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**Unformatted text preview: **Department of Economics M. Majumdar
Cornell University Monopoly A monopolist is the single seller of a good: thus, there is no point in
distinguishing between “a firm” and “an industry”. Let q= —f(p) where dq/dp < 0 be the demand function faced by the monopolist.
Its inverse is denoted by F, i.e., p= F(q) is the inverse demand function .
- ' Recall that dp/dq =1/[dq/dp]; hence we also have dp/dq <0. The total revenue function of the monopolist can be written as a function
of q through appropriate substitution; TR =pq=F(q)q Hence, MR= p+ q[dp/dq] (i) MR < p [compare this with the price-taking firm studied earlier] (ii) MR = p[1-(1/e)] ‘ (1)
Example: p = a-bq; 7 Compute TR and MR as functions of q alone. Verify (i) and (ii) Profit Maximization Let C(q) be the total cost function; then the proﬁt 11 of the monopolist can
be expressed as a function of q alone: n (q) = TR(q) - C(q)
The ﬁrst order condition for profit maximization at q~ > 0 is n' (q~)=0 so we get “MR=MC" [what was the condition for a competitive firm?] “Think of the second order condition carefully: is the second order condition consistent with TR”<0
Will a monopolist maximize profit at q‘ where the demand curve is inelastic ? ** Since MR < p for a monopolist, it is often said that “the market price of a
monopolist is a markup over marginal cost, the amount of the markup depends on the elasticity of demand” [see (1)] Suppose that a monopolist has a production function given by:
Q= h(X1,X2)- Express 11 as a function of x1 and xz as follows: 11 = TR(h(x1,xz)) —[ mm + r2X2 +A] Interpret the first order condition as “MRP=price of input” Taxing a monopolist One can think of : (i) a lump sum tax “T” (ii) a tax that takes a fraction “t” of the profit
(iii) a tax per unit of output sold In case (i), “(q) = TR(q) - C(q) -T
In case (ii), 11(q)= [TR(q)- C(q)] — t[TR(q) - C(q)] where O<t<1
In case (iii),rr(q) = TR(q) - C(q) — Bq where i5>0 **How does each tax affect the price, output and profit of a
monopolist?** Price Discrimination :v m? “a: A monopolist need not always sell its entire output in a single market at a-
single price (per unit). In some cases it may be possible for it to sell in
two or more distinct markets and thereby increasing its total profit. Such
“price discrimination” is possible only if buyers are unable to purchase the product in one market and resell it in a high—price market at a profit, and thereby equalize prices in all markets.
If a monopolist practices price discrimination in two distinct markets, its profit is given by : Ti = R1(Cl1) + R2(CI2) - C(Ch + Q2) where q1 and q2 are the quantities it sells in the two markets, R,(qi) [i=1,2]
is the total revenue function in market i , and C(q1+q2) is the total cost
function. If profit is maximized at q, >0, the first order condition leads to:
6n/6q1= (dR1/dq1) -C’ (Ch +qz) =0 (2)
OTT/BCIF (dRz/dCIz) “ C'(Cli+qg) =0 Thus, the MR ineach market must equal the MC of total output. ** Equality of the MRS does not necessarily imply that the
prices in both the markets are equal** [ you may think of the condition MR1: MR2: MC this way: if MR1 is not
equal to MR2, the monopolist can increase its total profit by shifting sales
from the lower MR market to the higher MR market, without affecting the
total cost] Monopsony Here we consider a single buyer. Suppose that the supply function of the
input “labor” is given by L=g(w) with g'(w)>0 for all w>0 or, take the inverse supply function w=G(L). The production function is given by: q=f(L),f’ >0, f’ ’<O. If the price of output is given by “p", the proﬁt of the monopsonist is
given by: 11 = pq -wL =pf(L) — G(L)L The first order condition for profit maximization is: dn/dL = pf'(L) - [w + G’(L)L] =0 or, VMP == w[1-w] where u=G’(L)L/w [interpret in terms of elasticity] ...

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