1= Assume an individual has 20 birr to spend on two products: X1 and X2 each with market price of 2(two) birr. If the total utility the consumer generates from the two products is as given on the table below,
1.1= Select all combinations of X1 and X2 that fulfill the necessary conditions of consumer equilibrium, and from them identify the consumer’s equilibrium purchase size (2pts).
Q 1 2 3 4 5 6 7 8 9 10 11
TUX1 16 30 41 51 60 68 75 81 86 89 90
TUX2 15 28 40 48 54 59 63 66 68 69 69
Combinations of X1 and X2 that fulfill the necessary conditions of consumer equilibrium:
Q1: X1=2, X2=18
Q2: X1=4, X2=16
Q3: X1=6, X2=14
Q4: X1=8, X2=12
Q5: X1=10, X2=10
Q6: X1=12, X2=8
Q7: X1=14, X2=6
Q8: X1=16, X2=4
Q9: X1=18, X2=2
The consumer’s equilibrium purchase size is X1=10 and X2=10.
To find the combinations of X1 and X2 that fulfill the necessary conditions of consumer equilibrium, we need to compare the total utility (TU) generated from each combination to see if it maximizes the consumer's satisfaction.
1.2= First, let's calculate the Marginal Utility (MU) for each product. The MU is the change in total utility when consuming one additional unit of the product.
To find MU for X1:
- The TU for X1 is given as follows:
- TU1 = 16, TU2 = 30, TU3 = 41, TU4 = 51, TU5 = 60, TU6 = 68, TU7 = 75, TU8 = 81, TU9 = 86, TU10 = 89, TU11 = 90
- Now, let's calculate MU for X1:
- MU1 = TU1 - 0 = 16 - 0 = 16
- MU2 = TU2 - TU1 = 30 - 16 = 14
- MU3 = TU3 - TU2 = 41 - 30 = 11
- MU4 = TU4 - TU3 = 51 - 41 = 10
- MU5 = TU5 - TU4 = 60 - 51 = 9
- MU6 = TU6 - TU5 = 68 - 60 = 8
- MU7 = TU7 - TU6 = 75 - 68 = 7
- MU8 = TU8 - TU7 = 81 - 75 = 6
- MU9 = TU9 - TU8 = 86 - 81 = 5
- MU10 = TU10 - TU9 = 89 - 86 = 3
- MU11 = TU11 - TU10 = 90 - 89 = 1
To find MU for X2:
- The TU for X2 is given as follows:
- TU1 = 15, TU2 = 28, TU3 = 40, TU4 = 48, TU5 = 54, TU6 = 59, TU7 = 63, TU8 = 66, TU9 = 68, TU10 = 69, TU11 = 69
- Now, let's calculate MU for X2:
- MU1 = TU1 - 0 = 15 - 0 = 15
- MU2 = TU2 - TU1 = 28 - 15 = 13
- MU3 = TU3 - TU2 = 40 - 28 = 12
- MU4 = TU4 - TU3 = 48 - 40 = 8
- MU5 = TU5 - TU4 = 54 - 48 = 6
- MU6 = TU6 - TU5 = 59 - 54 = 5
- MU7 = TU7 - TU6 = 63 - 59 = 4
- MU8 = TU8 - TU7 = 66 - 63 = 3
- MU9 = TU9 - TU8 = 68 - 66 = 2
- MU10 = TU10 - TU9 = 69 - 68 = 1
- MU11 = TU11 - TU10 = 69 - 69 = 0
1.3= Now, let's compare the MU for X1 and X2 with the market price of 2 birr for each product.
- The market price for X1 and X2 is 2 birr.
Comparing MU1 for X1 (16) with the market price (2), MU1 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU2 for X1 (14) with the market price (2), MU2 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU3 for X1 (11) with the market price (2), MU3 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU4 for X1 (10) with the market price (2), MU4 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU5 for X1 (9) with the market price (2), MU5 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU6 for X1 (8) with the market price (2), MU6 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU7 for X1 (7) with the market price (2), MU7 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU8 for X1 (6) with the market price (2), MU8 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU9 for X1 (5) with the market price (2), MU9 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU10 for X1 (3) with the market price (2), MU10 > Price. Therefore, the consumer should consume 1 unit of X1.
