The game proceeds in rounds, with one decision problem per round. We created 8 problems, from which you can choose any subset. Use the Add Round to add a problem, and the X to remove it from the list.
The analytical solution for each problem yields integer quantities. The player choosing the piece offering higher utility per dollar will find the optimum.
Note: To ensure marginal utilities are well defined, we start each problem with the player having chosen one of each piece. The problems fit together to highlight various features of utility in general and the Cobb Douglas utility function in particular.
Problem 1
u(x,y)=40x0.5y0.5, with Px=$1, Py=$1 and Budget=$12, yielding x*=6 and y*=6. Equal exponents summing to 1 with equal prices makes this a straightforward first problem since students spend 0.5/(0.5+0.5)=50% of their allocated budget on both x and y.
Problem 2
u(x,y)=40x0.5y0.5, with Px=$2, Py=$1 and Budget=$12, yielding x*=3 and y*=6. We take problem 1 and double the price of x. As the player still optimally spends half of her budget on each good, she purchases half as much x.
Problem 3
u(x,y)=40x0.5y0.5, with Px=$4, Py=$2 and Budget=$24, yielding x*=3 and y*=6. Taking problem 2 and doubling both prices and the budget does not change the utility-maximizing quantities.
Problem 4
u(x,y)=40x0.25y0.75, with Px=$1, Py=$1 and Budget=$8, yielding x*=2 and y*=6. Unequal exponents make this slightly more challenging, although prices are equal and the exponents still sum to one.
Problem 5
u(x,y)=80x0.25y0.75, with Px=$1, Py=$1 and Budget=$8, yielding x*=2 and y*=6. Taking problem 4 and doubling all utility values (by doubling the coefficient) does not affect the optimal budget allocation.
Problem 6
u(x,y)=200x0.2y0.8, with Px=$1, Py=$2 and Budget=$20, yielding x*=4 and y*=8. This problem forms the basis for a second set of problems with unequal exponents AND unequal prices. Exponents summing to one make it easy to see that the student optimally spends 20% on x and 80% on y.
Problem 7
u(x,y)=200x0.1y0.4, with Px=$1, Py=$2 and Budget=$20, yielding x*=4 and y*=8. Taking problem 6 and halving both exponents does not change the utility-maximizing allocation. The optimal allocation still entails spending 0.1/(0.1+0.4)=20% on x.
Problem 8
u(x,y)=200x0.3y1.2, with Px=$1, Py=$2 and Budget=$20, yielding x*=4 and y*=8. Taking problem 7 and tripling both exponents does not change the utility-maximizing allocation because the ratio of an exponent to the sum of exponents does not change.
