III. A common utility function used to illustrate economic examples is the Cobb-Douglas function where U(X,...

III. A common utility function used to illustrate economic examples is the Cobb-Douglas function where U(X, Y)= XαYβ, where α and β are decimal exponents that sum to 1.0 (for example, 0.3 and 0.7).

a. For this utility function, the MRS is given by MRS = MUX=MUY = αY/βX. Use this fact together with the utility-maximizing condition (and that α+ β =1) to show that this person will spend the fraction of his other income on good X and the fraction of income on good Y— that is, show PXX/I = α, PYY/I = β.

b. Use the results from part a to show that total spending on good X will not change as the price of X changes so long as income stays constant.

c. Use the results from part a to show that a change in the price of Y will not affect the quantity of X purchased.

d. Show that with this utility function, a doubling of income with no change in prices of goods will cause a precise doubling of purchases of both X and Y.

Expert Solution

Rahul Sunny answered 2 years ago

Consider the following Cobb-Douglas utility function: U(x,y) = 10 + 22xy, where x denotes the quantity...

Consider the following Cobb-Douglas utility function: U(x,y) = 10 + 22xy, where x denotes the quantity of apples and y denotes the quantity of pears. Income is m; the price for one apple is px and the price for one pear is py. b) By applying the Lagrangian method, derive the optimal demands for apples and pears. c) What is the impact of an increase in the price of apples on the optimal demand for pears? What is the impact...

Assume a Cobb-douglas utility function of 2 goods x and y given by U = x...

Assume a Cobb-douglas utility function of 2 goods x and y given by U = x 0.5y 0.5 and an initial income I of 100. Let initial price be px = 4 and py = 1. Now vary the price of x from 1 to 7 in steps of 1. So you have 7 prices for x. px = {1, 2, 3, 4, 5, 6, 7} For each of these px, py REMAINS the SAME at 1. In the excel...

Substitution and Income Effect (Calculation) Suppose that the consumer's utility function is Cobb-Douglas U(x; y) =...

Substitution and Income Effect (Calculation) Suppose that the consumer's utility function is Cobb-Douglas U(x; y) = xy. She initially has $100 income and the prices that she faces are px = py = $2: a. Compute her demands for x and y. b. Now suppose that px decreases to $1 while py remains unchanged. Compute her new demands for x and y. c. What is the total (price) effect on the consumption of x? d. What is the substitution effect...

Which of the following utility functions is a Cobb-Douglas? Group of answer choices U(x,y) = ln(x)...

Which of the following utility functions is a Cobb-Douglas? Group of answer choices U(x,y) = ln(x) + 4y U(x,y) = 2x + 4y U(x,y) = ln(x) + 4ln(y) U(x,y) = min{x, 4y} None of the above

Solve the optimization problem for a Cobb-Douglas utility function with three goods: U(x1, x2, x3) =...

Solve the optimization problem for a Cobb-Douglas utility function with three goods: U(x1, x2, x3) = x c 1x d 2x f 3 , where c, d, f > 0.

Katherine has an estimated Cobb-Douglas utility function of U = q1^0.25q2^0.75 for food, q1, and housing,...

Katherine has an estimated Cobb-Douglas utility function of U = q1^0.25q2^0.75 for food, q1, and housing, q2. The price for food is arbitrarily set at $1 per unit and the average monthly rent near the University of Chicago , p2, is a dollar fifty per square foot. Caroline, like the average University of Chicago student spend $750 on food and housing per month. (a) Using calculus, solve for Katherine's optimal quantities of housing and food. Provide the marginal utility...

Marvin has a Cobb-Douglas utility function, U=q10.5q20.5 his income is Y=$900, and initially he faces prices...

Marvin has a Cobb-Douglas utility function, U=q10.5q20.5 his income is Y=$900, and initially he faces prices of p1=$44 and p2=$2. If p1 increases from $4 to $5, what are his compensating variation (CV), change in consumer surplus (ΔCS), and equivalent variation (EV)? Marvin's compensating variation (CV) is $nothing. (Enter your response rounded to two decimal places and include a minus sign if necessary.) Marvin's change in consumer surplus (Upper DeltaΔCS) is $nothing. (Enter your response rounded to two decimal places...

Consider the Cobb-Douglas production function ?=??^??^??^? where ?, ?, ?, ? are positive constants and ?+?+?<1....

Consider the Cobb-Douglas production function ?=??^??^??^? where ?, ?, ?, ? are positive constants and ?+?+?<1. Let ? be the amount of labor, ? the amount of capital, and ? be the amount of other materials used. Let the profit function be defined by ?=?−(??+??+??) where the costs of labor, capital, and other materials are, respectively, ?, ?, and ?. Determine whether second order conditions for profit maximization hold, when the profit function is defined by ?=?−(30?+20?+10?) with ?=5?^0.3?^0.4?^0.2.

Suppose that the utility function of a consumer is U(x,y) = x ¼y ¾, where x...

Suppose that the utility function of a consumer is U(x,y) = x ¼y ¾, where x and y are the quantities of the good X and good Y consumed, respectively. The consumer's income is 400. (a) What is the demanded bundle when the price of good X is 10 and the price of good Y is 10? (b) Redo part (a) when the price of good X is doubled? (c) Redo part (a) when the price of good Y is...

Given Barbara's estimated Cobb-Douglas utility function, U(q 1, q 2) = q 1^0.2 q 2^0.8, for...

Given Barbara's estimated Cobb-Douglas utility function, U(q 1, q 2) = q 1^0.2 q 2^0.8, for CDs, q 1, and DVDs, q 2, derive her Engel curve for movie DVDs. Illustrate in a figure. Let p be the price of DVDs, $1.00 be the price of CDs, and Y be income. Barbara's Engel curve for movie DVDs is Y= ? (Please provide answer) (Properly format your expression using the tools in the palette. Hover over tools to see keyboard...

Question