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  3. assume that the supplier of a good x faces the...

Question: assume that the supplier of a good x faces the...

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Assume that the supplier of a good, a, faces the demand curve p(z) a tr The cost function for the supplier is c(z) ca. The suppliers objective is to choose r to maximise profits, T(z) p(r)a (z), and the supplier is only constrained to choices such that r 0. Throughout, assume that a 2c. 1. rite out the optimisation problem mathematically and check suppliers that the objective function is concave in the choice variable. 2. Find and solve the first-order condition 3. Denote by v a, c) the indirect profit function. write out the exact expres- sion for the indirect profits of the supplier Now suppose that the supplier is able to invest money into a new cost-saving technology. Specifically, if t x y dollaria are invested then the cost function becomes c(z) car Of course the only possible choices for y are non-positive my 0. 1+y profits of the supplier can now be Notice that the indirect expressed as v a, ty, where v is the same function of question 3 above. 1. Show that the indirect profit function is concave in y 5. Find the expression for the first-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 find the sign (positive or negative) of the comparative static effects of t and c upon yAssume 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 § .

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