The Life Cycle/Permanent Income Benchmark

The standard approach to households' behavior (incorporated in NK-DSGE models) is based on “two-stage budgeting.” In the first stage, the representative infinitely lived household maximizes expected lifetime utility subject to the intertemporal budget constraint, determining the optimal size of consumption expenditure $C_{h,t}$, which we will sometimes refer to hereafter as the consumption budget.

In the second stage the household determines the composition of $C_{h,t}$, i.e., the fraction $C_{i,h,t}/C_{h,t}$ for each variety $i=1,2,\dots ,F_c$ where $F_c$ is the cardinality of the set of C-goods (and of C-firms).²³

As far as the first stage is concerned, optimal consumption turns out to be a function of expected future consumption $E_t{C}_{h,t+1}$ and the real interest rate r (consumption Euler equation).

Notice now that consumption expenditure is equal by definition to permanent income, $C_{h,t}=Y_{h,t}^p$. Hence, after some algebra, we get

(1) $C_{h,t}=\hat{r}(W_{h,t}^h+W_{h,t}^f)$

where $\hat{r}=\hat{R}-1$ is the (net, real) interest rate, $W_{h,t}^h$ is human capital, and $W_{h,t}^f$ is financial wealth. Eq. (1) can be interpreted as a benchmark Life Cycle/Permanent Income (LCPI) consumption function. In this framework, by construction, the consumption budget is equal to the annuity value of total (financial and human) wealth.

Human capital, in turn, is defined as the discounted sum of current income and expected future incomes accruing to the household:

(2) $W_{h,t}^h=\frac{1}{\hat{R}}\sum _{s=0}^{\infty }{\left(\frac{1}{\hat{R}}\right)}^s{E}_t{Y}_{h,t+s}=\frac{1}{\hat{R}}{Y}_{h,t}+\frac{1}{\hat{R}}\sum _{s=1}^{\infty }{\left(\frac{1}{\hat{R}}\right)}^s{E}_t{Y}_{h,t+s}.$

Substituting (2) into (1), the LCPI consumption function becomes

(3) $C_{h,t}=\frac{\hat{r}}{\hat{R}}{Y}_{h,t}+\frac{\hat{r}}{\hat{R}}\sum _{s=1}^{\infty }{\left(\frac{1}{\hat{R}}\right)}^s{E}_t{Y}_{h,t+s}+r{W}_{h,t}^f.$

In words, consumption is a linear function of current and expected future incomes and of financial wealth.

By definition, in the LCPI benchmark, $W_{h,t+1}^f=\hat{R}{W}_{h,t}^f+Y_{h,t}-C_{h,t}$. Therefore savings, i.e., the change in financial wealth $S_{h,t}=W_{h,t+1}^f-W_{h,t}^f$, turns out to be equal to $S_{h,t}=\hat{r}{W}_{h,t}^f+Y_{h,t}-C_{h,t}$. Since consumption is equal to permanent income, savings can be specified as follows:

(4) $S_{h,t}=\hat{r}{W}_{h,t}^f+(Y_{h,t}-Y_{h,t}^p)$

where the expression in parentheses is transitory income. All the income in excess of permanent income will be saved and added to financial wealth. If, on the other hand, current income falls short of permanent income, the household will stabilize consumption by decumulating financial wealth.

As far as the second stage is concerned, it is easy to show that in a Dixit–Stiglitz framework the (optimal) fraction of each good in the bundle is

(5) $\frac{C_{i,h,t}}{C_{h,t}}={\left(\frac{P_{i,t}}{P_t}\right)}^{-\epsilon }$

where $P_{i,t}$ is the price of the ith variety, $P_t$ is the general price level,²⁴ and ε is the absolute value of the price elasticity of demand. In words, the fraction of the consumption budget allocated to each variety is a decreasing function of the relative price $\frac{P_{i,t}}{P_t}$.

