04 Jun uestion 1. Suppose a consumer has preferences that sat
uestion
1. Suppose a consumer has preferences that satisfy A1-A4 (all 4 of our axioms of consumerchoice). The consumer’s income is $100. The price of good x is px = 10 for the first 5 units of x and is px = 5 for the any additional units of x above 5. The price of good y is py = 10 for the first 5 units of y and is py = 5 for the any additional units of y above 5.
(a) Draw the consumers budget constraint:
0 5 10 15
(b) Is it possible for this consumer to have multiple utility maximizing bundles? If yes, sketch a possible indifference curve (satisfying A1-A4) demonstrating this. If not, explain why.
2. For the following problem, the consumer has Leontief preferences, as given below:
Max (min[3x,y + 1])
s.t. pxx + pyy ? m, x ? 0, y ? 0
Solve for the demand functions (neatly show work and circle answer):
(a) x?(px,py,m) =
(b) y?(px,py,m) =
Question 2 continued: Calculate the following elasticities for demand of good x (neatly show work and circle answer): (c) Price elasticity of demand, ex,px =
(d) Cross price elasticity of demand, ex,py =
(e) Income elasticity of demand, ex,m =
Solve for the compensated demand functions (neatly show work and circle answer):
(f) xc(px,py,U¯) =
(g) yc(px,py,U¯) =
(h) Graph the demand function for x?(px,py,m) when py = 1 and m = 100 (place x on the x-axis and px on the y-axis).
(i) Graph the compensated demand function for xc(px,py,U¯) when py = 1 and U¯ = 15 (place x on the x-axis and px on the y-axis).
3. Given a consumer with linear preferences:
u(x,y) = 4x + 5y
Graph the demand function for x* when M = 160 and py = 20. Again, place x on the x-axis and px on the y-axis.
0 5 10 15 20
4. Given a consumer with Cobb-Douglas preferences, facing prices px > 0, py > 0 and income m > 0:
u(x,y) = x2/5y3/5
(a) Concisely state a condition that explains why Cobb-Douglas preferences will guaranteean interior solution:
(b) Solve for the marginal utilities and the Marginal Rate of Substitution:
•MUx =
•MUy =
•MRS =
(c) Solve for the demand function for x and y (show your work):
•x?(px,py,m) =
•y?(px,py,m) =
(d) What fraction of income does the consumer spend on y?
(e) Calculate the following elasticities for demand of good y (show your work):
• Price elasticity of demand, ey,py =
• Cross price elasticity of demand, ey,px =
• Income elasticity of demand, ey,m =
(f) Solve for the compensated demand functions (show your work):
•xc(px,py,U¯) =
•yc(px,py,U¯) =
(g) Solve for the expenditure function (show your work):
•E(px,py,U¯) =
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