Caleb consumes only two goods, X and Y, and faces the following utility function: U=XY. His initial budget is $800, and the prices of X and Y are $12.5 and $2.
What is the marginal utility for X?
What is the marginal utility for Y?
**Most answers should be round numbers. Answer everything to 1 decimal place, if need be**
What are the amounts of X and Y that will maximize Caleb's utility?
X =
Y =
How many X and Y will Caleb choose if the price of X suddenly fell to $8?
X =
Y =
How many X and Y are in the substitution bundle?
X =
Y =
What are the break-downs of the price change on the amount of X purchased?
Substitution effect =
Income effect =
What is the compensating variation?
Solution:
Utility function: U = X*Y
Budget constraint: Px*X + Py*Y <= M ; where Px is price of good X, Py is price of good Y, and M is the income of the consumer.
So, using the given information, budget line becomes: 12.5*X + 2*Y = 800
Marginal utility of X, MUx = = Y
Marginal utility of Y, MUy = = X
Utility maximizing bundle is the one which satisfies the optimal condition of: MUx/MUy = Px/Py
So, Y/X = 12.5/2 or in other words, Y = 6.25*X
Substituting this in the budget line, we get 12.5*X + 2*6.25*X = 800
25*X = 800
X = 800/25 = 32 units
So, Y = 6.25*32 = 200 units
With change in price of good X:
Px'= $8
Again using the optimal condition above, we now have, Y/X = 8/2, so Y = 4*X
Substituting this in the new budget line (with changed prices): 8*X + 2*Y = 800
8*X + 2*4*X = 800
X = 800/16 = 50 units
So, Y = 4*50 = 200 units
Finding the substitution bundle:
Substitution bundle is the bundle which is attained at new prices, such that the utility remains unchanged.
Initial utility, U1 = 32*200 = 6400
At new prices, we know at optimum, Y = 4*X
Unchanged utility = 6400 = X*(4*X)
X2 = 6400/4 = 1600
X = 40 units
So, Y = 4*40 = 160 units
(NOTE that this is Hicksian substitution bundle, and not Slutsky substitution bundle, since nothing is specifically asked for. In case of otherwise, please let know in the comments)
Break-down of price change:
Substitution effect = Substitution bundle - old bundle
S.E. = (40, 160) - (32, 200) = (+8, -40), so with decrease in price of good X, under substitution demand for good X has increased.
Income effect = New bundle - substitution bundle
I.E. = (50, 200) - (40, 160) = (+10, +40)
So, for good Y, substitution and income effect cancel out each other, while for good X, both effects move in same direction (indicating that good X is a normal good).
Compensating variation is the amount of extra income (in case of price increase; in case of price decrease it is reduction in income) required to reach the same utility level as the old one. In other words, it is same as extra (or reduction in) income required to reach at substitution bundle from the old bundle.
Income required at substitution bundle (at new prices, of course) = 8*40 + 2*160 = $640
So, compensating variation = compensated income - actual income
C.V. = 640 - 800 = -$160
Caleb consumes only two goods, X and Y, and faces the following utility function: U=XY. His...
Marta consumes only goods X and Y and faces the following utility function: U=7 X+4 Y. The marginal utility for X is MUX=7 and the marginal utility for Y is MUY=4 . The price of X is $10 and the price of Y is $50. Marta has an initial budget of $200. How many of X and Y will Marta buy given her utility function, her budget, and the prices? X= Y= Suppose that the government places a restriction on X...
Imagine you consume two goods, X and Y, and your utility function is U = XY. Your budget is $100, the price of Good X is $4, and the price of Good Y is $25. So, the optimal bundle for you to consume is (12.5, 2). Now the price of good X increases to $10. The compensated price bundle is (7.91, 3.16). What is the income effect on X?
4. Andy's utility is represented by the function U(X,Y) - XY. His marginal utility of X is MUx = Y. His marginal utility of Y is MUY = . He has income $12. When the prices are Px - 1 and Py -1, Andy's optimal consumption bundle is X* -6 and Y' = 6. When the prices are Px = 1 and P, = 4, Andy's optimal consumption bundle is X** = 6 and Y* 1.5. Suppose the price of...
5. Douglas consumes two goods, x and y. His utility function is u(x) = Vx+y Let the price of good x be $2 and the price of good y be $2. Furthermore, assume that Douglas has $420.00 to spend on these two goods. Find the demand for good x and y. Now suppose that the price of good x decreases to $1.00. What is the income effect and substitution effect of this price change on the demand for x?
3. (10 points) Income and substitution effects A consumer's utility is given by U(x, y) = xy. Income is m and prices are given by p and Py (a) Find the demand functions for x and y. (b) What is demand for each good if p 2 and py 1 and income is m = (c) If price of x fell to pa 1, what is the consumer's new bundle? (d) How much of the response in the consumption of...
An individual’s utility is expressed by the function u(x,y) = xy The person’s income is ten dollars (I = $10) The price of item x is $1. The price of item y is $1. Maximize this consumer’s utility subject to a budget constraint using the Lagrange Multiplier method. At what point does the marginal rate of substitution equal the price ratio?
John has the following utility function that represents his preferences over food (x) and housing (y) (his only two expenses) and marginal utilities: มุ4 for a level of wealth W and prices of food and housing P y respectively. Using the results from the previous homework answer the following questions Write down the Engel Curve for both goods and graph them 2) Assume W-10 and the price of food changes from 1 to 3 while the price of housing remains...
Suppose that a consumer’s utility function is U=xy with MUx=y and MUy=x. Suppose the consumer‘s income is $480. For this question you may need to use the following approximations: sqrt(2) is approximately 1.4, sqrt(3) is approx. 1.7 and sqrt(5) is approx 2.2. a) Initially, the price of y is $4 and the price of x is $6. What is the consumer’s optimal bundle? b) What is the consumer's initial utility? Now suppose that price of x increases to $8 and...
Initial Price of X can be assumed to be $1 2. Justin ‘s utility function for two goods, x and y, is U(x, y) = xy . The marginal utility of good x is y + 1 and the marginal utility of y is x. Initially, the price of good y is S4 per unit. Justin's income is $100. (a) If the price of good x rises to S5 per unit, what is the income effect with respect to good...
06 Question (3 points) e See page 149 Douglas consumes two goods, x and y. His utility function is u(x, y) = (x + y. In the questions below, give your answers to two decimal places. 1st attempt Part 1 (1 point) See Hint Let the price of good x be $2 and the price of good y be $10. Furthermore, assume that Douglas has $360.00 to spend on these two goods. How much of good x does Douglas demand?...