Problem 5.1 For the case of Cobb-Douglass utility (u(x, y) = xay with a, ß > 0), write the demand functions x(px, Py, I) and y(pa, Py, I). Then analyze dx/dI and Oy/I. Are both goods normal, or can one be inferior? (Assume pr>0 and py > 0.)
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- Suppose that a fast-food junkie derives utility from three goods-soft drinks (x), hamburgers (y), and ice cream sundaes (z)− according to the Cobb-Douglas utility function U(x,y,z)=x0.5y0.5(1+z)0.5. Suppose also that the prices for these goods are given by px=1,py=4, and pz=8 and that this consumer's income is given by I=8.a. Show that, for z=0, maximization of utility results in the same optimal choices as in Example 4.1 . Show also that any choice that results in z>0 (even for a fractionalz ) reduces utility from this optimum.b. How do you explain the fact that z=0 is optimal here?c. How high would this individual's income have to be for any z to be purchased?Problem 2: Satiation point(s) There two goods, candy and soda, available in arbitrary non-negative quan- tities (so the consumption set is R). A consumer has preferences over con- sumption bundles that are represented by the following utility function: u(x, y) = -|4 – x| – |4 – y| where x is the quantity of candy (in grams), y is the quantity of soda (in liters), and |.| denotes the absolute value: for any real number r E R, if r >0 |r| = if r 0 Dirhams. The price of candy is p > 0 Dirhams/gram, and the price of soda is q > 0 Dirhams/liter. (a) Calculate the utility of the following consumption bundles: (4, 4), (4,5), (5,4), (4, 3), (3,4), and a(4, 5) + (1 — а) (3,4) for a € [0, 1]. (b) In an appropriate diagram, illustrate the consumer's map of indiffer- ence curves. (c) Are the consumer's preferences monotone? Provide an explanation for your answer. (d) Find the demand for candy and soda as a function of wealth w > 0 for the following specific prices, explaining how you arrived at…Suppose that U(f,c) = f + 8c^(1/2)is a utility function that describes Amelia’s preferences over two goods: fish(f)and custard (c). For the following, think of fish as the good graphed on the horizontal axis.a. Derive an expression for her marginal utility (Uf)from a small increase in f holding c fixed. Also find themarginal utility for custard (Uc).b. What is Amelia’s marginal rate of substitution (MRS)? Give a brief (2 sentences maximum) intuitivedescription of what MRS represents. If Amelia has 4 units of custard, holding her utility constant, howmany units of custard would she be willing to give up in order to get one more unit of fish?c. Graph Amelia’s indifference curve for a utility level of 40. Be sure to specify at least 3 bundles of goodson the indifference curve.d. Does the fact that Amelia’s indifference curve intersects with the custard axis violate any of the 5properties of indifference curves? Briefly support your answer.e. Give another utility function that represents…
- 14.3 (0) Quasimodo consumes earplugs and other things. His utility function for earplugs x and money to spend on other goods y is given by u(x, y) = 100x 2:² 2 +y. (a) What kind of utility function does Quasimodo have? (b) What is his inverse demand curve for earplugs?. (c) If the price of earplugs is $50, how many earplugs will he consume? (d) If the price of earplugs is $80, how many earplugs will he consume? (e) Suppose that Quasimodo has $4,000 in total to spend a month. What is his total utility for earplugs and money to spend on other things if the price of earplugs is $50?Moe's income is $320 per week and he spends it on two goods, X and Y. Good X costs $8 and good Y costs $4 per unit. His utility function is U = 4.5XY. (a) Calculate Moe's utility-maximizing purchases of X and Y. (b) Calculate Moe's constrained utility- maximum if his income decreases by $2.00? (c) If the price of Y doubles, with no change in the price of X, by how much would his income have to increase to enable him to maintain his initial level of utility (as in part (a) above)?This question will let you examine/explore a more interesting utility func- tion than the simple example discussed in class as there will be both cross-price elas- ticity and an inferior good. Suppose you are told a consumer has the following utility function: U (9x9₂) = 9x + √√9₂ +9₂ You should assume income is Y, the price of good z is Pr, and the price of good z is P₂. This question will ask about several concepts discussed in lecture. (a) What is the Marshallian demand for goods x and z? I.e. find (9) for both interior solutions and corner solutions. Note: the outcome is "ugly" for the interior section and both corner should include constraints, i.e. limits using Y relative to f(P2, P₂). Hint: if solving using MRS = MRT, can use this information again in part (d).
- An individual values two goods, X and Y, according to the following utility function: -1.33 U(X,Y)= 9 › (0.33 X−0.75 +0.67 Y− The price of X is Px=2, and the price of Y is Py=3. The individual has 13 dollars. What is the Lagrangean for this problem? Please substitute the given values for variables where applicable. L= 9(0.33*X^(-0.75)+0.6*Y^(-0.75) +X[ 13-2*X-3*Y Y-0.75) What are the first order conditions? Type L for X. ƏL: əx ƏL. ΟΥ ƏL. Əx Y= Number = Number Number Number How much X and Y should this individual buy? (Round answers to the nearest two decimals. For example, 1.666 rounds to 1.67.) X= NumberSuppose there are two agents Ahmet and Berk in an economy, and both consume two goods X and Y. Also assume that price of X is 2 YTL and Y is the numeraire good, thus price of Y is 1 YTL. Ahmet and Berk has the following utility functions:UAhmet (XA,YA)= 5ln(XA)+ln(YA)UBerk (XB,YB)= XB0,5 YB0,5a. Now assume that both X and Y are private goods. Write down the optimality condition for both agents. Then, write down the optimal level of X as a function of Y for both agents.b. Now assume that X is a public good, but Y is a private good. Write down the optimality condition for good X. Then, write down the optimal level of X as a function of Y for both agents.c. Now compare the consumption levels for X in parts a and b.A consumer has GH¢600 to spend on two commodities, A and B. Commodity A costs GH¢20 per unit and Commodity B costs GH¢30 per unit. Suppose that the utility derived by the consumer from x units of Commodity A, and y Commodity B is given by the Cobb-Douglas utility functionU (x, y) = 10x0.6y0.4a. How many units of each commodity should the consumer buy tomaximize utility?b. Is the budget constraint binding?
- If the utility function of an individual takes the form: U = U ( x 1, x2) = (x1 + 2) 2 (x2 + 3) 3 Where U is total utility, and x1 and x2 are the quantities of two commoditiies consumed: (a) Find the marginal-utility function of each of the two commodities (b) Find the value of the marginal utility of the first commodity when 3 units of each commodity are consumed.Question 1. For the case of two goods, give an example of a utility function U(₁, ₂) that represents the preferences of a consumer who regards the two goods as perfect complements. Next, take the transformation f(U)= U³ of the your example utility function and explain if this newly gener- ated function represents the original preferences. Further, provide clear arguments supporting or rejecting the claim that "f(U (₁, ₂)) must be strongly increasing in (₁, ₂)."The daily total utility per quantity of consumption is shown in the figure. This shows that the total utility of consuming 1 product is $1.60, the total utility of 2 products is $3.00, and so on. Using marginal analysis, for which market price (per unit) is the optimal daily quantity of consumption 4 units? Quantity (units) 1 2 3 4 5 Total utility $1.60 $3 $4.20 $5.20 $6.00 a) $5.10 b) $0.90 c) $1.10 d) $1.30 e) $5.30
