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**Case Study - Consumer Theory**

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**Section 1 - Preferences and Utility: Make a table containing utility values for each house (bundle) that are consistent with these preferences.**

Preferences and utility:

Combining the given conditions, we get C < A < D ~ B < F ~ E

Table:

S. NO. |
House |
Utility |

1 |
E,F |
F(30,6)=f(20,7) |

2 |
B,D |
F(25,3) = f(13,6) |

3 |
A |
F(10,5) |

4 |
C |
F(20,2) |

Utility function = f(x,y)

**Section 2 - Indifference Curves**

**a) Create a graph with Size on the x-axis and Amenities on the y-axis. Plot a point on the graph for each of the houses in the table above. Draw 4 indifference curves that are consistent with the preferences (and utilities) identified in Section 1.**

**b) In general, do we expect the marginal utility of size to be higher for house D or for house E? Explain.**

Answer - Marginal utility of size is greater for D since the slope is higher at D as compared to E.

**c) Is the marginal rate of substitution higher for house B or for house D? Is the marginal rate of substitution higher for house E or for house F? Explain.**

Answer - Rate of substitution is greater for D since the slope is higher at D as compared to B.

Rate of substitution is greater for F since the slope is higher at F as compared to E.

Slope = Marginal Rate of substitution

**Section 3 - Consumer Maximization**

Budget: 25000 x + 10000 y = B

Equations:

25000 y + 10000 x = 200,000 => Points (20,0) and (0,8)

25000 y + 10000 x = 400,000 => Points (40,0) and (0,16)

25000 y + 10000 x = 600,000 => Points (60,0) and (0,24)

**a) Create a graph with Size on the x-axis and Amenities on the y-axis. Amenities cost $25,000 and Size costs $10,000. Draw the budget curve for a consumer with a budget of $200,000. Also draw the budget curves for a consumer with a budget of $400,000 and a consumer with a budget of $600,000. When drawing the curves be sure to indicate a couple points on each curve.**

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**b) Now take the graph with your indifference curves from section 2 and add in a couple budget curves. You can put the budget curves in any of several places, but they should meet a couple constraints. The budget curves should be parallel because they are for constant prices. Each budget curve should be tangent to a single indifference curve. The resulting optimal combination of features (amenities and size) should be marked. This point need not correspond to one of the 6 houses in the original table.**

**Section 4 - Indifference Curves and Demand**

S= size, A = Amenities

Utility function = U(S,A) = S^{½} A^{½}

**a) Take the derivative with respect to Size (S) to find the general form of the consumer's marginal utility of Size (S).**

Answer - General form of consumer's marginal utility of size, dU/dS = A^{½}/ 2S^{½}.

**b) Take the derivative with respect to Amenities (A) to find the general form of the consumer's marginal utility of Amenities (A).**

Answer - General form of consumer's marginal utility of amenities, dU/dA = S^{½}/2A^{½}.

**c) Use the marginal utilities to find the marginal rate of substitution (MRS).**

Answer - MRS , Marginal Rate of substitution = Marginal utility of size/ Marginal utility of amenities

= dU/dS / dU/dA = A/S

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**d) Suppose the consumer has an income (M) of 10. Assume the price of Amenities (p _{A}) is 2.5 and the price of Size (p_{S}) is 1. Write the consumer's budget equation by substituting the prices and income into the general budget equation: M = p_{S}S + p_{A}A.**

Answer - Consumer's Budget Equation: M = p_{S}S + p_{A}A

10 = 1 x S + 2.5 x A

S + 2.5 A = 10

**e) What amounts of size and amenities will the consumer choose? **

S + 2.5 A = 10

MRS = Price Ratio

A/S = 2.5 / 1 = 2.5

A= 2.5 S

S + 2.5 x 2.5 S = 10

S = 10/7.25 = 1.38

A = 2.5 S = 2.5 x 1.38 = 3.45

Hence Size = 1.38 and Amenities = 3.45 must be chosen.

**f) Finally, find the aggregate demand curve for the two consumers for prices of 1, 2, 3, and 4.**

Aggregate Demand Curve:

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