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Answer :
We have utility function as u(x, y) = 3 ln x + ln y. Now, to draw an indifference curve, we need to have a table showing different combinations of x and y that give the same level of utility.
Let's find the level of utility corresponding to (1,1)u(1,1) = 3 ln 1 + ln 1 = 0Now, let's find the level of utility corresponding to (e, e^2)u(e, e^2) = 3 ln e + ln e^2 = 3 + 2 ln e = 5.22To find the level of utility corresponding to (e^2, e^4)u(e^2, e^4) = 3 ln e^2 + ln e^4 = 6 + 4 ln e = 8.44 Similarly, we can find more such values and plot the points on the graph as shown below: Representative Indifference Curve for u(x,y) = 3 ln x + ln y Now, we are given $500 to spend and wine costs $20 per bottle and cheese costs $10 per pound. Let's assume we buy x bottles of wine and y pounds of cheese.
Total cost of buying x bottles of wine and y pounds of cheese = 20x + 10yWe need to maximize our utility subject to this constraint. So, the problem can be stated as follows :Maximize u(x,y) = 3 ln x + ln y Subject to the constraint 20x + 10y <= 500Also, we can't buy a fractional number of bottles of wine.
Therefore, x should be an integer. Let's try to convert the constraint into a form that can be more easily used for optimization:20x + 10y <= 50010y <= 500 - 20x y <= 50 - 2xNow, we have an inequality that expresses y in terms of x. So, we can substitute this expression for y in the utility function to get a function of x alone that we can optimize.
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