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Assume a consumer has a utility function U(x1, x2) = x1x2 where x1, x2 represents the amount of two goods 1 and 2 consumed in a given time period, find the utility-maximizing consumption function subject to the budget constraint
5x1 + 4x2 <= 50
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- The utility function of a consumer is u = √x + 2y. (a) Calculate how much MRS of that consumer when x = 4. (b) If the consumer currently consumes 4 units of goods x and the price ratio of Px / Py = 1/4, then the consumer should increase or decrease the consumption of goods x?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 +XbA consumer has the following utility function: U(X,Y)= 400X2/5Y3/5 Suppose that the price of a spa day (x) is £200 and the price of a city break (y) is £300 Find the number of spa days and city breaks that minimise expenditure if 12,800 units of utility are obtained. Also calculate the consumer's minimised expenditure
- Find marginal utility MU for a consumer consuming x amount of a given commodity, where his total utility function is TU = x²ln (x² + 1).3) The utility function of a consumer who consumes quantities x and y of two goods is defined by the expression U(x, y)=√xy subject to the constraint 7x+3y=84 where $84 is the consumer's overall budget. Assuming marginal utilities U,,U>0, maximize the utility function of the consumer. What are the optimal values of x, y and U(x,y) ?Question 1 Consider a consumer with a utility function of U(X,,X2, ... , Xn) = II-1 X , a; > 0, i = 1,2,3, ...,n. P; is a price of 1 unit of the i-th good. Assuming that the consumer has the total budget of $M, derive his optimal consumption bundles in the following procedure. (a) Set up the consumer's utility maximization program with a Lagrangean multiplier. (b) What is an economic intuition of the Lagrangean multiplier? (c) Assuming an interior solution, get all the FOCS. (d) Find an optimal consumption bundle while specifying each step required to get the solution. (e) Characterize the maximized utility as a function of $M and the parameters. What does it look like? (f) Find an exact value of the Lagrangean multiplier.
- For the utility function U=Qx0.370,(1-0.37) and the budget 194 140x+6Qy find the CHANGE in optimal consumption of X if income increases by a factor of 1.4. Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 1.2, 1.14=1.1).A consumer has utility function U(x1, X2, X3, X4, X5, X6, X7) = min{/x1x2, /X3X4, VX5X6X7} Find the demand function. 3)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?
- A 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.A consumer is maximising her utility function: U(r. y) = (r05 +y0.5)2, subject to the budget constraint 4x + 2y = 108. (a) Set up the Lagrangian function of this utility maximisation problem and first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the multiplier. (c) What are the utility maximising amounts of x and y if the budget constrain to x+ 2y = 36? How would change? Explain your reasoning. (Hint: Yc need to calculate A, rather comment on how it would change and why.)A consumer has a utility function U (C₁, C₂) and her expenditure function is E = P₁C₁ + P₂C₂ where the prices of goods one and two are p₁ and p2 respectively. She wants to find the minimum level of expenditure, E* (P₁, P2, U), that allows her to achieve a certain level of utility U, i.e., so that U(C₁, C₂) ≥ U. Suppose that U(C₁, C₂) = (c₁)^(1/2)+ a(c₂)^(1/2) a) will constraint U(C₁, C₂) ≥ U bind. And what is the langranian for this problem B)Write down the Lagrangian conditions and find the ratio C1/C2 in terms a,p1or p2 C)Assume that a=1,p1 = pand p2 =1.Find C1 and C2. Substituting these values into the expenditure function, find E*(p, Ū), the minimum level of expenditure required to achieve the required level of utility Ū. Use this to prove thatdE*/dp = C1