Bank Manager can reduce interest rate risk by ____ the duration of the bank’s assets to increase their rate sensitivity or, alternatively, ____ the duration of the bank’s liabilities:

(A) shortening; shortening
(B) lengthening; lengthening
(C) lengthening; shortening
(D) shortening; lengthening
(E) None of the above

For which of the following is the change in reserves necessarily different from the change in the monetary base?

(A) Open market purchases from an individual who cashes the cheque
(B) Open market purchases from a bank
(C) Open market purchases from an individual who deposits the cheque in a bank
(D) Open market sale to a bank
(E) None of the above.

According to the expectations theory of the term structure

(A) when the yield curve is downward sloping, short-term interest rates are expected to remain relatively stable in the future.
(B) yield curves should be equally likely to slope downward as slope upward.
(C) investors have strong preferences for short-term relative to long-term bonds, explaining why yield curves typically slope upward.
(D) when the yield curve is steeply upward sloping, short-term interest rates are expected to remain relatively stable in the future.
(E) None of the above.

Suppose the central bank unexpectedly allows a sharp appreciation of the NTD in period $t+1$. Which of the following statements are correct?（答案可能不只一項）

(A) The realized domestic-currency return on pension savings is lower than expected.
(B) The marginal utility of consumption in old age increases relative to expectations.
(C) The pension system is unaffected because U.S. Treasuries are risk-free.
(D) The old-age budget constraint becomes tighter in NTD terms.
(E) Retirees experience a positive wealth effect.

Suppose Taiwan delays exchange-rate adjustment for many years, allowing pension funds to accumulate large USD asset positions. When appreciation finally occurs, which statements are correct?（答案可能不只一項）

(A) Retirees at the time of appreciation bear significant welfare losses.
(B) Young workers can fully offset losses through labor supply adjustment.
(C) Losses arise because realized returns differ from expected returns.
(D) Exporters are the primary losers in the OLG sense.
(E) The policy redistributes welfare across cohorts.

請考慮一個沒有不確定性的兩期消費模型（$t=1,2$）。各期消費分別為 $c_1,c_2>0$。A 的偏好可以用兩期加總效用代表：

$U(c_1,c_2)=u(c_1)+\beta u(c_2),\quad 0<\beta<1.$

其中單期效用函數為：

$u(c)=\frac{c^\alpha-1}{\alpha},$

$c$ 為該期消費，參數 $\alpha\ne 0$。

A 在兩期的所得分別為 $y_1,y_2>0$。兩期間 A 可以固定利率 $r$ 借貸，跨期預算限制為：

$c_1+\frac{c_2}{1+r}=y_1+\frac{y_2}{1+r},\quad r>-1.$

(a) 請推導 A 在選擇兩期消費以極大化效用之下的跨期替代彈性，其定義為：

$\frac{\partial\ln(c_2^*/c_1^*)}{\partial\ln(1+r)}.$

其中 $c_1^*,c_2^*$ 為讓兩期效用最大的消費組合。

（1）____

提示：可從效用極大化問題的一階條件計算此彈性。

(b) 假設 $\alpha=-1$、$\beta=5/22$、$r=0.1$、$y_1=36$、$y_2=66$。請求解 $(c_1^*,c_2^*)$。

（2）____

填答範例：若解出 $c_1^*=55,c_2^*=65$，則填寫答案為 $(55,65)$。

一位買方準備購買一間房屋。房屋的真實市場價值可能為 $\theta\in\{0,1,2\}$，分別代表低價、中價與高價房屋，且三種狀態發生機率皆為 $1/3$。

買方無法直接觀察 $\theta$。買方期望效用函數為：

$u(a,\theta)=-(a-\theta)^2,$

其中 $a\in\{0,1,2\}$，代表買方的出價。此效用函數設定代表出價低於或高於真實市場價值都會帶來損失。例如出價過高容易成交但買貴了，出價過低可能無法成交而錯失機會。

(a) 假設買方選擇出價以極大化期望效用，請根據先驗機率（即三種狀態發生機率皆為 $1/3$）求解買方的最適出價。

（3）____

(b) 假設房仲可觀察到 $\theta$，並給買方建議 $m\in\{L,H\}$。$L$ 代表房屋真實價值偏低，$H$ 代表真實價值偏高。假設房仲建議策略為：$\theta=0$ 時給 $L$；$\theta=1,2$ 時給 $H$。

另假設買方在不同的出價之下期望效用相同時，將選擇其中最高的出價。請求買方在得到建議為 $L$ 與 $H$ 之下的最適出價 $(a^L,a^H)$ 為何？

（4）____

填答範例：若解出 $a^L=1,a^H=2$，則填寫答案為 $(1,2)$。

(c) 假設房仲偏好買方的出價相較於真實價值略高一級，反映其希望買方「較積極出價」以提高成交機率與佣金收入的想法。房仲期望效用函數為：

$v(a,\theta)=-(a-(\theta+1))^2.$

假設買方依 (b) 小題所算出的最適策略行動。在給定房仲觀察到 $\theta=1$ 的狀況下，請計算房仲依照 (b) 小題的設定給出 $H$ 建議的期望效用 $v^H$，以及背離 (b) 小題的設定給出 $L$ 建議的期望效用 $v^L$。

（5）____

填答範例：若解出 $v^H=0,v^L=-1$，則填寫答案為 $(0,-1)$。

提示：在本題設定下，不論真實的 $\theta$ 值為何，只要買方依照 (b) 小題答案的策略出價，房仲都不會想要背離 (b) 小題所給的策略。因此本小題答案中 $v^L$ 將小於或等於 $v^H$，代表房仲背離原先策略的效用不會比較高。

(d) 請計算有房仲時買方的預期效用較無房仲時增加或減少多少？

（6）____

填答範例：若無房仲時買方效用期望值為 $-2$，有房仲時為 $-1$，代表有房仲時效用期望值增加 $1$，則填寫答案為 $1$。若答案為負值則代表效用減少。

假設一債券持有人的期望效用函數為：

$u(x)=\frac{x^\alpha-1}{\alpha},$

其中 $x$ 為一有違約風險的零息債券到期時支付的金額，參數 $\alpha\ne 0$。

(a) 定義 Arrow-Pratt 相對風險趨避係數：

$RRA=-\frac{xu''(x)}{u'(x)}.$

請根據債券持有人的效用函數推導 $RRA$。

（7）____

提示：答案為 $\alpha$ 的函數。

(b) 定義確定等值（certainty equivalent，$CE$）須滿足下式：

$u(CE)=E(u(x)).$

請根據債券持有人的效用函數推導 $CE$，並將 $CE$ 寫為 $E(x)$ 與 $RRA$ 的函數。

（8）____

(c) 假設該債券到期時若沒有違約（機率為 $0.75$）將支付債券持有人 $100$ 元，若違約（機率為 $0.25$）僅能支付 $50$ 元。

債券持有人考慮購買由銀行推出的信用違約交換（credit default swap，簡稱 CDS）。CDS 在債券違約狀態下將支付投資人 $50$ 元，在未違約狀態下支付 $0$。另假設 $\alpha=-1$，無風險利率為 $0$。請計算債券持有人對於 CDS 的最高願付價格。

（9）____

(d) 承上題，假設銀行為風險中立，CDS 市場為完全競爭。此外，銀行經辦 CDS 業務時除了可能給投資人的支出外，每份合約尚有固定成本 $5$ 元。請求出市場均衡時的 CDS 價格。

（10）____