# assignment 3

November 24, 2016

Carleton University, Economics 2400C
Assignment 3
Due October 21st, 2015 at the beginning of the class tutorial. The assignment consists of 5
questions (which are all to be completed). Provide a concise but clear explanation of your answers.
A reasonable attempt at a solution results in a passing grade. You are encouraged to work together,
but you must submit your own work.
1. Find the extreme values of the following functions and state in each case whether they are a
local maximum or minimum. Sketch the functions.
(a) y = 2x
3 ? 0.5x
2 + 2
(b) y = 4x
2 ? 5x + 10
(c) y = 6x/(x
4 + 2)
(d) y = 0.5x
4 ? 5x
3 + 2x
2
2. We wish to consider an individuals labour supply decision. Utility is a function of units
of labour supplied l and consumption c. Consumption is defined by after tax income c =
wl(1 ? ? ), where the wage is given by w and the income tax rate is ? . The utility function is
u(c, l) = c
?
?
?
l
1+?
1 + ?
,
where ? ? (0, 1).
(a) Find the optimal labour supply l
?
(w, ? ). Ensure to check the second-order condition.
(b) Calculate dl?
dw and interpret the sign of this derivative.
(c) Calculate dl?
d? and interpret the sign of this derivative.
(d) Derive the elasticity of labour supply with respect to the tax rate. Interpret.
3. Show that a profit maximizing monopolistâ€™s output is unaffected by a proportional profit tax,
but is reduced by a tax of \$t per unit of output. Explain this.
4. A firm in a competitive market has the total cost function
C = 10lnx,
where x is output. Explain to the extent you can, in both mathematical and economic terms,
why there may be a breakdown of perfect competition in this market.
5. A competitive firm produces output y using two inputs, labour L and capital K. The firm
faces a product price p and input prices w and r per unit, and has production function
y = AL?K?
, where ?, ? > 0.
1
(a) Characterize the optimal inputs using the first order conditions.
(b) What further restrictions on ? and ? can we make to ensure that gives a maximum?
(c) Let y = L
0.2K0.6
, p = 100, w = 10, and r = 20. Solve for L
? and K? and show that this
is in fact a true optimum.
2

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