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Distinguish between: Total utility, Marginal utility and The law of Diminishing Marginal utility.

Economics

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6 parts
i

88 utils88 \text{ utils}

ii

2 Pizzas and 4 Milkshakes2 \text{ Pizzas and } 4 \text{ Milkshakes}

iii

140 utils140 \text{ utils}

iv

MUpizzaPpizza=MUmilkshakePmilkshake=12\frac{MU_{pizza}}{P_{pizza}} = \frac{MU_{milkshake}}{P_{milkshake}} = 12

v

5 Pizzas and 3 Milkshakes5 \text{ Pizzas and } 3 \text{ Milkshakes}

vii

elastic\text{elastic}

Step-by-step solution

14 steps

Here are the solutions to question 12:

12. The above table represents a consumer buying two goods, Pizza and Milk shakes. The price of a Pizza is $2.00 per unit and a Milkshake is $1.00 per unit. The consumer's income is $8.00 and is all spent. Answer the following questions.

Part i· 2 steps

What is this consumer's total satisfaction from consuming 1 pizza and two milk shakes?

  1. Identify the marginal utility for each item.

    From the table:
    MU of 1st Pizza = 36 utils
    MU of 1st Milkshake = 30 utils
    MU of 2nd Milkshake = 22 utils

  2. Sum the marginal utilities to find total satisfaction.

    Total satisfaction = MU(1st Pizza) + MU(1st Milkshake) + MU(2nd Milkshake)
    Total satisfaction = 36+30+22=8836 + 30 + 22 = 88 utils.
    The total satisfaction is 88 utils\boxed{88 \text{ utils}}.

Part ii

2 Pizzas and 4 Milkshakes2 \text{ Pizzas and } 4 \text{ Milkshakes}

Show the best way for this consumer to spend his income in order to maximize his satisfaction.

To maximize satisfaction, the consumer must allocate their income such that the marginal utility per dollar spent is equal for both goods, and all income is spent. The condition is MUpizzaPpizza=MUmilkshakePmilkshake\frac{MU_{pizza}}{P_{pizza}} = \frac{MU_{milkshake}}{P_{milkshake}}.

  1. Calculate the marginal utility per dollar for each unit of Pizza (P_{pizza} = \2.00$).

    • Q=1: 362=18\frac{36}{2} = 18
    • Q=2: 242=12\frac{24}{2} = 12
    • Q=3: 202=10\frac{20}{2} = 10
    • Q=4: 182=9\frac{18}{2} = 9
    • Q=5: 162=8\frac{16}{2} = 8
    • Q=6: 102=5\frac{10}{2} = 5

  2. Calculate the marginal utility per dollar for each unit of Milkshake (P_{milkshake} = \1.00$).

    • Q=1: 301=30\frac{30}{1} = 30
    • Q=2: 221=22\frac{22}{1} = 22
    • Q=3: 161=16\frac{16}{1} = 16
    • Q=4: 121=12\frac{12}{1} = 12
    • Q=5: 101=10\frac{10}{1} = 10
    • Q=6: 41=4\frac{4}{1} = 4

  3. Allocate income by choosing units with the highest MU/P until all income ($8.00) is spent.

    RankGoodQMU/PCost per unitCumulative Cost
    1Milkshake130$1.00$1.00
    2Milkshake222$1.00$2.00
    3Pizza118$2.00$4.00
    4Milkshake316$1.00$5.00
    5Pizza212$2.00$7.00
    6Milkshake412$1.00$8.00

    At this point, the consumer has spent all 8.00.Theconsumershouldbuy8.00. The consumer should buy \boxed{2 \text{ Pizzas and } 4 \text{ Milkshakes}}$.

Part iii

140 utils140 \text{ utils}

What will be his total satisfaction in utils?

  1. Sum the marginal utilities for the optimal consumption bundle (2 Pizzas, 4 Milkshakes).

    MU for 2 Pizzas = MU(1st Pizza) + MU(2nd Pizza) = 36+24=6036 + 24 = 60 utils.
    MU for 4 Milkshakes = MU(1st Milkshake) + MU(2nd Milkshake) + MU(3rd Milkshake) + MU(4th Milkshake) = 30+22+16+12=8030 + 22 + 16 + 12 = 80 utils.

  2. Calculate the total satisfaction.

    Total satisfaction = 60+80=14060 + 80 = 140 utils.
    His total satisfaction will be 140 utils\boxed{140 \text{ utils}}.

Part iv

MUpizzaPpizza=MUmilkshakePmilkshake=12\frac{MU_{pizza}}{P_{pizza}} = \frac{MU_{milkshake}}{P_{milkshake}} = 12

State his equilibrium condition.

The equilibrium condition is when the marginal utility per dollar spent is equal for both goods.
For 2 Pizzas, MUpizzaPpizza=242=12\frac{MU_{pizza}}{P_{pizza}} = \frac{24}{2} = 12.
For 4 Milkshakes, MUmilkshakePmilkshake=121=12\frac{MU_{milkshake}}{P_{milkshake}} = \frac{12}{1} = 12.
The equilibrium condition is MUpizzaPpizza=MUmilkshakePmilkshake=12\boxed{\frac{MU_{pizza}}{P_{pizza}} = \frac{MU_{milkshake}}{P_{milkshake}} = 12}.

