Multi-product firms are firms that produce multiple goods, and therefore have to deal with allocating inputs more properly in order to attain higher production levels. This is a greater problem than the one single-product firms face, the production maximisation problem, since multiproduct firms must allocate their factors not only to produce one good, but multiple goods.

Indeed, production analysis can be extended to include multiproduct firms, as it can be derived from F. Y. Edgeworth‘s “Mathematical Psychics”, 1881­­. In this case, a company produces two goods, X and Y, using the same two factors, L and K, and subject to the following restrictions:

In this first figure, which is known as an Edgeworth box, if we take A as a starting point, we can see that it is far from an optimum, as we could increase our production of X whilst maintaining production of Y constant by moving from A to B, and maintain production of X constant whilst increasing production of Y by moving to point C. These optimum points pertain to the contract curve, along which production is efficient. Along all these points, the marginal rate of technical substitution for both products is equal: MRTS_x=MRTS_Y.

From this Edgeworth box and, more precisely, the contract curve that takes all efficient production levels into account, we can derive the production possibility frontier (PPF), shown in the figurebelow. This is just another way to look at the same problem: how to better allocate our resources in order to obtain the maximum output possible. However, by using the PPF, we concentrate more on the amounts being produced, rather than the inputs being used.

Of course, what is missing here to complete the analysis would be the cost of both inputs and how much is required to produce each type of good. In this case, by factoring in the selling price of each, we would have a closed optimisation problem that we would be able to resolve analytically, determining an efficient production function that could optimise the ratio of production of one good to another.

New Keynesian Economics argue that menu costs are the reason for price stickiness. Price stickiness, the suboptimal adjustment of prices in response to demand shocks, can result in business cycles. Gregory Mankiw proved in 1985 in his article “Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly” that sticky prices can be both privately efficient and socially inefficient. In fact, as Mankiw points out, even small menu costs can cause large welfare losses.

Mankiw uses a static model of a monopoly firm’s pricing decision, which starts by setting its price in advance and changes it ex post, but incurring in a small menu cost. The monopoly faces a constant cost function and an inverse demand function:

C = kqN

where

C: total nominal cost of production

q: quantity produced

k: constant

P = f(q)N

where

P: nominal price

N is a nominal scale variable, which denotes the exogenous level of aggregate demand. It can be regarded, for instance, as the overall price level. C and P increase proportionally to the level of nominal demand N.

Now, let c = C/N and p = P/N, which will turn the firm’s problem independent of aggregate demand:

c = kq

p = f(q)

In the figure below, we see the producer surplus (profits earned by the firm), equal to the rectangle between k and p_m. On the other hand, the excess value over the price paid represents consumer surplus, the triangle above profits. Total surplus, which represents welfare, is the sum of consumer and producer surplus.

The firm needs to set its price one period ahead based on expectations about future aggregate demand, being this price p_mN^e. If expectations are correct ex post, the observed price p₀ is p_m. If not, the observed price is p_m(N^e/N).

First scenario

The first case Mankiw examines is when aggregate demand N is lower than expected, and therefore p₀ is higher than p_m, as shown in the figure below. The producer surplus is lowered by B-A (since profits as seen before were equal to rectangle B plus the rectangle to its left), which is positive because p_m is by definition the profit-maximising price. Social welfare (or total surplus) is reduced by B+C, and therefore the reduction in welfare due to the contraction in aggregate demand is larger than the loss in the surplus of the firm.

Now, let’s suppose that the firm is capable of changing its price ex post, at a menu cost of z. The firm can then reduce its price from p₀ to p_m and obtain additional profits of B-A, which the firm will do if B-A > z. However, from the point of view of a social planner, the firm should lower its prize if and only if B+C > z. Let’s see Mankiw’s propositions on different outcomes:

-Proposition 1. Following a contraction in demand, if the firm cuts its price, then doing so is socially optimal (if it cuts it, is because B-A > z and therefore B+C > z+A+C > z).

-Proposition 2. Following a contraction in demand, if B+C > z > B-A, then the firm does not cut its price to p_m, even if it would be socially optimal (the inefficiency results because printing new menus result in an external benefit of C+A).

