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List of currently published content:

- Study Pages (notes and accompanying interactive graphs)
- Practice Pages (user defined test-like questions with an answer validator and step by step solutions)
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Class Notes and interactive content to help make the concepts more engaging

The Function you entered appears as:

U(x,y) =

MUx, the derivative of U(x,y) with respect to x is:MUx =

MUy, the derivative of U(x,y) with respect to y is:MUy =

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Class Notes and interactive content to help make the concepts more engaging

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Note that the income expansion path is the collection of all of the utility maximizing bundles subject to a budget constraint at every income level. Recall how we find a utility maximizing bundle subject to a budget constraint: we set MRSxy = the slope of the budget line. Then we used systems of equations with the budget line to find the utility maximizing bundle. The first step to constrained optimization: setting MRSxy = the slope of the budget line gives us the relationship between x and y that maximizes our utility independant of our budget. This means that setting MRSxy = the slople of the budget line is the equation that gives us the income expansion path.

The paramaters governing the dynamics for the Engel Curves are taken from the Income Expansion Path paramaters since they are so intertwined.

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An Engel plot is simply the income expansion path with just one good placed on the x axis and income placed on the y axis. It is useful foor identifying income ranges over which goods are normal or inferior.

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In the above graph, the original budget line is given by the dark blue line and the original income expansion path is given by the dark green line. After the change in prices and/or income the utility maximizing bundle has shifted. The bright green line is the new income expansion path and the solid light blue line is the new budget line. The dashed light blue line is a ficticious budget line parallel to the new budget line. It is tangent to the original indifference curve at the same place the new budget line intersects the old indifference curve. That point where those three lines: the ficticious budget line, the new income expansion path, and the indifference curve corresponding the the utility realized in the original bundle is the ficticious point we use to calculate the income and substitution effects. The differnce between it and the original bundle is the substitution effect, and the income effect is the difference between the new utility maximizing bundle and it. Note: the income effect takes place along the new income expansion path and the substitution effect takes place along the indifference curve corresponding to the original utility. When only income changes, there is no substitution effect, only a income effect.

In the example in the plot above, the substitution effect was: the income effect was: and the total effect was:

Plot Supply and Equilibrium

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Class Notes and interactive content to help make the concepts more engaging

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The following inputs are for part b

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Class Notes and interactive content to help make the concepts more engaging

Class Notes and interactive content to help make the concepts more engaging

Be sure not to omit the multiplication sign (*) when inputting functions

Inputs for part a

Inputs for part b

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Be sure not to omit the multiplication sign (*) when inputting functions

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Be sure not to omit the multiplication sign (*) when inputting functions

The following options are for the Cournot Question

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