Microeconomic Theory
12th Edition
ISBN: 9781337517942
Author: NICHOLSON
Publisher: Cengage
expand_more
expand_more
format_list_bulleted
Question
error_outline
This textbook solution is under construction.
Students have asked these similar questions
Consider a consumer with the utility function U (x1, x2 ) = 10x12/3x21/3 −50. Suppose the prices of x1 and x2 are 10 and 2 respectively and the consumer has an income of 150.
(a) Write out the consumer’s constrained optimization problem. Specifically, write out the objective function and constraint for the problem (e.g. max _?_ subject to _?__).
(b) Write the Lagrangian equation corresponding to the constrained optimization problem. Derive the Necessary First Order Conditions.
(c) Use the NFOCs to solve for the consumer’s optimal bundle.
(d) Show that at the solution you found in (c), the tangency condition is satisfied: MRS = p1 / p2.
(e) How did the ‘50’ in the utility function influence the optimal con- sumption bundle? How did the ‘10’ in the utility function influence the optimal consumption bundle? (i.e., how would the optimal bun- dle change if these coefficients were to change?). How would the optimal bundle change if the utility function was x12x2? Lastly, how would…
Answer d and e.
Consider a consumer with the utility function U (x1, x2 ) = 10x12/3x21/3 −50. Suppose the prices of x1 and x2 are 10 and 2 respectively and the consumer has an income of 150.
(a) Write out the consumer’s constrained optimization problem. Specifically, write out the objective function and constraint for the problem (e.g. max _?_ subject to _?__).
(b) Write the Lagrangian equation corresponding to the constrained optimization problem. Derive the Necessary First Order Conditions.
(c) Use the NFOCs to solve for the consumer’s optimal bundle.
(d) Show that at the solution you found in (c), the tangency condition is satisfied: MRS = p1 / p2.
(e) How did the ‘50’ in the utility function influence the optimal con- sumption bundle? How did the ‘10’ in the utility function influence the optimal consumption bundle? (i.e., how would the optimal bun- dle change if these coefficients were to change?). How would the optimal bundle change if the utility function was x12x2? Lastly, how…
Consider a consumer’s constrained optimization problem. max u(n,y)=4n+3y subject to wn+y=24w+Yo Where N = non-work (leisure) hours, Y = total income (consumption), w = the wage rate and H = hours worked with N + H = 24. a. Assuming an interior solution, use the first order conditions from the Lagrangean to solve for the utility maximizing choice of N* and Y*. How many hours, H*, does this consumer work? (Any of these answers might be functions of w.) when w=1 and Yo=50. No hand written solution and no image
Knowledge Booster
Similar questions
- The consumer choice is not restricted to the choice of consumptiongoods. In fact, it can apply to all our decisions that involve trade-offs. Suppose Mary has awage per hour of 10 euros. With her earned income she consumes. That isC=wH per day.She also decides how many hours to work of take leisure time each day.H=24-N, whereHis work and N is leisure. Her utility is given by (picture) Solve for the optimal decision of labor/leisure. Plot the budget constraint and the indif-ferent curve. What is the labor supply function?arrow_forwardA stadium can seat 60,000 people. Assume 35% of seats are not occupied (not for sale) because of Covid-19 and social distancing requirements. You are told that the current price of tickets is $70 and that the demand is linear and that the demand function (Qd) = 60,000 - 250P. examine the market for tickets for games at the stadium during these Covid times. You should Illustrate themarket for tickets using a demand and supply diagram. Show all your calculations and properly label the diagram. At $70 a ticket the cap on the number of seats available (because of Covid restrictions) is creating a shortage. How many people miss out on tickets when they are priced at $70 per ticket?arrow_forwardTrue/False/Uncertain: 1. In taking an exam, Atack, a rational student, allocates his time to the various questions so as to equalize hismarginal point utility per minute on all questions. 2. The marginal utility of food to Zecher depends only on the amount of food (and not on the amount ofhousing) and the marginal utility declines as more food is consumed; likewise for housing. Therefore,both food and housing are normal goods. (Hint: Express the optimality condition for Zecher’s [UMP], MUF/MUH = PF/PH. Notice that PF/PH is fixed. If all of an increase in income is spent on F, can the equality be maintained?)arrow_forward
