Microeconomic Theory

Quiz 5 :Demand Relationships Among Goods

Heidi receives utility from two goods, goat's milk (m) and strudel (s), according to the utility function a. Show that increases in the price of goat's milk will not affect the quantity of strudel Heidi buys; that is, show that b. Show also that c. Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the derivatives in parts (a) and (b) are identical. d. Prove part (c) explicitly using the Marshallian demand functions for m and s.
Free
Essay

Utility function is a mathematical function which represent the individual preference for the goods.
Substitution effect: It measures the change in demand due to change in relative prices keeping purchasing power constant.
Income effect: It measures the change in demand due to change in due to change in purchasing power keeping relative price constant.
a )
The given utility function is shown below:
Utility function:
Where, m denotes the goat's milk and s denote the strudel.
The condition of consumer equilibrium can be easily understood with the help of Lagrange multiplier.
Budget constraint can be for this problem is shown below:
By using above information, the following lagragian function is obtained which is shown below:
Taking first order condition:
By using above, the following result is obtained which is shown below:
Taking ratio of equation (1) and equation (2), the following result is obtained which is shown below:
Now, substituting the value of M in the budget constraint, the following result is obtained which is shown below:
Thus, from above value of s the following result is obtained which is shown below:
This shows that increases in the price of goat's milk does not affected the quantity of strudel.
b )
The lagragian function for the given problem is shown below:
Taking first order derivative, the following result is obtained which is shown below:
By using above function, the following result is obtained which is shown below:
Taking ratio of equation (1) and equation (2) the following result is obtained which is shown below:
Substitute the M value in budget constraint:
Thus, from above value of M the following result is obtained which is shown below:
The above result shows that increase in the price of strudel does not affect the quantity of goat's milk.
c )
In two goods case the income and substitution effects from the change in the price of one good on the demand for another good usually work in opposite directions. Its substitution effect is positive but its income effect is negative.
Slutsky equation:
Where,
Uncompensated demand is represented by
,
Compensated demand or Hickson demand is represented by
.
To generate
create the expenditure function and take the first derivative:
The Hickson demand function is obtained by taking the derivative of expenditure function with respect to their prices as shown below:
Now, taking the derivative of
with respect to their prices the following result is obtained which is shown below:
Since,
and
It shows that
Hence proved.
d )
The Marshallian demand function shows changes in the price of y do not affect x purchases.
That is,
Thus, by using this the following result is obtained which is shown below:
Hence Proved.

Hard Times Burt buys only rotgut whiskey and jelly donuts to sustain him. For Burt, rotgut whiskey is an inferior good thatexhibits Giffen's paradox, although rotgut whiskey and jelly donuts are Hicksian substitutes in the customary sense. Develop anintuitive explanation to suggest why an increase in the price of rotgut whiskey must cause fewer jelly donuts to be bought. Thatis, the goods must also be gross complements.
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Essay

Hard Times buys only rotgut whiskey and jelly donuts. Rought whiskey and jelly donuts are Hicksian substitute goods in customary sense. Since rotgut whiskey is an inferior good that exhibits giffen's paradox. Law of demand does not work in case of giffen good.
As we know that in case of giffen good, demand of the good is increased as price of the good increases and vice-versa. As price of rotgut whiskey increases, thus demand of rotgut whiskey is also increase. Hard Times' income remains the same. Now Hard Times expend more income to buy whiskey than that of earlier. Simultaneously he reduces his expenditure to buy jelly donuts. Therefore, an increase in the price of rotgut whiskey must cause fewer jelly donuts to be bought.

