For each example below, draw a set of three indifference� curves that represent the given preferences. Be certain to show the direction of increasing utility. Also write down a utility function that would be consistent with the given preferences and use that utility function to get expressions for the marginal utility

of each good.

c)Carl likes both burgers and hotdogs but experiences diminishing marginal utility of hotdogs and diminishing marginal utility of hamburgers.

d) Dennis likes both co�ffee and cigarettes and both are addictive goods, meaning
that the more co�ffee Dennis consumes, the more he's willing to pay to get an
additional cup of coffee and the more cigarettes he smokes, the more he is willing to pay for an additional cigarette.

Attempt:

c) For diminishing marginal utility, the indifference curves should look something like y = sqrt(x), but I'm not sure this is the correct function. Increasing utility should be up and to the left?

d) This one I have no idea how to do

See if any of these sites will help:

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Sra

c) To represent Carl's preferences with diminishing marginal utility for both hamburgers and hotdogs, we can draw a set of three indifference curves.

To begin, let's define a utility function consistent with Carl's preferences. Let's assume Carl's utility function to be U(x,y) = √(x) + √(y), where x represents the quantity of hamburgers consumed and y represents the quantity of hotdogs consumed.

To draw the indifference curves, we can choose various combinations of x and y values and substitute them into the utility function to get the corresponding utility levels. For example, if we choose x = 1 and y = 1, the utility level would be U(1,1) = √(1) + √(1) = 2.

Now, let's choose three different utility levels and solve for the corresponding combinations of x and y values.

1) If we want to represent the utility level U = 4, we can set up the equation as follows:
4 = √(x) + √(y)
Solving for y, we get y = (4 - √(x))^2. By choosing various values for x and substituting them into this equation, we can get the corresponding values of y. This will help us plot points on the graph.

2) If we want to represent the utility level U = 6, the equation becomes:
6 = √(x) + √(y)
Again, solve for y, and by choosing different values of x, substitute them to get the corresponding values of y.

3) Similarly, we can choose another utility level, let's say U = 8, and solve for y.

Plotting the points obtained from solving these equations on a graph will give us three indifference curves. Ensure that each curve represents a higher utility level from the one below it. The curves should slope downward and to the right, indicating diminishing marginal utility for both hamburgers and hotdogs.

d) In this scenario, Dennis likes both coffee and cigarettes, and both goods have addictive properties. The more he consumes of each good, the more he is willing to pay for an additional unit.

To represent this situation, we can draw indifference curves that are upward-sloping and bowed inward or convex to the origin.

To get the utility function consistent with Dennis's preferences, we can use the concept of Cobb-Douglas utility function:

U(x,y) = (x^α) * (y^β)

Here, x represents the quantity of coffee consumed, and y represents the quantity of cigarettes consumed. α and β are parameters that determine the shape of the indifference curves.

The marginal utility of coffee (∂U/∂x) can be calculated by differentiating U with respect to x:

∂U/∂x = α * (x^(α-1)) * (y^β)

Similarly, the marginal utility of cigarettes (∂U/∂y) is:

∂U/∂y = β * (x^α) * (y^(β-1))

By replacing α and β with suitable values, we can define the specific utility function for Dennis's preferences. Then, differentiate it partially to find the expressions for the marginal utilities of coffee and cigarettes.