- Consider Alexei who makes $1,400 per week and just won a ‘set for life’ lottery ticket whichinvolves a fortnightly payment of $10,000 for 20 years. Currently he works 35 hours per week. Nowdo the following:• Illustrate the effect of the lottery win on Alexei’s budget constraint using a fully labelleddiagram where i) the horizontal axis represents the hours of free time per week, and ii) thevertical axis represents Alexei’s weekly consumption.• Illustrate on the same diagram Alexei’s optimal decisions before and after the lottery winalong with his indifference curves.arrow_forwardDefine a weakly concave on the Cobb-Douglas Functionarrow_forwardHello! The problem states: Given the estimated function, Equation 2.2, for coffee, Q=8.56-p-0.3p(s)+0.1Y, use algebra (or calculus) to show how the demand curve shifts as per capita income, Y, increases by $10,000 a year. Illustrate this shift in a diagram. Other parts of the chapter state $0.20/lb as the substituted value for p(s) for example problems. My question is: I need a value for p(s) (which is the price of sugar) to solve for this right? The problem doesn't explicitly state a value for me to input for p(s), but I am thinking to use 0.20 for p(s) to simplify the demand function. Thanks in advance!arrow_forward
- Anna has endowment 1500 now and 500 later. Internet rate is 2.0%. She prefers smooth consumption to time (i.e., u0=u1=u). a. Assume utility function, u(c)= log c. What are the optimal consumption c0and c1if Anna's beta=1, and she wants to maximize her utility? b. Now assume that the utility function, u(c)=c0.5. If everything else remains the same as Problem 1(a), what are the optimal consumption c0and c1if Anna wants to maximize her utility?arrow_forwardA penurious graduate student has a food budget of $100.00/week. To survive with sufficient energy to attend classes, he knows that he needs to consume 50 protein units per week. The only two foods he can stand to eat on a regular basis are beans and hamburger. He derives twice as ich pleasure per protein unit from eating hamburger as he does from beans. a. Assume that hamburger costs $3.00 per protein unit and beans cost $1.00 per protein unit. Formulate the student's diet problem as a linear program. (You can assume he wants to maximize his "total utility" from his diet and that he gets I utile from each protein unit of beans he consumes and 2 utiles from each pro- tein unit of hamburger.) What is the optimal consumption of beans and ham- burger in this case? b. Plot the student's price-response curve for beans as the price of beans goes from $0.01 to $2.00, assuming that everything else (including the price of hamburger) stays constant. Note that his individual price-response…arrow_forwardCan you please solve 8 e knowing that the awnser to 8 d is found written below. Please also see attached question Awnser to 8d: U = (X1)^(1/2)*(X2)^(1/3) We have budget constraint as, m = P1*X1+P2*X2 Let's set up the Lagrange maximizing function as below. L = U(X1, X2) - λλ(P1X1+P2X2-m) L = (X1)^(1/2)*(X2)^(1/3)- λλ(P1X1+P2X2-m) Finding first order condition, dL/dX1 = (0.5)*(X2)^(1/3)/(X1)^(1/2)-λλP1 dL/dX2 = (1/3)*(X1)^(1/2)/(X2)^(2/3)-λλP2 Equating these equations to zero to get, (0.5)*(X2)^(1/3)/(X1)^(1/2)-λλP1 = 0 (1/3)*(X1)^(1/2)/(X2)^(2/3)-λλP2 = 0 (0.5)*(X2)^(1/3)/(X1)^(1/2)/P1 = (1/3)*(X1)^(1/2)/(X2)^(2/3)/P2 (3/2)*(X2/X1) = P1/P2 X1 = (1.5P2*X2)/P1 dL/dλλ = P1X1+P2X2-m=0 P1X1+P2X2 = m P1*(1.5P2*X2)/P1+P2X2=m 2.5*P2X2=m X2 = 0.4*m/P2 X1 = 0.6*m/P1 Hence, Marshallian demand function for good q and good 2 respectively are X1 = 0.4*m/P1 X2 = 0.6*m/P2arrow_forward
- (a) Dividing MUX and MUY by their respective prices compute the weighted marginal utility or marginal utility of money expenditure and draw up Table 2 showing diminishing returns for the consumption of the 6 units? (b) Briefly state what you can infer on this equation; MUX/PX = MUY/PY as far as equi marginal principle and managerial economics is concerned? C)Given that MUX/PX and MUY/PY are equal to 6 when 5 units of X and 3 units of Y are purchased. By purchasing these combinations of X and Y, calculate the amount the consumer will spend and derive the maximum satisfaction from combination of these units. (d) Applying the above principle illustrate in terms of a figure, and draw marginal utility curves for goods X and Y. You need to use marginal utility and price i.e. Marginal utility per Kwacha spent on good X = MUX/PX, and that of Y = MUY/PY. The MUX/PX curve should be shown in figure (a) while the MUY/PY curve should be shown in Fig (b). Please do not draw negative…arrow_forwarda) You are given a utility function described as:U= U(K, L, M) = 10LogK + SLogL. + 2LogMWhere K, L and M are goods consumed. The individual has K100 to spend on the threeitems. Furthermore, you are informed that monthly salary is $500 and the prices of K, Land M are $2, $10 and $4 respectively. Calculate the consumer's optimal bundle.D.b) Provided a demand curve for Beans is as followsO = 4000 - 255PWhere Q is the quantity demanded and P is the price. Find the elasticity at K7, K10 and K15.arrow_forwardYou decide to sell pairs of running shoes. Assume that the quantity demanded is alinear function of the price. It is known that if you charge $40 per pair of shoes, thenyou will be able to sell 400 pairs of shoes in a month. If you charge $60 per pair ofrunning shos, then you can sell 300 pairs of shoes in a month. It is also known thatyour monthly fixed cost is $100 and pairs of shoes cost $20 per shoe to produce. Findwhat price you should charge for a pair of shoes in order to maximize profit in amontharrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you