We know that n differentiable functions x₁(p, m), ..., xn(p,m) of prices and income are generated by some utility maximizing consumer if and only if each of them is homogeneous of degree zero in prices and income, i.e. Σ¿pi xi(p, m)=m, and whose matrix of substitution terms (the Slutsky matrix) is negative semi-definite. This makes Afriat's Theorem and GARP unnecessary Comment.
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- Derive Ryan's demand function for q₁, given his utility function is where o = = (9₁) P + (9₂)P, 1 1-p The demand curve for q₁ as a function of P₁, P2, and Y is (Properly format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ 9₁ = character.) U= Let the price of q₁ be p₁, let the price of q2 be p2, and let income be Y.An individual’s utility function is given by where is the amount of leisure measured in hours per week and is income earned measured in cedis per week. Determine the value of the marginal utilities, when = 138 and = 500. Hence estimate the change in utility if the individual works for an extra hour, which increases earned income by GH¢15 per week. Does the law of diminishing utility hold for this function?The utility that Elena receives by consuming food F and clothing C is given by U(F,C) = FC + F. Food costs £1 per unit, and clothing costs £2 per unit. Elena's income is£22. MUF = C+1 and MUc = F.(i) Define the term Marginal Rate of Substitution (MRS). (ii) Find the utility maximizing values of C and F for the numerical values given above(Hint: You can assume interior solution). (ili) Now suppose that price of C and Elena's income remain unchanged at £2 per unitand £22, respectively, while the price of F varies. Find the equation of the demand forF as a function of the unit price of F (PF).
- Now assume that the utility function takes the function form u(c) = ln(c). Note that the derivative of the natural logarthm is u'(c) = 1/c .The marginal utilities derived from the consumption of goods X and Y by a given consumer are given by MUx=40-6x and MUy=80-10y. If the prices of X and Y are Gh2 and Gh10 respectfully. determine the number of units of the commodities that the consumer must consume to optimize satisfaction granted that he/she earns an income of GH40 within the period.A utility function is called separable if it can be written as U(x, y) = U (x) + U,(y), where U > 0, U" < 0, and U1, U, need not be the same function. a. What does separability assume about the cross-partial derivative U„? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to con- clude definitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.
- 1. For each of the following utility function, compute the Hicksian demand function h(p, v), the expenditure function e(p, v) and the Slutsky (Substitution) matrix S(p, w). In your com- putation procedure, feel free to use your (or my) answers in Assignment 5. 3 a) u(*1, 12) = VI, + ¤2 b) u(x1, 12) = log(1) +3 log r2 c) u(r1, 12) = min(r1, 12) d) u(x1,r2) = max(r1, r2) %3D %3D %3D %3DAyana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain.Ayana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain. (b) Calculate consumers’ income elasticity as a function of the parameters. (c) Anaya is targeting a market whose income is expected to double over the next five years.Explain whether she should expect the demand for chargers to increase or decrease, andby how much, using the language of normal, inferior, luxury, and necessity goods. (d) Calculate the cross-price elasticity ofCwith respect toPF. (e)…
- A consumer's utility function is given by the expression: U = (0.6.x"s + 0.4Y"s)*. %3D • 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.a good is normal, then an increase in the price of the good will lead to which of the following to be true for this good? (Assume that there are only two goods, the individual's preferences lead to well-behaved preferences with strictly convex indifference curves and an interior solution for all budgets). Let SE = substitution effect, IE = income effect) (a) The magnitude of the IE for this good must be larger than the magnitude of the SE (b) The magnitude of the SE for this good must be larger than the magnitude of the IE (c) The good could be a Giffen good d) The good must be an ordinary good ( (e) None of the aboveFi Consider the utility function: u(x1,x2) = Axf x}-« (-a where 0 0. (a) Compute the Marshallian demand functions. (b) Derive the indirect utility function. O Type here to :