Question: assume that the supplier of a good x faces the...
Assume that the supplier of a good, x, faces the demand curve p(x) = a° 1 2 x. The cost function for the supplier is c(x) = cx. The supplierís objective is to choose x to maximise proÖts, º(x) = p(x)x ° c(x), and the supplier is only constrained to choices such that x ¸ 0. Throughout, assume that a > 2c.
1. Write out the supplierís optimisation problem mathematically and check that the objective function is concave in the choice variable.
2. Find and solve the Örst-order condition.
3. Denote by v(a; c) the indirect proÖt function. Write out the exact expression for the indirect proÖts of the supplier.
Now suppose that the supplier is able to invest money into a new cost-saving technology. SpeciÖcally, if t £ y dollars are invested, then the cost function becomes c(x) = cx 1+y . Of course the only possible choices for y are non-positive values, y ¸ 0. Notice that the indirect proÖts of the supplier can now be expressed as v ³ a; c 1+y ¥ ° ty, where v is the same function of question 3 above.
4. Show that the indirect proÖt function is concave in y.
5. Find the expression for the Örst-order condition for an optimal choice of y. (Note; you are not asked to explicitly solve this equation)
6. Use the implicit function theorem to Önd the sign (positive or negative) of the comparative static e§ects of t and c upon y § .