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Q: Consider the following utility functions. G(x,y) = −1 / [ min(6x,2y)+1] H(x,y) = min(6x,6y) L(x,y) =…
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A:
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- 2. Consider an individual who not only consumes goods x and y, but also needs a mini- mum amount of x = xo to survive. Her (Stone-Geary) utility function is given by U(x, y) = (x — xo)¹/²y¹/² Let prices be pa, Py and income is I > Pro. (a) Show that this individual maximizes utility by spending (I-Pro) + Pro on good rand(I - Pro) on good y. Provide an economic interpretation for this result. [Hint: It might help to define z = x - xo and I = I - Pro. Then find the optimal z (and therefore x = z + xo) and y]1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.Adrienne consumes three goods, x, y, and z. Her consumption preference is given by the utility function: u(x, y, z) = x*y} z#, (Pz-Py, Pz) where x, y, and z are the quantities she consumes (respectively). Let p = be the prices of the goods. Answer the following questions, and justify your answers carefully and mathematically. (a) Last year p = sumption bundle last year. (1, 10, 10) and Adrienne's income was 35. Find her optimal con- (b) This year, the prices of x and z have both risen 3.5% while the price of y has risen 7%, and Adrienne got an income raise of 3%. Find the (approximate) percentage changes of her consumption in the three goods due to these changes in prices and income. Does her raise adequately compensate her for the changed prices? What additional percentage change of her income would be needed if we were to restore Adrienne to last year's exact utility level?
- VI. Utility Maximization 3. Consider an individual with the following utility function, U(x, x2) = ax5x95. The current price %3D of x, is 2 and the current price of x, is 4. 1. What is this consumer's budget constraint? 2. Calculate the optimal consumption of x, and x, if the consumer has income m.Question 2 David spends his budget on chocolate and chip. His utility function is given by ?(?1,?2)= 2?1?2, where ?1 is the number of chocolates he consumers per week, and ?2 is the number of chips he buys per week. A chocolate costs 10 SEK, and a chip costs 20 SEK. David’s weekly budge for consuming on these two goods is 120 SEK. (1) What is David’s budge line? Draw the budget line on a graph with chocolate amounts on the horizontal axis and chip amounts on the vertical axis. Write explicitly at which points budget line crosses the axis. (2) What is David’s marginal utilities for the two goods, respectively? What is his marginal rate of substitution between the two goods? (3) What is David’s optimal choice? Calculate the numerical answer for the optimal bundle. Also draw an indifference curve for David on the same graph as question(1) and show the optimal bundle.0.8 A consumer's utility function is given by the expression: U = (0.6X0.5 +0.4Y0.5) ². Determine the marginal utility functions for each commodity. Does marginal utility decrease when consumption increases? Assuming that the price of good X is Rs 15 and the price of Y is Rs 6, write the equation of the budget line and plot it when income is Rs 450. What is its slope? What does it indicate? Calculate the marginal rate of substitution of Y for X and interpret its economic meaning. Write the equation showing consumer's equilibrium condition. Obtain the equilibrium values of X and Y. Find the expressions for change in MUx due to increase in Y and change in MUy due to increase in X.
- QUESTION 1 For the utility function U = Qx0.50Qy(1-0.50) and the budget 122 = 8Qx + 14Qy find the CHANGE in optimal consumption of Y if the price of Xincreases by a factor of 1.7. Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 1.2, 1.14 = 1.1).please answer (a)A consumer with income I=120 facing prices pX = 4 and pY = 8 for two goods X and Y (for each good she prefers more to less) chooses optimally to consume 12 units of X. If the prices change and now pX = 6 and pY = 4, what is the possible range for her optimal X consumption? (like, x >/ 7 or 10 >/ x >/7…etc. Use indifference curve analysis on a graph to reason about the possible locations of the new optimal bundle.)
- 2 COOKIES AND TEA Consider a consumer, Jennie, who only consumes tea (t) and cookies (c). Jennie's utility func- tion is U(t, c) = min{5t, c}. The min function gives you the smaller of the two arguments.¹ For example, the min{1,3} = 1 since 1 is less than 3. She has $10 to spend, the price of a cup of tea is normalized to $1, and the price per cookie is $2. Assume she can buy tea and cookies in fractions. a. Graph the set of the tea and cookie pairs that yield the utility level of 5 with tea on the x-axis and cookies on the y-axis. Now graph an indifference curve associated with the level of utility equal to 7. b. Given the market prices that Jennie faces and her income, how much tea and how many cookies will Jennie choose to consume? Show her optimal choice graphically with cookies on the x-axis and tea on the y-axis. [Note: You cannot use the standard Lagrangian technique here because the slope of the indifference curve is not defined at the kink. Try working with the graph of the…What happens to the original budget line if the income of the consumer falls by 25%? (i.e., Will there be a change in the slope and/or intercepts of the original budget line? If yes, what are those changes?) Draw the new budget line after the fall in income.Suppose a consumer has utility function U(X, Y) = X2/3Y1/3. Suppose the consumer has M =$90 to spend and the prices of goods X and Y are PX = $15 and PY = $6 . a. Carefully express the consumer’s choice problem, using the given information (this is whereyou write out the max operator, the choice variables, the objective function, and the budget constraint). b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it todetermine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape.To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether|MRS(X, Y)| is diminishing along an indifference curve. c. If the indifference curves are C-shaped write out the budget line and the equal slopes conditionthat characterize an interior solution to the consumer’s choice problem. Use the particulars for the givenconsumer. Solve these conditions to find the interior solution. On the…