Donald, a frugal graduate student, consumes only coffee ( c ) and buttered toast ( bt ). He buys these items at the university cafeteria and always uses two pats of butter for each piece of toast. Donald spends exactly half of his meager stipend on coffee and the other half on buttered toast. a. In this problem, buttered toast can be treated as a composite commodity. What is its price in terms of the prices of butter ( p b ) and toast ( p t )? b. Explain why . c. Is it also true here that are equal to 0?
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Essay

a.
Mr. D uses two pats of butter for each piece of toast.
The price of buttered toast can be shown in terms of price of butter and price of toast as given below:
Where,
b.
Mr. D spends exactly half of his meager stipend on coffee (c) and the other half on buttered toast (bt).
Differentiating value of bt and c with respect to each other prices as shown below:
is zero because demand of coffee is not a function of price of butter.
It means that change in price of butter toast does not affect the demand of coffee.
c.
Mr. D uses two pats of butter for each piece of toast.
It is true that
and
are equal to zero as demand for coffee doesn't depend on the prices of toast and butter. Mr. D spends a fixed proportion of his income on the coffee. So price of toast and butter will not affect his demand for coffee.

Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by where each letter stands for miles traveled by a specific mode. Suppose that the ratio of the price of train travel to that of bus travel ( p t / p b ) never changes. a. How might one define a composite commodity for ground transportation? b. Phrase Sarah's optimization problem as one of choosing between ground ( g ) and air ( p ) transportation. c. What are Sarah's demand functions for g and p ? d. Once Sarah decides how much to spend on g , how will she allocate those expenditures between b and t?
Essay
Suppose that an individual consumes three goods, x 1 , x 2 , and x 3 , and that x 2 and x 3 are similar commodities (i.e., cheap and expensive restaurant meals) with -that is, the goods' prices have a constant relationship to one another. a. Show that x 2 and x 3 can be treated as a composite commodity. b. Suppose both x 2 and x 3 are subject to a transaction cost of t per unit (for some examples, see Problem 6.6). How will this transaction cost affect the price of x 2 relative to that of x 3 ? How will this effect vary with the value of t? c. Can you predict how an income-compensated increase in t will affect e x penditures on the composite commodity x 2 and x 3 ? Does the composite commodity theorem strictly apply to this case? d. How will an income-compensated increase in t affect how total spending on the composite commodity is allocated between x 2 and x 3 ?
Essay
Apply the results of Problem 6.5 to explain the following observations: a. It is difficult to find high-quality apples to buy in Washington State or good fresh oranges in Florida. b. People with significant babysitting expenses are more likely to have meals out at expensive (rather than cheap) restaurants than are those without such expenses. c. Individuals with a high value of time are more likely to fly the Concorde than those with a lower value of time. d. Individuals are more likely to search for bargains for expensive items than for cheap ones. Note: Observations (b) and (d) form the bases for perhaps the only two murder mysteries in which an economist solves the crime; see Marshall Jevons, Murder at the Margin and The Fatal Equilibrium. Reference: Problem 6.5 Suppose that an individual consumes three goods, x 1 , x 2 , and x 3 , and that x 2 and x 3 are similar commodities (i.e., cheap and expensive restaurant meals) with -that is, the goods' prices have a constant relationship to one another. a. Show that x 2 and x 3 can be treated as a composite commodity. b. Suppose both x 2 and x 3 are subject to a transaction cost of t per unit (for some examples, see Problem 6.6). How will this transaction cost affect the price of x 2 relative to that of x 3 ? How will this effect vary with the value of t? c. Can you predict how an income-compensated increase in t will affect e x penditures on the composite commodity x 2 and x 3 ? Does the composite commodity theorem strictly apply to this case? d. How will an income-compensated increase in t affect how total spending on the composite commodity is allocated between x 2 and x 3 ?
Essay
In general, uncompensated cross-price effects are not equal. That is, Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good regardless of relative prices. (This is a generalization of Problem 6.1.)
Essay
Example 6.3 computes the demand functions implied by the three-good CES utility function a. Use the demand function for x in Equation 6.32 to determine whether x and y or x and z are gross substitutes or gross complements. b. How would you determine whether x and y or x and z are net substitutes or net complements?
Essay