-Proposition 3. A contraction in aggregate demand reduces social welfare unambiguously, as shown by the sum of producer and consumer surplus. If the producer reduces its price, then the contraction only has the menu cost z. If the firm does not cut its price, then the contraction has the cost of B+C (probably much larger than z).

Second scenario

The second case Mankiw analyses is an expansion in aggregate demand (N > N^e), and therefore p₀ <p_m. Firstly, let’s see what happens when p₀ > k (N/N^e < p_m/k), as shown in our third figure. In this case, producer surplus is reduced by D-F, which is positive (p_m maximises the firm’s profits) and social welfare increases by E+F. The firm will reset its price if the increase in profits beats the menu cost. In other words, the firm will change its price if D-F > z.

-Proposition 4. If there is an expansion in demand and if the firm resets its price, social welfare decreases by the menu cost. If it doesn’t, total surplus increases by E+F.

Third scenario

Now, let’s see what happens if N/N^e > p_m/k, and therefore p₀ < k. Then, social welfare decreases by a positive or negative amount of I-J, which makes for an uncertain welfare effect. The firm’s profits, which are now negative, have been reduced by G+H+I. If G+H+I > z, then the firm will reset its price to p_m. Doing so would be socially optimal if I-J > z.

-Proposition 5. If the firm resets its price following the expansion in demand mentioned, social welfare (total surplus) will decrease by the menu cost z. If it doesn’t, total surplus does not decrease by more than the menu cost (if the firm doesn’t reset its price is because G+H+I < z, which implies that I-J < z-J-G-H < z and therefore the social welfare reduction I-J < z).

-Proposition 6 (as a summary of previous propositions). An expansion in aggregate demand may either increase welfare or reduce it, but never more than the menu cost. On the other hand, a contraction in aggregate demand will reduce welfare, possibly in an amount much larger than the menu cost.

Conclusion

To sum up, Mankiw demonstrates that private incentives ensures a high price adjustment when aggregate demand expands, but a small adjustment following a contraction in aggregate demand. From a social planner’s point of view, prices may be stuck too high, but never too low, which translates into downward price stickiness, although not into an upward rigidity. Mankiw also points out that a more complete model (general equilibrium) will probably show higher degrees of price stickiness, since interfirm purchases will exacerbate price rigidity. Therefore, the main conclusion of this paper (that small menu costs can translate into large inefficiency effects) would certainly remain in a general equilibrium model.

Regarding economic policies required to alleviate these problems, Mankiw explains how an active monetary policy is required. More particularly, he mentions policies that aim at the pricing mechanism, such as tax-based incomes policy and other supply-side policies.

Menu costs are costs that result from price changes. An easy way to understand menu costs is by means of a typical example: restaurants. When a restaurant manager wants to change prices, the cost of changing the menus (in order to show the new prices) must be taken into consideration. Therefore, the manager will need to assess whether the increase in prices will cover for the cost of printing new menus.

As Gregory Makiw demonstrates in his article “Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly”, 1985, this menu costs may lead to a non-socially optimal situation. Indeed, Mankiw’s menu costs model shows how higher menu costs may lead to a situation where prices remain unchanged, thus provoking so called price stickiness.

Contrary to what economists from the New Classical Macroeconomics think, which is that business cycles are originated by shifts in supply (which originate mainly from technological shocks), New Keynesian economists believe that business cycles are originated by price stickiness.  This price stickiness might be explained by menu costs, since the slow adjustment of the economy may explain business cycles.

There is a great debate over the following question: are these menu costs really large enough to cause business cycles? An article from Daniel Levy et al., “The Magnitude of Menu Costs: Direct Evidence from Large U.S. Supermarket Chains”, 1997, demonstrates, using empirical evidence, that these menu costs are indeed large enough to cause business cycles. The authors use store-level data from five multistore supermarket chains to directly measure menu costs. They find that the menu costs per store at those chains average more than 35% of net profit margins, which may be forming a barrier to price changes. When applying these findings to theoretical models of menu costs, the authors conclude that these menu costs are large enough to be capable of having macroeconomic significance. Furthermore, they believe that these menu costs may cause considerable nominal rigidity in other industries or markets, thus amplifying the effects on business cycles. These menu costs, they say, vary largely due to local and regional legislation, which may require, for instance, a separate price tag on each item, thus increasing menu costs.