Comparing MU11 for X1 (1) with the market price (2), MU11 < Price. Therefore, the consumer should not consume any additional units of X1.
Comparing MU1 for X2 (15) with the market price (2), MU1 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU2 for X2 (13) with the market price (2), MU2 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU3 for X2 (12) with the market price (2), MU3 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU4 for X2 (8) with the market price (2), MU4 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU5 for X2 (6) with the market price (2), MU5 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU6 for X2 (5) with the market price (2), MU6 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU7 for X2 (4) with the market price (2), MU7 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU8 for X2 (3) with the market price (2), MU8 > Price. Therefore, the consumer should consume 1 unit of X2.
Comparing MU9 for X2 (2) with the market price (2), MU9 = Price. It is unclear whether the consumer should consume an additional unit of X2. Further analysis is needed.
Comparing MU10 for X2 (1) with the market price (2), MU10 < Price. Therefore, the consumer should not consume any additional units of X2.
Comparing MU11 for X2 (0) with the market price (2), MU11 < Price. Therefore, the consumer should not consume any additional units of X2.
1.4= From the above analysis, we find that the consumer's equilibrium purchase size is as follows:
For X1: The consumer should purchase 10 units of X1 to maximize their utility.
For X2: The consumer should purchase 9 units of X2 to maximize their utility.
To find the consumer's equilibrium purchase size, we need to identify the combinations of X1 and X2 that fulfill the necessary conditions of consumer equilibrium. The conditions for consumer equilibrium are:
1. The consumer's budget constraint: The consumer should not exceed their budget of 20 birr.
2. The consumer's utility maximization: The consumer should allocate their budget in a way that maximizes their total utility.
First, let's check the budget constraint:
The market price of each product, X1 and X2, is 2 birr.
The consumer has a budget of 20 birr.
Since the market price of each product is 2 birr, the consumer can buy a maximum of 10 units of each product with a budget of 20 birr.
Now, let's analyze the total utility generated from different combinations of X1 and X2. Here are the given combinations of TUX1 and TUX2:
Q 1 2 3 4 5 6 7 8 9 10 11
TUX1 16 30 41 51 60 68 75 81 86 89 90
TUX2 15 28 40 48 54 59 63 66 68 69 69
To fulfill the condition of consumer equilibrium, we need to find the combinations where the total utility is maximized without exceeding the budget of 20 birr.
Let's calculate the total utility for each combination by adding TUX1 and TUX2:
Q 1 2 3 4 5 6 7 8 9 10 11
TUX1 16 30 41 51 60 68 75 81 86 89 90
TUX2 15 28 40 48 54 59 63 66 68 69 69
Total 31 58 81 99 114 127 138 147 154 158 159
Now, let's identify the combinations that meet the condition of not exceeding the budget of 20 birr:
Q 1 2 3 4 5 6 7 8 9 10 11
TUX1 16 30 41 51 60 68 75 81 86 89 90
TUX2 15 28 40 48 54 59 63 66 68 69 69
Total 31 58 81 99 114 127 138 147 154 158 159
Budget <20 <20 <20 <20 <20 <20 >20 >20 >20 >20 >20
Based on the calculations, the combinations that fulfill the condition of not exceeding the budget of 20 birr are:
Combination 1: Q1 (X1 = 1, X2 = 1) - Total Utility = 31
Combination 2: Q2 (X1 = 2, X2 = 3) - Total Utility = 58
Combination 3: Q3 (X1 = 3, X2 = 4) - Total Utility = 81
Combination 4: Q4 (X1 = 4, X2 = 5) - Total Utility = 99
These are the combinations that fulfill the necessary conditions of consumer equilibrium.
From these combinations, we can identify the consumer's equilibrium purchase size. The combination with the highest total utility and does not exceed the budget is Combination 4: Q4 (X1 = 4, X2 = 5) with a total utility of 99.
Therefore, the consumer's equilibrium purchase size is 4 units of X1 and 5 units of X2.