Part v

5 Pizzas and 3 Milkshakes5 \text{ Pizzas and } 3 \text{ Milkshakes}

The price of both goods now cos $1.00 each. How should this consumer spend his income?

New prices: P_{pizza} = \1.00,, P_{milkshake} = $1.00.Income. Income I = $8.00$.

  1. Calculate the marginal utility per dollar for each unit of Pizza (P_{pizza} = \1.00$).

    • Q=1: 361=36\frac{36}{1} = 36
    • Q=2: 241=24\frac{24}{1} = 24
    • Q=3: 201=20\frac{20}{1} = 20
    • Q=4: 181=18\frac{18}{1} = 18
    • Q=5: 161=16\frac{16}{1} = 16
    • Q=6: 101=10\frac{10}{1} = 10

  2. The marginal utility per dollar for Milkshake remains the same as P_{milkshake} = \1.00$.

    • Q=1: 30
    • Q=2: 22
    • Q=3: 16
    • Q=4: 12
    • Q=5: 10
    • Q=6: 4

  3. Allocate income by choosing units with the highest MU/P until all income ($8.00) is spent.

    We need to find a combination of Pizzas and Milkshakes such that Q_{pizza} \times \1.00 + Q_{milkshake} \times $1.00 = $8.00,whichmeans, which means Q_{pizza} + Q_{milkshake} = 8.Also,. Also, \frac{MU_{pizza}}{P_{pizza}} = \frac{MU_{milkshake}}{P_{milkshake}}.BycomparingtheMU/Pvalues,welookforacommonvaluewherethetotalquantityis8.If. By comparing the MU/P values, we look for a common value where the total quantity is 8. If \frac{MU}{P} = 16:ForPizza,MU/P=16correspondstoQ=5.ForMilkshake,MU/P=16correspondstoQ=3.Totalquantity=: For Pizza, MU/P = 16 corresponds to Q=5. For Milkshake, MU/P = 16 corresponds to Q=3. Total quantity = 5 + 3 = 8.Totalcost=. Total cost = (5 \times $1.00) + (3 \times $1.00) = $5.00 + $3.00 = $8.00.Theconsumershouldbuy. The consumer should buy \boxed{5 \text{ Pizzas and } 3 \text{ Milkshakes}}$.

Part vi

Sketch the Demand curve for Pizza.

From part (ii), when P_{pizza} = \2.00,thequantitydemandedis2Pizzas.Frompart(v),when, the quantity demanded is 2 Pizzas. From part (v), when P_{pizza} = $1.00,thequantitydemandedis5Pizzas.Thesetwopointsare, the quantity demanded is 5 Pizzas. These two points are (P=$2.00, Q=2)andand(P=$1.00, Q=5)$.

\begin{tikzpicture} \begin{axis}[ xlabel={Quantity of Pizza}, ylabel={Price of Pizza (\$)}, xmin=0, xmax=6, ymin=0, ymax=3, xtick={1,2,3,4,5,6}, ytick={0.5,1,1.5,2,2.5,3}, grid=both, grid style={line width=.1pt, draw=gray!10}, major grid style={line width=.2pt,draw=gray!50}, axis lines=left, legend pos=north east ] \addplot[ color=blue, mark=*, line width=1pt, domain=0:6 ] coordinates { (2,2) (5,1) }; \node[above right] at (axis cs:2,2) {A}; \node[below right] at (axis cs:5,1) {B}; \end{axis} \end{tikzpicture}

Part vii

elastic\text{elastic}

Is Pizza elastic or inelastic at the above price points? Explain.

We use the midpoint formula for price elasticity of demand (EdE_d) between the two points: (P_1, Q_1) = (\2.00, 2)andand(P_2, Q_2) = ($1.00, 5)$.

Ed=Q2−Q1(Q1+Q2)/2P2−P1(P1+P2)/2E_d = \frac{\frac{Q_2 - Q_1}{(Q_1 + Q_2)/2}}{\frac{P_2 - P_1}{(P_1 + P_2)/2}}

  1. Calculate the percentage change in quantity.

    ΔQ=5−2=3\Delta Q = 5 - 2 = 3.
    Average quantity = 2+52=3.5\frac{2 + 5}{2} = 3.5.
    Percentage change in quantity = 33.5=67\frac{3}{3.5} = \frac{6}{7}.

  2. Calculate the percentage change in price.

    \Delta P = \1.00 - $2.00 = -$1.00.Averageprice=. Average price = \frac{$2.00 + $1.00}{2} = $1.50.Percentagechangeinprice=. Percentage change in price = \frac{-1}{1.5} = -\frac{2}{3}$.

  3. Calculate the price elasticity of demand.

    Ed=67−23=67×(−32)=−1814=−97≈−1.286E_d = \frac{\frac{6}{7}}{-\frac{2}{3}} = \frac{6}{7} \times \left(-\frac{3}{2}\right) = -\frac{18}{14} = -\frac{9}{7} \approx -1.286

  4. Interpret the elasticity.

    The absolute value of the price elasticity of demand is ∣−9/7∣≈1.286|-9/7| \approx 1.286.
    Since ∣Ed∣>1|E_d| > 1, the demand for Pizza is elastic\boxed{\text{elastic}}.
    Explanation: A change in the price of Pizza leads to a proportionally larger change in the quantity demanded. In this case, a 50% decrease in price (from 2.00to2.00 to 1.00) led to a 150% increase in quantity demanded (from 2 to 5).

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