Determine the maximum utility if the production function is P = 20 – x2 + 10x – 2y2 + 5y, the prices of inputs x and y are 2 and 1, respectively, and the unit price of product P is 5.
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unconstrained optimization application exercises
Determine the maximum utility if the production function is P = 20 – x2 + 10x – 2y2 + 5y, the prices of inputs x and y are 2 and 1, respectively, and the unit price of product P is 5.
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- Given U(c, t)=3c+4t expalin what the utility function says about the marginal rate of substitution.Find marginal utility for each consumption level for both products and please show your work in Table.Find marginal utility MU for a consumer consuming x amount of a given commodity, where his total utility function is TU = x²ln (x² + 1).
- Consider the utility function: U(x,y)= -4(3x-12)² - (2y-10)² Determine the marginal utility for the two commoditiesA utility function is given by the equation U = 20xe¬0.1x, where x is the number of glasses of wine consumed. (a) Show that this utility function has a maximum value and calculate the maximum utility. (b) Describe how marginal utility changes for glasses of wine consumed after the maximum utility is reached. Do you consider this reasonable? Give an explanation.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…
- Total Utility derived from the consumption of a product to a consumer is given by U = (20-0.50)'Q where Q is the number of units consumed Using a graphical representation, show what is the nature of variation in the utility with the increasing consumption of the product? Is it in line with the law of diminishing marginal utility? Explain.Explain the technique of constrained optimization in relation to utility functions and how it relates to consumer equilibrium.For the following utility function find the MRS at (x, y) = (2, 3) a) U(x, y) =2x +y b) U(x, y) =x^3 y^2 c) U(x, y) =min(x, 2y) Solve subparts (c )correctly
- The utility of a consumer derives from consuming the two goods X and Y can be assumed to be determined by the utility functionU(X,Y)= 4X03Y05. If good X costs 10 birr a unit and good Y costs 15 birr a unit and the consumer's income is 120 birr. What combination of X and Y that will maximize utility?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 +XbPlease solve all parts (d and e) Find the marginal rate of substitution of y for x [MRS(x,y)] for the following utility functions
