Consider an economy consisting of many imperfectly...

Consider an economy consisting of many imperfectly competitive, pricesetting firms. The profits of the representative firm, firm $i,$ depend on aggregate output, $y,$ and the firm's real price, $r_{i}: \pi_{i}=\pi\left(y, r_{i}\right),$ where $\pi_{22} < 0$ (subscripts denote partial derivatives). Let $r^{*}(y)$ denote the profit-maximizing price as a function of $y ;$ note that $r^{*}(y)$ is characterized by $\pi_{2}\left(y, r^{*}(y)\right)=0$. Assume that output is at some level $y_{0},$ and that firm i's real price is $r^{*}\left(y_{0}\right)$. Now suppose there is a change in the money supply, and suppose that other firms do not change their prices and that aggregate output therefore changes to some new level, $y_{1}$. (a) Explain why firm i's incentive to adjust its price is given by $G=\pi\left(y_{1}\right.$ $\left.r^{*}\left(y_{1}\right)\right)-\pi\left(y_{1}, r^{*}\left(y_{0}\right)\right)$. (b) Use a second-order Taylor approximation of this expression in $y_{1}$ around $y_{1}=y_{0}$ to show that $G \simeq-\pi_{22}\left(y_{0}, r^{*}\left(y_{0}\right)\right)\left[r^{* \prime}\left(y_{0}\right)\right]^{2}\left(y_{1}-y_{0}\right)^{2} / 2$. (c) What component of this expression corresponds to the degree of real rigidity? What component corresponds to the degree of insensitivity of the profit function?

Consider an economy consisting of many imperfectly competitive, pricesetting firms. The profits of the representative firm, firm $i,$ depend on aggregate output, $y,$ and the firm's real price, $r_{i}: \pi_{i}=\pi\left(y, r_{i}\right),$ where $\pi_{22} < 0$ (subscripts denote partial derivatives). Let $r^{*}(y)$ denote the profit-maximizing price as a function of $y ;$ note that $r^{*}(y)$ is characterized by $\pi_{2}\left(y, r^{*}(y)\right)=0$.
Assume that output is at some level $y_{0},$ and that firm i's real price is $r^{*}\left(y_{0}\right)$.
Now suppose there is a change in the money supply, and suppose that other firms do not change their prices and that aggregate output therefore changes to some new level, $y_{1}$.
(a) Explain why firm i's incentive to adjust its price is given by $G=\pi\left(y_{1}\right.$ $\left.r^{*}\left(y_{1}\right)\right)-\pi\left(y_{1}, r^{*}\left(y_{0}\right)\right)$.
(b) Use a second-order Taylor approximation of this expression in $y_{1}$ around $y_{1}=y_{0}$ to show that $G \simeq-\pi_{22}\left(y_{0}, r^{*}\left(y_{0}\right)\right)\left[r^{* \prime}\left(y_{0}\right)\right]^{2}\left(y_{1}-y_{0}\right)^{2} / 2$.
(c) What component of this expression corresponds to the degree of real rigidity? What component corresponds to the degree of insensitivity of the profit function?

Key Concepts

Profit Maximization in Imperfect Competition

Firms in imperfectly competitive markets set prices to maximize their profits. In this context, each firm’s profit depends not only on its own price but also on an aggregate variable such as output. When the firm chooses its price optimally given the state of the economy, the first?order condition (i.e. setting the derivative with respect to its own price equal to zero) characterizes the profit-maximizing price. This concept is central because deviations from the optimum, due to, for example, aggregate shocks, create an incentive to adjust the price in order to restore optimal profitability.

Incentive to Adjust Prices

The incentive to adjust prices arises because when there is a change in the aggregate economic environment (such as a change in output), the profit function – being dependent on both the individual price and aggregate output – no longer matches the condition of profit maximization. The expression G = ?(y1, r*(y1)) ? ?(y1, r*(y0)) captures the loss in profit incurred by not adjusting the price to the new optimum, thereby providing the rationale for the firm's pricing adjustment decision in response to changes in aggregate output.

Taylor Series Approximation

The Taylor series approximation is a mathematical technique used to approximate the value of a function near a given point by expanding it in a series whose terms depend on the derivatives of the function. In this context, a second-order Taylor approximation is used to approximate the change in the firm’s profit when moving from the original aggregate output level to a nearby level. This approach helps decompose the profit loss into components that are quadratic in the change in output, enabling the identification of factors such as the responsiveness of the optimal price to output changes and the curvature of the profit function.

Real Rigidity

Real rigidity refers to the degree to which a firm’s optimal price responds to changes in real economic variables, such as output. In the derived expression, the term [r*'(y0)]² reflects the degree of real rigidity: if this derivative is small, the optimal price is relatively insensitive to changes in aggregate output, meaning that prices are rigid or slow to adjust, whereas larger values indicate more responsiveness. Thus, this component provides a measure of the adjustment lag in a rich real environment.

Profit Function Insensitivity

The insensitivity of the profit function is captured by the curvature parameter, specifically ??22 (which is positive because ?22 < 0). This second derivative signifies how sharply profits decline as the firm deviates from the optimal price. A small absolute value of ?22 implies that profits decline slowly, making deviations less costly, while a large absolute value implies that even small deviations from the optimal price result in significant profit losses. Therefore, this curvature term quantifies the sensitivity of the profit outcome to changes in the firm's real price.

Transcript

00:02 Marginal cost is the derivative of total cost.

00:06 So total cost is the integral of marginal cost plus a constant.

00:09 And that constant is fixed cost because it doesn't change with q.

00:17 And that's equal to q plus 1.

00:21 So the total cost of q would be the integral of q plus 1dq, which is one half q squared plus q plus f.

00:31 That is the constant of integration.

00:33 It's a fixed cost since it's the cost when q is zero.

00:37 Price -taking firm produces where p is equal to the marginal cost of q and the break -even means pi is equal to zero so p -q is equal to the total cost of q so p -k equals the marginal cost of q which would mean 15 is equal to q plus one which would mean q is equal to 14 so the break -even pie is equal to p -k minus the total cost of q it's got to be equal to zero which means p -k -k is equal to the total of q.

01:08 So 15 times 14 would be equal to one half times 14 squared plus 14 plus f.

01:15 That would be 210 equals 98 plus 14 plus f and subtracting will give you f equals 98.

01:28 For part c, recompute the optimal q at the new price and then compute the profit.

01:33 So 20 is equal to q plus 1, which means q is 19.

01:37 So the total cost at 19 would be one half times 19 squared plus 19 plus 98, which would be one half of 361 plus 117, which is 297 .5.

01:55 Pi of 20 is 20 times 19 minus 297 .5, which is 82 .5...

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