Economics 101. Lecture 7 - Monopoly and Oligopoly

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1 Economics 0 Lecture 7 - Monooly and Oligooly Production Equilibrium After having exlored Walrasian equilibria with roduction in the Robinson Crusoe economy, we will now ste in to a more general setting. Consider the case of two goods denoted by x and y. Suose we have a consumer with quasi-linear utility over these goods, given by u(x, y) = x + v(y) for some function v satisfying v > 0 and v < 0. We can think about y as a articular good of interest and x as some conglomeration of everything else eole consumer (or just money). On the roduction side, for now we assume there is one, rice-taking firm with a technology that can roduce y units using C(x) goods. Let this function satisfy C > 0 and C > 0, meaning it exhibits decreasing returns to scale. Normalizing the rice of x to and y to, we can find the consumer s otimal choice for general v. Using the budget constraint, we can substitute in to make this a choice urely over y. Thus, the consumer will maximize u(y) = e y + v(y) where e is the endowment of x and we assume the endowment of y is zero. Taking a derivative of the above, we find that v (y) = On the roduction side, we can write the firm s rofit as a function of the amount of y roduced as π(y) = y C(y)

2 Taking the derivative to maximize rofits, we find the condition = C (y) Combining the otimality conditions from the firm side and the consumer side, we arrive at one equation characterizing the equilibrium level of y v (y) = C (y) So the level of y will equate the marginal utility of the consumer with the marginal cost of the firm. Indeed, we will find the same condition if we look at the efficient allocation. To do this, we assume that the consumer oerates the roduction technology, solving the maximization roblem max e C(y) + v(y) y which yields the same condition as above. So the equilibrium is efficient.. Examle Now let s fix secific utility functions and cost functions and see how things lay out. We ll use v(y) = 2 y for the utility function and C(y) = 2 y2 for the cost function. This can be equivalently reresented by the roduction function f(x) = 2x/. As before, we can reresent utility as a function of y Taking a derivative, we get which yields the demand function u(y) = e y + 2 y u y = + y = 0 y D () = ( ) 2 This tells us, for a given rice, how much the consumer will wish to urchase. Alternatively, we can think about the inverse of this function as a ricing function D (y) = y 2

3 which tells us what the rice must be for a consumer to demand quantity y. On the firm side, rofit can be exressed as Taking a derivative yields So the suly function is given by π(y) = y 2 y2 π y = y = 0 y S () = This tells us, for a given rice, how much the firm chooses to roduce. Now we simly set suly equal to demand to find the equilibrium rice y S () = y D () = ( ) 2 = 2 /3 /3 This imlies an equilibrium roduction level of 2 Monooly y = ( ) 2/3 We ve been assuming so far that firms behave cometitively, that is, they take rice as given when making roduction decisions. However, given that we have only one firm at this oint, this is robably not a good assumtion. Let s redo the firm maximization taking the consumer s demand into account. First, a firm chooses a rice to charge. Then using the consumer s demand function, it knows how much it will be able to sell at that rice, meaning it can calculate it s rofit. This is written as π(y) = y D () C(y D ()) 3

4 We could have also thought about this as choosing a level of roduction and using the inverse demand function to find the rice charged. Plugging in the secific functional forms we ve been using yields π(y) = ( ) 2 2 ( ( ) ) 2 2 = Taking the derivative and solving for rice yields π y = = 0 M = 2 /3 2 /3 /3 > From here we can find the quantity roduced using the demand function ( ) 2/3 y M = 2 /3 < y So the monoolist sets a higher rice and roduces less than the cometitive equilibrium level. 2. Monooly Power In the above examle, we fixed the shae of the utility function and scaled it u and down using. As a result, the monooly roduction level is simly a fixed fraction of the cometitive level. That is, the ratio of monooly to cometitive roduction is a constant indeendent of and. What we ll do in this section is introduce a curvature arameter into the utility function, which is given according to v(y) = ( γ ) y γ This arameter γ (0, ) controls how utility changes with y. Our revious examles were secial cases of this form with γ = /2. The standard otimality condition imlies v (y) = y γ = ( ) y D () = 4

5 Plugging this into the monoolists rofit equation ( π(y) = Taking the derivative yields ( π γ y = = ) ( γ γ ( ) = 2 / ( M = γ This leads to roduction of ) ) y M = γ 2 ( ) γ ( ( 2 ( ) ) 2 ) 2 ) ( ) 2 γ ( ) < y ( ) 3 γ = 0 where the cometitive equilibrium level of roduction is given according to y = ( ) and can be found using the general equation we derived in the beginning of the lecture. So the ratio of monooly roduction to equilibrium roduction is y M y = γ Therefore, monooly distortion is worst when γ is close to zero, meaning the utility function has high curvature. When γ is close to one, which is the linear case, there is almost no monooly distortion. 5

6 3 Oligooly Now we will continue in the not-fully-cometitive environment, but instead of one monoolistic firm, we will have multile, strategically interacting firms. We ll jum straight to the more general case with N firms. We ll also use secific functional forms for utility and roduction. To make things simle, let v(y) = log(y) and, as before, C(y) = 2 y2. Denote the roduction choice of firm i {,..., N} by y i. Let the sum of all roduction by firms be denoted by Y = i y i. Denote the consumtion by consumers with Y as well. The utility of the consumer can be exressed as u(y ) = e Y + log(y ) Taking a derivative, we then find the demand function for the consumer Y D () = This imlies an inverse demand function given by D (Y ) = Y Now, given a set of roduction choices y i for each firm, we can then find the total roduction Y, through which we can find the rice from the inverse demand function. So firm i s rofit is given by π i = P D (Y )y i 2 y2 i = y i i y i 2 y2 i Taking the derivative with resect to y i, we can find the otimal choice π i = i y i y i y i ( i y i) 2 y i = 0 Y y i Y 2 y i = 0 6

7 This equation will hold for each firm. Therefore, we can sum this over all i N Y Y Y = 0 (N ) Y = Since each firm has the same roduction technology, by symmetry, we must have y i = y = Y/N for all i. Thus ( ) ( ) ( ) N y O = N N Similarly, the rofits will the same across i as well. So π O = N 2 y2 = (N ) N 2N ( 2 ) ( ) N + = 2N N So the er-firm rofit declines with N. If we think about the usual notion of rofit margins being rofit over revenue, then this is simly π R = ( ) N + 2 N which converges to /2 as N grows large. Consider the otimal roduction lan. Here the consumer chooses the er-firm roduction y u(y) = e NC(y) + v(ny) = e N 2 y2 + log(ny) The derivative here yields u y = Ny + y = 0 ( ) ( ) y E = N 7

8 So the ratio of oligooly roduction to efficient roduction is (N ) y O y = E N N Thus in the limit, the level of roduction in oligooly is efficient. Another thing to note is that we might find it reasonable for firms to have a fixed cost of roduction F. In this case, firms will enter until the firm rofit falls below their fixed cost. The number of firms entering will then be the largest number N such that π(n ) > F. 8

5.1 THE ROBINSON CRUSOE ECONOMY

5.1 THE ROBINSON CRUSOE ECONOMY Essential Microeconomics -- 5 THE ROBINSON CRUSOE ECONOMY Ke ideas: Walrasian equilibrium allocation, otimal allocation, invisible hand at work A simle econom with roduction Two commodities, H consumers,