Economic Theory and Policy: Problems and Solutions
University of Westminster
BEQM601 Economic Theory and Policy
Solution guide for problem set 3A
Problem 1
(a) Because Holly spends 40% of her income for good 1 no matter what happens to her income or to the prices, her optimal expenditure share is fixed. So for Holly
[pic]. For Holly, the optimal demands are therefore:
[pic]. For Holly, optimal demand for good 1 depends on the income and the price of good 1, but does not depend on the price of good 2. Optimal demand for good 2 depends on its own price and Holly’s income, but is not affected by the price of good 1. As can be seen from her optimal demand function, Holly’s demand for both goods increase if there is an increase in her income. Optimal demand for good 1 falls if price of good 1 increases. Optimal demand for good 2 falls in price of good 2 increases.
For Ben, optimal shares are
[pic]
SO Ben’s optimal demands are
[pic]
We can simplify it to
[pic]
Ben’s optimal demand for good 1 is a function of his income, price of good 1 and price of good 2. His optimal demand for good 2 is also a function of the same three. As we can see in these functions for Ben, Ben’s optimal demand for both goods increase if his income increases. However, the effect of a price increase for Ben’s demand is ambiguous.
(b) For a consumption tax of [pic] on per unit price of good 2, price of good 2 is now equal to [pic].
For Holly, her shares for two goods remain unchanged, because these are fixed. For Holly, the optimal demand equations are now:
[pic]
Because price of good 2 is now higher, Holly will reduce the consumption of good 2 such that the total expenditure on good 2 stays the same as before (the tax). Because there is no change in Holly’s income, and also because Holly’s demand for good 1 does not depend on the price of good 2, Holly’s optimal demand for good 1 will remain unchanged. All adjustment will therefore be in the consumption of good 2. Holly will reduce consumption of
BEQM601 Economic Theory and Policy
Solution guide for problem set 3A
Problem 1
(a) Because Holly spends 40% of her income for good 1 no matter what happens to her income or to the prices, her optimal expenditure share is fixed. So for Holly
[pic]. For Holly, the optimal demands are therefore:
[pic]. For Holly, optimal demand for good 1 depends on the income and the price of good 1, but does not depend on the price of good 2. Optimal demand for good 2 depends on its own price and Holly’s income, but is not affected by the price of good 1. As can be seen from her optimal demand function, Holly’s demand for both goods increase if there is an increase in her income. Optimal demand for good 1 falls if price of good 1 increases. Optimal demand for good 2 falls in price of good 2 increases.
For Ben, optimal shares are
[pic]
SO Ben’s optimal demands are
[pic]
We can simplify it to
[pic]
Ben’s optimal demand for good 1 is a function of his income, price of good 1 and price of good 2. His optimal demand for good 2 is also a function of the same three. As we can see in these functions for Ben, Ben’s optimal demand for both goods increase if his income increases. However, the effect of a price increase for Ben’s demand is ambiguous.
(b) For a consumption tax of [pic] on per unit price of good 2, price of good 2 is now equal to [pic].
For Holly, her shares for two goods remain unchanged, because these are fixed. For Holly, the optimal demand equations are now:
[pic]
Because price of good 2 is now higher, Holly will reduce the consumption of good 2 such that the total expenditure on good 2 stays the same as before (the tax). Because there is no change in Holly’s income, and also because Holly’s demand for good 1 does not depend on the price of good 2, Holly’s optimal demand for good 1 will remain unchanged. All adjustment will therefore be in the consumption of good 2. Holly will reduce consumption of