microAssn3

1) Suppose a consumer has preferences between two goods that can be represented by the utility function U = xy.

MRS=y/x ## a) On a graph, draw the indifference curve associated with the utility level U1 = 128. Then answer the following questions:

# Set up the values for x and utility levels
x <- seq(1, 20, length = 100)

# Utility levels U1 = 128 and U2 = 200
U1 <- 128
U2 <- 200

# Calculate y for both utility levels
y_U1 <- U1 / x
y_U2 <- U2 / x

# Plot the indifference curves
plot(x, y_U1, type = "l", col = "blue", lwd = 2, ylim = c(0, 25),
     xlab = "Good X", ylab = "Good Y", main = "Indifference Curves for U = xy")
lines(x, y_U2, col = "red", lwd = 2)

# Add a legend
legend("topright", legend = c("U1 = 128", "U2 = 200"), col = c("blue", "red"), lwd = 2)

# MRS is -y/x, and as x increases, MRS decreases in magnitude, confirming it's diminishing.

i) Does the indifference curve intersect either axis?

No, the indifference curve does not intersect the axes. As either x or y approaches zero, the utility level goes to zero, so the curve approaches but never touches the axes.

ii) Does the shape of the indifference curve indicate that MRSx, y is diminishing?

Yes, the Marginal Rate of Substitution (MRS) = -y/x is indeed diminishing. As x increases, y decreases along the indifference curve, which means the absolute value of MRS, which is y/x, decreases. This implies that the curve becomes flatter as x increases. To elaborate, the shape of the indifference curve—being a hyperbola—illustrates this diminishing MRS clearly. As you move along the curve from left to right (increasing x and decreasing y), the slope of the curve (which represents MRS) becomes less steep. This diminishing MRS indicates that the consumer is willing to give up less of good y to obtain an additional unit of good x as they have more of x and less of y.

b) On the same graph draw a second indifference curve, U2 = 200. Show how MRSxy depends on x and y, and use this information to determine if MRSx, y is diminishing for this utility function.

Please see graph drawed above MRSxy =− (∂U/∂y) / (∂U/∂x) = |−y/x| The Marginal Rate of Substitution (MRS) = |-y/x|. As we move along the indifference curve, increasing x and decreasing y, the ratio y/x decreases. This shows that MRS depends on x and y and is diminishing, meaning the consumer’s willingness to substitute x for y decreases as they have more x and less y.

Since both MUx and MUy are positive when the consumer has positive amounts of x and y, the indifference curves are negatively sloped. This means that as x increases along an indifference curve, y must decrease to keep utility constant.

2) Nick has a preference ordering that can be represented by the utility function u(x,y)=xy^2 . What is the general expression for Nick’s Marginal Rate of Substitution between x and y? What is the value of Nick’s MRS if he is consuming a bundle which is on the indifference curve for which u(x,y)=32, and he is consuming 4 units of y?

MRS=(du/dx)/(du/dy)=y^2/2xy=y/2x. ->MU_x = y^2, MU_y = 2xy. u(x, y) = 32 = xy^2, y = 4 -> x = 2 MRS = 1

3) Do the following questions from the textbook - Ch 3: Q1, Q3, Q7, and Q10. (see attached if you are using a different edition of the textbook)

Q1 Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the MRS declines as x increases)

See code below for graph ## a. U(x, y) = 3x + y MRS = MU_x/MU_y = 3 constant ## b. U(x, y) = sqrt(x*y) MRS = MU_x/MU_y = y/x convex, diminishing ## c. U(x, y) = sqrt(x) + y MRS = MU_x/MU_y = 0.5x^(-0.5) diminishing ## d. U(x, y) = sqrt(x^2 - y^2) MRS = MU_x/MU_y = x/y increasing ## e. U(x, y) = xy/(x+y) MRS = MU_x/MU_y = y^2 / x^2 convex, diminishing

# Set up the values for x and y
x <- seq(0.1, 10, length = 100)
y <- seq(0.1, 10, length = 100)
X <- outer(x, y, FUN = function(x, y) x)
Y <- outer(x, y, FUN = function(x, y) y)

# a. U(x, y) = 3x + y
U_a <- function(x, U) U - 3 * x

# b. U(x, y) = sqrt(x * y)
U_b <- function(x, U) (U^2) / x

# c. U(x, y) = sqrt(x) + y
U_c <- function(x, U) U - sqrt(x)

# d. U(x, y) = sqrt(x^2 - y^2) (for x > y)
U_d <- function(x, U) sqrt(x^2 - U^2)

# e. U(x, y) = xy / (x + y)
U_e <- function(x, U) (U * x) / (x - U)

# Plot indifference curves
par(mfrow = c(2, 3))

# Plot a. U(x, y) = 3x + y
contour(x, y, outer(x, y, function(x, y) 3*x + y), levels = 3*c(1, 2, 3, 4), col = 1:4,
        xlab = "x", ylab = "y", main = "U(x, y) = 3x + y")

