1.1 Derive the marginal utility of good X (MUx) and marginal utility of good Y (MUy) from the utility function: U = f(x; y) = x²y³ - 10x using partial differential calculus. 1.2 Derive the MP₁ and MPK equations, if TP = f (L; K) = 10L² +20L²K² +3K² using partial differential calculus.
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- Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…A consumer can consume two goods, A and B, and has the utility function U-15A1/281/2, The consumer's budget is $900, the price of good A is $15 per unit and the price of good B is $45 per unit. (Assume A is the horizontal axis good and B is the vertical axis good. Both goods are infinitely divisible, but round numerical answers to 2 decimal places as necessary) What is the consumer's marginal rate of substitution? What is the consumer's marginal rate of transformation? What is the formula for the consumer's budget constraint? What is the consumer's utility-maximizing bundle given the utility function and budget constraint? A B H4.13 CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by U(x, y) = (x° +y®)'/8 [in this function the elasticity of substitution o = 1/(1 – 6)]. a. Show that the indirect utility function for the utility function just given is V = I(p, + p,)¬/", where r = 8/(ò – 1) = 1 – 0. b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by E = V(p', + p,)''". f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.
- 1) Suppose that a person consumes two goods, x and y, in fixed proportions. He or she always consumes 1 unit of x together with 3 units of y no matter what the relative prices are.a) What is the mathematical form for this person's utility function?b) Calculate the Marshallian demand functions for both goods for this person.c) Calculate the indirect utility function and the expenditure function for this person.d) In class we discussed why expenditure functions are concave in prices. Is the expenditure function you calculated in part (c) concave in ???= Suppose you are trying to minimize your expenditures E utility level (20.5 + x9.5)² = 175. 4x1 + 3x2 subject to the target maximum • Set up the Lagrangean function for this expenditure-minimization problem with constraint • Compute for the expenditure minimizing quantities of ₁ and ₂. x2. • What is the minimum expenditure you can incur given the level of utility you have to attain?Summer break is approaching! Suppose you derive utility from days spent traveling on vacation domestically, D, and days spent traveling on vacation in a foreign country, F. Your utility function over these two “goods” is: U(D, F) : 4D0.25F 0.75 Let your budget constraint be I(D, F) = pDD + pF F; where I is your income, pD is the price of domestic travel per day, and pF is the price of foreign travel per day. a. Determine the demand functions for domestic travel and foreign travel. Make sure you show your work – show the steps used. b. Suppose that you’ve saved $800 for your summer travel, the price of domestic travel per day is $25 and the price of foreign travel per day is $100. How many days of each type of travel will you embark on? c. Illustrate the indifference curve, budget constraint, and the utility maximizing bundle associated with (b). Make sure you show the level of utility, the budget constraint intercepts, and the optimizing equilibrium.
- 1 Consumption utility functions Consider the following utility functions defined over the outcome space X = R². 1. U(1, ₂) = 1X2 2. U(1, ₂) = 2x₁ + x₂ 3. U(1, 2) = min{2x1,4x2} 4. U(₁, ₂) = 100 - (₁ - 10)²(x₂ - 10)² 5. U(x₁, x₂) = 1 + ln(x₂ + 1) 6. U(x₁, x₂) = x² + x² For each function: (a) Draw two indifference curves and an arrow indicating the direction of increasing prefer- ence. (b) Shade the upper contour set above the more-preferred indifference curve you drew in the previous part. (c) Indicate which of the following properties are satisfied by the underlying preference relation: convexity, monotonicity, local nonsatiation. (A given preference may satisfy more than one!) Explain why each property does or does not hold.If a and ß are just parameters and if the direct utility function is u (x1, X2) = a · In(x1) + ß · In(x2) %3D then what is the marginal utility of x1 and what is the marginal utility of x2 ? Demonstrate whether these marginal utilities are increasing, diminishing, or constant. Assume that the following indifference curve belongs to the direct utility function specified above. u - 1.565 G (5, 4) 4 Н (10, 0.25) 0.25 5 10 1 What is the value for M RS12 evaluated at bundle G(5,4) if a = 0.8 and B = 0.2 ? What is the value for M RS12 evaluated at bundle H(10, 0.25) ? Discuss what these calculated values for M RS12 imply regarding whether preferences are convex or not (regardless of what the figure depicts above). Show your work!