Intermediate Microeconomics · free preview
Corners and Kinked Budgets
When the tangency equation has no sensible solution the optimum sits at a boundary, corners arise with substitutes and satiation, and the budget itself can kink, under rationing, quantity discounts, or vouchers, moving the calculus to the kink's neighborhood.
1. The boundary optimum
The boundary optimum
2. Another corner worked
Another corner worked
3. Budgets that kink
Budgets that kink
4. Why economists watch boundaries
Why economists watch boundaries
5. Diagnosing the corner
For u equal to 2 x1 plus x2 with income 20 and prices 1 and 1, why is the optimum (20, 0)?
6. The linear substitution check
For u equal to 3 x1 plus x2, income 30, prices 2 and 1, what does substituting the budget into utility show?
7. Where the voucher bends
A 12-dollar voucher usable only on good 1, with income 30 and prices 1 and 1, creates what budget?
8. The candidate list on a kinked budget
What is the complete list of candidates when optimizing on a kinked budget with smooth convex preferences?
9. A corner's utility
For u equal to 3 x1 plus x2 with income 30 and prices 2 and 1, compute the optimal utility, and write the number.
10. The other corner for comparison
Same consumer, compute the utility of spending all 30 on good 2, and write the number.
11. The winning corner in units
For u of 3 x1 plus x2 with income 30 and prices 2 and 1, what is the optimal x1?
12. The voucher's kink coordinates
With income 30, prices 1 and 1, and a 12-dollar voucher for good one, where does the budget kink?
13. Cobb-Douglas on a voucher budget
For u of x1 x2 on the voucher budget, income 30 cash plus the 12-dollar voucher, where is the optimum?
14. The corner's signature
At the corner optimum (20, 0) for u of 2 x1 plus x2 with equal prices, what does the MRS of 2 against the price ratio of 1 tell you?
15. Utility on the voucher budget
For u of x1 x2 with income 30 plus the 12-dollar voucher, compute the utility at the optimum, and write the number.
16. All-in on the voucher good
For u of 3 x1 plus x2 with income 30 plus the 12-dollar voucher and equal prices, compute the optimal utility, and write the number.
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