# Plot b. U(x, y) = sqrt(x * y)
contour(x, y, outer(x, y, function(x, y) sqrt(x * y)), levels = 3*c(1, 2, 3, 4), col = 1:4,
        xlab = "x", ylab = "y", main = "U(x, y) = sqrt(x * y)")

# Plot c. U(x, y) = sqrt(x) + y
contour(x, y, outer(x, y, function(x, y) sqrt(x) + y), levels = 3*c(1, 2, 3, 4), col = 1:4,
        xlab = "x", ylab = "y", main = "U(x, y) = sqrt(x) + y")

# Plot d. U(x, y) = sqrt(x^2 - y^2)
x_d <- seq(1, 10, length = 100)  # Restrict x to be greater than y
y_d <- seq(0.1, 9, length = 100)
contour(x_d, y_d, outer(x_d, y_d, function(x, y) sqrt(x^2 - y^2)), levels = 3*c(1, 2, 3, 4), col = 1:4,
        xlab = "x", ylab = "y", main = "U(x, y) = sqrt(x^2 - y^2)")

Warning in sqrt(x^2 - y^2): NaNs produced

# Plot e. U(x, y) = xy / (x + y)
contour(x, y, outer(x, y, function(x, y) x * y / (x + y)), levels = 3*c(1, 2, 3, 4), col = 1:4,
        xlab = "x", ylab = "y", main = "U(x, y) = x * y / (x + y)")

Q3 Consider the following utility functions:

a. U(x, y) = xy

MU_x = y, MU_y = x, MU_xx = 0, MU_yy = 0 MRS = y/x

b. U(x, y) = x^2 y^2

MU_x = xy^2, MU_y = 2x^2 y, MU_xx = 2y^2, MU_yy = 2x^2 MRS = y/x

c. U(x, y) = lnx + lny

MU_x = 1/x, MU_y = 1/y, MU_xx = -1/x^2 < 0 , MU_yy = -1/y^2 < 0 MRS = y/x MU may affected but it won’t affect MRS

Show that each of these has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?

See above

Q7

Note: M=px + qy, MRSxy = p/q

a. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve?

seems she tends to trade 3x for 1y, where it should be a linear indifference curve and MRS should be 1/3

b. A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb–Douglas of the form U(x, y) = x^α * y^β, where α and β are positive constants, what is the utility function for this consumer?

U(x, y) = x^α * y^β = 8^α * 1^β = 4^α * 4^β α = 2β (is the info above means that MRS = 2?) U(x, y) = x^2β * y^β = 8^2β * 1^β = 64^β * 1^β = 4^2β * 4^β = 16^β * 4^β (if so, 2= β/α)

c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function?

Yes, there was redundancy in the previous question. Both MRS conditions at the two different bundles provide more information than necessary to determine the Cobb-Douglas utility function. Specifically, you only need one MRS condition and the indifference between the two bundles to find the utility. The minimum information required is: (1) one MRS condition (either the willingness to trade 4 units of x for 1 unit of y at (8,1) or 1 unit of x for 2 units of y at (4,4)), and (2) the fact that the consumer is indifferent between the two bundles. This is enough to determine the parameters α and β in the utility function.

Q10 Example 3.3 shows that the MRS for the Cobb–Douglas function

\[ U(x,y) = x^α y^β \] # is given by \[ MRS = α/β* (y/x) \]

a. Does this result depend on whether α + β = 1? Does this sum have any relevance to the theory of choice?

MRS = (∂u/∂x) / (∂u/∂y) = (α x^(α-1) * y^β)/ (β x^α * y^(β-1)) = α/β* (y/x) No, the result MRS=α/β⋅(y/x) does not depend on whether α + β = 1. However, when α + β = 1, the utility function is homothetic, meaning proportional changes in both goods lead to proportional changes in utility, which is relevant for constant marginal utility of income.

b. For commodity bundles for which y = x, how does the MRS depend on the values of α and β? Develop an intuitive explanation of why, if α > β, MRS > 1. Illustrate your argument with a graph.

When y = x, MRS = α/β. If α > β, MRS > 1, meaning the consumer values x more than y and needs more y to compensate for giving up x. This happens because the consumer’s preference for x is stronger, making indifference curves flatter when α > β.

c. Suppose an individual obtains utility only from amounts of x and y that exceed minimal subsistence levels given by x_0 , y_0 . In this case,

\[ U(x, y) = (x-x_0)^α (y-y_0)^β \] ## Is this function homothetic? (For a further discussion, see the Extensions to Chapter 4.)

No, the function is not homothetic because the MRS depends on the absolute levels of x and y (through x - x_0 and y - y_0), not just their ratio.