消費者偏好與效用概念

內容綱要 無異曲線(Indifference Curves ) 邊際替代率(The Marginal Rate of Substitution) 效用函數(The Utility Function) 邊際效用(Marginal Utility) 特殊函數型態(Some Special Functional Forms)

無異曲線 定義: 一條無異曲線，乃是兩種物品產生同樣總效用水準的所有不同組合的軌跡。 定義: 一條無異曲線，乃是兩種物品產生同樣總效用水準的所有不同組合的軌跡。 定義:無異曲線圖(indifference curve map)是由許多條形狀相同但偏好程度不同的無異曲線所組成的。  

無異曲線圖的特性 完整性(Completeness) 每一個消費組何只能位在一條無異曲線上。 單調性(Monotonicity) 無異曲線是負斜率。 無異曲線不是一條厚的線。

y 單調性 • A x

y 單調性 偏好優於 A • A x

y 單調性 偏好優於A • A 偏好劣於A x

y 單調性 偏好優於A • A 偏好劣於A IC1 x

y 無異曲線不厚 B • • A IC1 x

無異曲線圖的特性 3. 遞移性(Transitivity) 任兩條無異曲線不相交。 4. 愈往右上方的無異曲線，其效用愈高 5. 平均優於臨界(Averages preferred to extremes) 無異曲線凸向原點。

y 無異曲線不能相交 假設一位消費者對A和C有相同偏好。 假設B優於A。 IC1 • B • A C • x

• • • 無異曲線不能相交 y 包含B和C的 IC是不可能的情形。 為什麼? 因為，根據IC的定義， 消費者是： A & C 一樣好。 因此A & B 也一樣好(遞移性) 。 => 矛盾。 IC2 IC1 • B • A C • x

y 平均優於臨界 A • • IC1 B x

y 平均優於臨界 A • (.5A, .5B) • C • IC1 B x

y 平均優於臨界 A • (.5A, .5B) • C IC2 • IC1 B x

y 平均優於臨界 A A & B 一樣好。 C 優於A。 C 優於B。 • (.5A, .5B) • C IC2 • IC1 B x

邊際替代率 邊際替代率的定義有許多方法 定義 1: 為了維持同一效用水準，當增加一個單位 X 物品的消費時，可以放棄的 Y 物品數量，就是以 X 替代 Y 的邊際替代率(MRS)。

邊際替代率 定義 2: 無異曲線的斜率是負的： (偏好固定)

邊際替代率遞減 無異曲線呈現邊際替代率遞減： 你擁有更多的財貨 x ，你所願意放棄的財貨y會愈少。 無異曲線 愈靠近橫軸愈平坦。 愈靠近縱軸愈平坦。

例子: 邊際替代率遞減

效用函數 Definition: The utility function measures the level of satisfaction that a consumer receives from any basket of goods.

The Utility Function The utility function assigns a number to each basket More preferred baskets get a higher number than less preferred baskets. Utility is an ordinal concept The precise magnitude of the number that the function assigns has no significance.

Ordinal and Cardinal Ranking Ordinal ranking gives information about the order in which a consumer ranks baskets E.g. a consumer may prefer A to B, but we cannot know how much more she likes A to B Cardinal ranking gives information about the intensity of a consumer’s preferences. We can measure the strength of a consumer’s preference for A over B.

Example: Consider the result of an exam An ordinal ranking lists the students in order of their performance E.g., Harry did best, Sean did second best, Betty did third best, and so on. A cardinal ranking gives the marks of the exam, based on an absolute marking standard E.g. Harry got 90, Sean got 85, Betty got 80, and so on.

The Utility Function Implications of an ordinal utility function: Difference in magnitudes of utility have no interpretation per se Utility is not comparable across individuals Any transformation of a utility function that preserves the original ranking of bundles is an equally good representation of preferences. eg. U = xy U = xy + 2 U = 2xy all represent the same preferences.

y Example: Utility and a single indifference curve 5 2 10 = xy 2 5 x

y Example: Utility and a single indifference curve 5 Preference direction 20 = xy 2 10 = xy 2 5 x

Marginal Utility Definition: The marginal utility of good x is the additional utility that the consumer gets from consuming a little more of x MUx = dU dx It is is the slope of the utility function with respect to x. It assumes that the consumption of all other goods in consumer’s basket remain constant.

