Lagrange Multipliers And Economics Worksheet

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Lagrange Multipliers and Economics
1. In class we considered how to optimize utility U (x, y) with budget constraint c
x + c
y = B.
x
y
(x is the number of units of good 1, y the number of units of good 2, c
the price of one unit
x
of good 1 and c
the price of one unit of good 2. Let’s suppose you are buying the goods, but
y
you could also suppose you are producing them and the c’s are production costs.) We found
that at optimum, U
/c
= U
/c
= λ.
x
x
y
y
Let’s get concrete. Suppose the utility of having x guns and y units of butter is U (x, y) = xy.
(The unit of butter might be a hundred pounds.) Suppose a gun costs $100 and a unit of butter
costs $50. Suppose the budget constraint is $400. Suppose currently the economy has 3 guns
and 2 units of butter (this does meet the constraint).
a) Which is currently bigger, U
/c
or U
/c
?
x
x
y
y
b) Show that with the same budget the society could have more utility. Give a specific
allocation choice that is better than (3,2); fractional purchases, like 2.5 guns, is allowed.
Did you increase the item with the higher marginal-utility-to-price ratio, or the one with
the lower ratio?
c) Find the optimal allocation given the budget.
d) Suppose society gets $100 more from somewhere. Give two new allocation plans you could
move to from the optimal answer in b). They should result is the same or almost the same
utility. Why?
2. Let’s generalize the model of the previous problem. It might be that the price of goods varies;
e.g., the price per item may go down as you buy more. So let us generalize the constraint to
C(x) + D(y) = B, where C(x) is the cost to buy x units of good 1, and similarly for D(y). The
utility function remains unchanged. C(x) and D(y) may be rather arbitrary functions (except
they must be differentiable).
a) Use Lagrange multipliers to show that the optimality condition is now
U
/C
(x) = U
/D
(y) = λ.
x
y
(Optimality condition is the name for the equations you get from the Lagrange approach
when you set the partials from the original variables equal to 0.)
b) What is the interpretation of the ratio U
/C
(x)? of U
/D
(y)?
x
y
3. It might be that the cost of the “bundle” (x, y) is not the sum of two separate functions C(x)
and D(x) as in problem 2. It might be even more general: a single function C(x, y). E.g.,
as you buy more of good 1, the price of good 2 might go down to be competitive. Find the
optimality condition now.
4. Another optimization situation in basic economics is maximizing production subject to a budget
for the costs of capital and labor. Suppose P (K, L) is how much you can produce if you have K
units of capital and L units of Labor. Suppose capital costs r per unit (economists call this cost
rent, hence the r). Suppose labor costs w per unit (w = wage). Then you want to maximize
P (K, L) subject to rK + wL = B.
March 4, 1997

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