Question 1
(a) Describe how we represent consumers' preferences in a world without constraints. Using two goods Y and X as an example, discuss the assumptions underlying consumers' preferences. (Total marks: 20)
(b) Define utility and marginal utility. Assuming your marginal utility for one of the goods from question (a) is always positive and diminishing. Draw a graph with total utility on the vertical axis and the quantity of the good on the horizontal axis. How does this relate to marginal utility? Graphically represent the marginal utility curve. (Total marks: 20)
(c) Considering both goods from question (a) Define the marginal rate of substitution. Suppose your utility for good Y and for good X can be represented as Uy0.5 x05 Draw the indifference curve that yields a utility level of 9. Calculate the marginal utility of good Y, the marginal utility
of good X and the marginal rate of substitution of Y for X (MRS) when X-3 (Total marks. 30)
(d) Suppose that the utility of the two goods can be describe with the following utility function: U 0.5Y+0.5X Draw the new indifference curve. What is the marginal rate of substitution Y for X when U? Are the two goods perfect substitutes or perfect complements? (Total marks: 25)
Question 2
a) Discuss how income can constram consumption choices over two goods Y and X using a graph
Indicate on the graph the utility maximizing quantities of the two goods (Total marks: 20) b) Describe graphically how the utility maximizing quantities change when the price of good X changes. Derive the price consumption curve and the demand curve for X
(Total marks: 20)
e) The utility you receive by consuming good Y and good X is given by LXXY)=XY+X. Find the
marginal utility for each good. (Total marks: 20)
d) Assume that X costs $1 per unit and Y costs 52 per unit. Your total income $44. Graph your budget line. Draw an indifference curve associated with a utility level of 72 and the indifference curve associated with a utility level of 144. Are the indifference curves com ex? (Total marks: 20)
What is the marginal rate of substitution of Y for X when utility is maximized? Show this graphically and algebraically (Total marks. 20)
Answer:
QUESTION 1:
A)
In a world without constraints, consumers' preferences can be represented using utility functions. A utility function assigns a numerical value to each possible consumption bundle of goods, representing the consumer's satisfaction or happiness associated with that bundle. The higher the utility value, the more preferred the bundle is to the consumer.
Let's consider two goods, Y and X. A typical utility function for a consumer's preferences over these two goods might be U(X,Y)=X^α Y^β, where α and β are positive constants that reflect the relative importance of X and Y in the consumer's preferences.
There are some assumptions underlying consumers' preferences that are implicit in this utility function. First, preferences are assumed to be complete, meaning that the consumer can express a preference between any two bundles of goods. Second, preferences are assumed to be transitive, meaning that if the consumer prefers bundle A to bundle B and bundle B to bundle C, then the consumer must also prefer bundle A to bundle C. Third, preferences are assumed to be continuous, meaning that small changes in the quantities of goods should result in small changes in the consumer's utility level.
B)
Utility is a measure of the satisfaction or happiness a consumer derives from consuming a particular combination of goods. It is typically represented by a utility function, which assigns a numerical value to each possible consumption bundle of goods.
Marginal utility, on the other hand, is the additional utility a consumer derives from consuming one additional unit of a particular good, holding the consumption of other goods constant. It is the derivative of the utility function with respect to the quantity of the good in question. Mathematically, marginal utility can be expressed as MU = dU/dX, where U is the utility function and X is the quantity of the good.
Assuming that the marginal utility for one of the goods from question (a) is always positive and diminishing, this means that as the consumer consumes more of that good, the additional satisfaction or happiness they derive from each additional unit decreases. This is because the consumer's needs and wants for that good are being increasingly satisfied, and they are less willing to pay the same price for an additional unit.
The marginal utility curve can be derived by taking the derivative of the total utility curve with respect to the quantity of the good. It will be downward sloping, reflecting the decreasing additional satisfaction the consumer derives from each additional unit of the good.
The curve labeled "TU" represents the total utility curve, while the curve labeled "MU" represents the marginal utility curve. As the quantity of the good increases, the total utility curve increases at a decreasing rate while the marginal utility curve decreases. At the point where the marginal utility curve intersects the horizontal axis (i.e., the quantity of the good where marginal utility is zero), the total utility curve reaches its maximum. This point is known as the consumer's optimal consumption level for that good.
C)
The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade one good for another while remaining at the same level of utility. It is equal to the ratio of the marginal utility of the two goods. Mathematically, MRS = MUx/MUy, where MUx is the marginal utility of good X and MUy is the marginal utility of good Y.
Suppose the utility function for a consumer's preferences over goods X and Y is U = Y^0.5 X^0.5. We can take the partial derivative of this utility function with respect to Y and X to obtain the marginal utility functions:
MUy = 0.5 Y^(-0.5) X^0.5
MUx = 0.5 Y^0.5 X^(-0.5)
To draw an indifference curve that yields a utility level of 9, we can set the utility function equal to 9 and solve for Y in terms of X:
9 = Y^0.5 X^0.5
Y = (9/X)^2
Plotting this equation on a graph with Y on the vertical axis and X on the horizontal axis, we obtain an indifference curve.
To calculate the marginal utility of good Y and good X when X=3, we can substitute X=3 into the marginal utility functions:
MUy = 0.5 (9/3)^(-0.5) 3^0.5
MUy = 0.5 (3)^(-0.5) 3^0.5
MUy = 0.5
MUx = 0.5 (9/3)^0.5 3^(-0.5)
MUx = 0.5 (3)^0.5 3^(-0.5)
MUx = 0.5 (3)^0.5/3^0.5
MUx =0.5
So the marginal utility of good Y is 0.5 and the marginal utility of good X is 0.5.
To calculate the MRS of Y for X when X=3, we can substitute the marginal utility of Y and the marginal utility of X into the formula for MRS:
MRS = MUx/MUy
MRS = 0.5/0.5
MRS = 1
Therefore, the MRS of Y for X when X=3 is 1.
D)
Suppose the utility function for a consumer's preferences over goods X and Y is U = 0.5Y + 0.5X. To draw an indifference curve for a certain level of utility U, we can set the utility function equal to U and solve for Y in terms of X:
U = 0.5Y + 0.5X
Y = U - 0.5X
Plotting this equation on a graph with Y on the vertical axis and X on the horizontal axis, we obtain an indifference curve.
To find the marginal rate of substitution (MRS) of Y for X at the utility level U0, we need to calculate the slope of the indifference curve at that point. The slope of the indifference curve is given by the absolute value of the ratio of the marginal utility of Y to the marginal utility of X:
MRS(Y,X) = |MU(Y)/MU(X)|
Since the utility function is U = 0.5Y + 0.5X, the marginal utility of Y is:
MU(Y) = dU/dY = 0.5
And the marginal utility of X is:
MU(X) = dU/dX = 0.5
Therefore, at the utility level U0, the MRS of Y for X is:
MRS(Y,X) = |MU(Y)/MU(X)| = |0.5/0.5| = 1
The MRS of 1 indicates that the two goods are perfect substitutes, which means that the consumer is indifferent between consuming one unit of Y or one unit of X, as they provide the same level of utility. The consumer is willing to substitute one unit of Y for one unit of X at a constant rate of 1:1, without affecting the overall level of utility.
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