Diminishing Marginal Utility Definition: The principle of diminishing marginal utility states that the marginal utility of a good falls as consumption of that good increases. Note: A positive marginal utility implies monotonicity.

Example: Relative Income and Life Satisfaction (within nations) Relative Income Percent > “Satisfied” Lowest quartile 70 Second quartile 78 Third quartile 82 Highest quartile 85 Source: Hirshleifer, Jack and D. Hirshleifer, Price Theory and Applications. Sixth Edition. Prentice Hall: Upper Saddle River, New Jersey. 1998.

Marginal Utility and the Marginal Rate of Substitution We can express the MRS for any basket as a ratio of the marginal utilities of the goods in that basket Suppose the consumer changes the level of consumption of x and y. Using differentials: dU = MUx . dx + MUy . dy Along a particular indifference curve, dU = 0, so: 0 = MUx . dx + MUy . dy

Marginal Utility and the Marginal Rate of Substitution Solving for dy/dx: dy = _ MUx dx MUy By definition, MRSx,y is the negative of the slope of the indifference curve: MRSx,y = MUx MUy

Marginal Utility and the Marginal Rate of Substitution Diminishing marginal utility implies the indifference curves are convex to the origin (implies averages preferred to extremes)

Example: U= (xy)0.5 MUx=y0.5/2x0.5 MUy=x0.5/2y0.5 Marginal utility is positive for both goods: => Monotonicity satisfied Diminishing marginal utility for both goods => Averages preferred to extremes Marginal rate of substitution: MRSx,y = MUx = y MUy x Indifference curves do not intersect the axes

y Example: Graphing Indifference Curves IC1 x

y Example: Graphing Indifference Curves 偏好方向 IC2 IC1 x

特殊函數型態 Cobb-Douglas (“標準例子”) U = Axy 這裡:  +  = 1; A, , 正的常數 特性: MUx = Ax-1y MUy = Axy-1 MRSx,y = y x

y 例子: Cobb-Douglas IC1 x

y 例子: Cobb-Douglas 偏好方向 IC2 IC1 x

特殊函數型態 2. 完全替代: U = ax + by 這裡: a,b 是正的常數 特性: MUx = a MUy = b MRSx,y = a (固定 MRS) b

例子: 完全替代 (鮮奶油 與 人工奶油) 鮮奶油 IC1 人工奶油

例子: 完全替代 (鮮奶油 與 人工奶油) 鮮奶油 IC1 IC2 人工奶油

例子: 完全替代 (鮮奶油 與 人工奶油) 鮮奶油 斜率 = -a/b IC1 IC2 IC3 人工奶油

特殊函數型態 3. 完全互補: U = min {x/a, y/b} 這裡: A,B 是正的常數 特性: MUx = A or 0 MUy = B or 0 MRSx,y = 0 or  or 無法認定

螺栓 例子: 完全互補 (螺帽 與 螺栓) IC1 螺帽

螺栓 例子: 完全互補 (螺帽 與 螺栓) IC2 IC1 螺帽

特殊函數型態 4. 準線性效用函數(Quasi-Linear Utility Functions): U = aV(x) + by 這裡: a,b 是正的常數, 而且 v(0) = 0 特性: MUx = av’(x) MUy = b MRSx,y = av’(x) (僅受x影響) b

y 例子: 準線性偏好(飲料的消費) IC1 • 飲料

y 例子: 準線性偏好(飲料的消費) IC2 IC在相同的飲料數量下，邊際替代率相同 IC1 • • 飲料

特殊函數型態 4. 中性偏好(neutral preference): U(x, y) =V(x) 這裡: v(0) = 0 特性: 或 U(x, y) =V(y) 這裡: v(0) = 0 特性: MUx = 0, MUy＞0 MUy = 0, MUx＞0

儲蓄 例子: 中性偏好 (守財奴，儲蓄 與 消費) IC2 IC1 消費

肉 例子: 中性偏好 (素食者， 蔬菜 與 肉) IC1 IC2 蔬菜