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Explain the technique of constrained optimization in relation to utility functions and how it relates to consumer equilibrium.
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- The utility function of a certain good is given by U(x) = 60x - x2 where x represents the number of units produced and sold during a month.How many units must be produced and sold to maximize utility? What is the maximum utility?If u(x, y) is a utility function, then v(u) = −u is a function that qualifies monotonic transformation. Do you agree with this statement? Explain!For a demand function Qd = f (P), a change in price causes a change in the quantity demanded. There are other five (5) variables, enumerate and explain each variable influencing demand in the general demand function. These are fixed at specific values for a given demand equation.
- Suppose that the utility derived by the consumer from x units of the first commodity and y units of the second commodity is given by the utility function U(x, y) = 10x0.6y0.4 The first commodity costs $ 20 per unit and the second $ 30 per unit. Both x and y are continuous variables. a) Find the function equation for the cost function C(x, y). Draw a contour plot of C (x, y) showing the contour lines of levels 500, 600 and 700. b) Add a contour plot of U(x, y) to this graph in which you draw contour lines of levels 100, 150 and 200. You are allowed to use the graphical calculator to graph a function from its explicit function equation. c) The consumer wants to maximize utility with a budget of $ 600. Indicate the correct option below: The maximum utility is below 100. The maximum utility is between 100 and 150. The maximum utility is larger than 150. Explain your answer based on the graph you made so far; d) Estimate how many units of each commodity the consumer should buy to minimize…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)…For each of the following utility functions get the marginal utility of consumption of each of the goods. 1/2 1/2 (a) u(a,b) = a b (b) u(x, y) = x³/4y1/4 (c) u(a, b) = n(a) + In(x) (d) u(x, y) = ln(x) + ln(y) (e) u(a, b) = 2xa +Xb
- 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.Suppose Vivi’s utility function over tea and crumpets are given by U(T, C) = min(T, 3C). (a) Graphically derive Vivi’s demand for tea when her income is $60 and the price of crumpets is $2. Do this by showing her optimal consumption (exact numbers) at three different values for the price of tea. (b) What is the formula for Vivi’s demand function for crumpets when her income is $60, the price of crumpets is pC, and the price of tea is pT ? Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.Consider the utility function: U(x,y)= -4(3x-12)² - (2y-10)² Determine the marginal utility for the two commodities
- buyers in the chocolate market demand 200 units of chocolate when price is zero and decrease the purchase of chocolates by 2 units for every one unit increase in price. write down the equation of the demand function in the form of Q= f(P)The demand function for potatoes per month is given by Q = 104 – 40p + 20pt + 0.01Y, where Q = the quantity of potatoes, p = price of potatoes, pt = price of tomatoes, and Y = income per month. Pt is currently $0.80 per pound). Derive the demand curve for potatoes from the data provided when income is $4,000 per month. Derive the new demand curve when income increases to $5,000 per month. ON a graph show the change in the demand curve resulting from the growth in income.Utility is a type of function that occurs in economics. When a consumer receives x units of a certain product, a certain amount of pleasure, or utility U, is derived. The graph represents a typical utility function. a) Find the average rate of change of U as x changes from 0 to 1; from 1 to 2; from 2 to 3; from 3 to 4. b) Why do the average rates of change decrease as x increases? a) The average rate of change of U as x changes from 0 to 1 is. (Type an integer or a decimal.) C Utility (pleasure units) AU 200- 150- 100- 50- 0 (1,67) (2,106) I (3,134) (4,156) 2 Number of units of product 4 X