(a) Suppose the indirect utility function v(p,m) is differentiable. Prove that dv(p, m) dv(p,m) Pi + m- дт = 0. (b) Consider a differentiable quasi-lincar utility function u(x) = x +(x2,X3), where x, can take on any value (including a negative value). Use the first-order conditions of the UMP and EMP for an interior solution to prove that dx,(p, m)/ap; = ah;(p, u)/dp; for i = 2 and 3. (c) Prove that the profit function (p, w) is convex in input prices w. ide
- 3. Consider the following utility function, u (21, x2) = min V#1, Varz), where a > 0 Derive the Marshallian demand functions. (Explain your derivation in details.) Does the Marshallian demand increase with price? Are the two consumption goods normal goods? Show two different ways to derive the Hicksian demand functions. (b) Does the Hicksian demand increase with price?Part 1: Illustrate general equilibrium and the Laffer curve in the context of a repre- sentative consumer with a utility function: U(C.I) = In(C) + In() that he or she maximises subject to a constraint: C= w(1 – t)(h – 1) + * where w, h,1, C,t and a are wages, hours of time available, leisure, consumption, tax rate, and dividend income. The production function for this economy is given by Y = C+G = A(h - 1)/2 Assume that h = 1, A =1 and that the government has a balanced budget. (a) Find the equilibrium by matching the Marginal Rate of Substitution to the Marginal Rate of Transformation and then substitute into the constraint. Also take into account that profits are non-zero for this setup. (b) Plot the government tax revenue for 0Part 1: Illustrate general equilibrium and the Laffer curve in the context of a repre- sentative consumer with a utility function: U(C,1) = In(C) + In(1) that he or she maximises subject to a constraint: C= w(1- t)(h – 1) + * where w, h,1, C, t and r are wages, hours of time available, leisure, eonsumption, tax rate, and dividend income. The production function for this economy is given by Y = C+G = A(h – 1)/2 Assume that h = 1, A = 1 and that the government has a balanced budget. (a) Find the equilibrium by matching the Marginal Rate of Substitution to the Marginal Rate of Transformation and then substitute into the constraint. Also take into account that profits are non-zero for this setup. (b) Plot the government tax revenue for 0SEE MORE QUESTIONSRecommended textbooks for youPrinciples of Economics (12th Edition)EconomicsISBN:9780134078779Author:Karl E. Case, Ray C. Fair, Sharon E. OsterPublisher:PEARSONEngineering Economy (17th Edition)EconomicsISBN:9780134870069Author:William G. Sullivan, Elin M. Wicks, C. Patrick KoellingPublisher:PEARSONPrinciples of Economics (MindTap Course List)EconomicsISBN:9781305585126Author:N. Gregory MankiwPublisher:Cengage LearningManagerial Economics: A Problem Solving ApproachEconomicsISBN:9781337106665Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike ShorPublisher:Cengage LearningManagerial Economics & Business Strategy (Mcgraw-…EconomicsISBN:9781259290619Author:Michael Baye, Jeff PrincePublisher:McGraw-Hill EducationPrinciples of Economics (12th Edition)EconomicsISBN:9780134078779Author:Karl E. Case, Ray C. Fair, Sharon E. OsterPublisher:PEARSONEngineering Economy (17th Edition)EconomicsISBN:9780134870069Author:William G. Sullivan, Elin M. Wicks, C. Patrick KoellingPublisher:PEARSONPrinciples of Economics (MindTap Course List)EconomicsISBN:9781305585126Author:N. Gregory MankiwPublisher:Cengage LearningManagerial Economics: A Problem Solving ApproachEconomicsISBN:9781337106665Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike ShorPublisher:Cengage LearningManagerial Economics & Business Strategy (Mcgraw-…EconomicsISBN:9781259290619Author:Michael Baye, Jeff PrincePublisher:McGraw-Hill Education