Generally, households are assumed to be identical. If household members are heterogeneous (for instance, because of the employment status, or the level and the source of income), in standard models complete markets are assumed so that idiosyncratic risk can always be insured. Within the household, “full consumption insurance” follows from the assumption that household members pool together their incomes (wages, unemployment subsidies, dividends) and consume the same amount in the same proportions. Thanks to this assumption, heterogeneity, albeit present, is irrelevant because the household may still be dealt with as a unique (representative) agent. If idiosyncratic income risk is uninsurable, then heterogeneity cannot be assumed away.²⁵

The Agent Based Approach to Consumption/Saving Decisions

A two stage procedure is also generally adopted in MABMs. In the first stage household h determines the consumption budget $C_{h,t}$, i.e., the amount of resources (income, wealth) to be allocated to consumption expenditure. In the second stage the household determines the composition of the bundle of consumption goods, i.e., the quantities $C_{i,h,t}$, $i=1,2,\dots ,F_c$ of the goods which enter the consumption bundle. Notice, however, that in general agents do not follow explicit optimization procedures. Markets are generally incomplete and within-household consumption insurance is ruled out.

The Choice of the Consumption Budget

As far as the first stage is concerned, the most general behavioral rule adopted to set the consumption budget in the MABM literature can be specified as follows:

(6) $C_{h,t}=c_h{W}_{h,t}^h+c_f{W}_{h,t}^f$

where $W_{h,t}^f$ is the household's financial wealth (deposited at the bank and/or invested in financial assets), $W_{h,t}^h$ is human capital, $c_h$ and $c_f$ are propensities to consume, both positive and smaller than one. In words, consumption is a linear function of human and financial wealth.

By definition, $W_{h,t}^f=\hat{R}{W}_{h,t-1}^f+Y_{h,t}-C_{h,t}$ and $S_{h,t}=W_{h,t}^f-W_{h,t-1}^f$. Consumption is defined as in (6). Hence, in a generic MABM savings turn out to be

(7) $S_{h,t}=Y_{h,t}+(\hat{r}-c_f)W_{h,t-1}^f-c_h{W}_{h,t}^h.$

If positive, savings increase financial wealth. In some MABMs saving can be involuntary: it may happen that the household cannot find enough consumption goods at the limited number of firms it visits. Savings will turn negative, i.e., the household will decumulate financial wealth, if the consumer does not receive income, for instance, because a worker becomes unemployed and/or financial income (interest payments on financial wealth) is too low. In most MABMs, household do not get into debt so that consumption smoothing is limited or absent (in the jargon of NK-DSGE models, asset market participation is limited).

Specifications of the general rule (6) differ from one MABM to another.

AGH set $c_h=c_f=c$. Moreover, they define human capital as the capitalized value of permanent income $W_{h,t}^h=k^h{Y}_{h,t}^p$ where $k^h$ is a capitalization factor.²⁶ Permanent income $Y_{h,t}^p$ is computed by means of an adaptive algorithm, $Y_{h,t}^p-Y_{h,t-1}^p=(1-\xi )(Y_{h,t-1}-Y_{h,t-1}^p)$ where $\xi \in (0,1)$ is a memory parameter. By iterating, it is easy to see that permanent income (and therefore human capital) turns out to be a weighted sum of past incomes only, $Y_{h,t}^p=(1-\xi ){\sum }_{s=0}^{-\infty }{\xi }^s{Y}_{h,t-s-1}$. Hence the AGH behavioral rule for consumption expenditure is

(8) $C_{h,t}=c{k}^h(1-\xi )\sum _{s=0}^{-\infty }{\xi }^s{Y}_{h,t-s-1}+c{W}_{h,t}^f.$

In words, consumption is a linear function of past incomes and financial wealth.

In the CATS/ADG framework, human capital is defined by the following adaptive algorithm: $W_{h,t}^h=\xi W_{h,t-1}^h+(1-\xi )Y_{h,t}$. By iterating, one gets $W_{h,t}^h=(1-\xi )Y_{h,t}+(1-\xi ){\sum }_{s=1}^{-\infty }{\xi }^s{Y}_{h,t-s}$, i.e., human capital is a weighted average of current and past incomes.²⁷ Substituting this definition into (6), the behavioral rule specializes to

(9) $C_{h,t}=c_h(1-\xi )Y_{h,t}+c_h(1-\xi )\sum _{s=1}^{-\infty }{\xi }^s{Y}_{h,t-s}+c_f{W}_{h,t}^f.$

While human capital in the neoclassical approach is a linear combination of current and expected future incomes, and therefore is formed in a forward looking way, the proxy for human capital in AGH and CATS/ADG is a linear combination of current and past incomes, i.e., it is determined by a backward looking algorithm.²⁸

Setting $\xi =0$, i.e., assuming that there is no memory, human capital boils down to current income, so that (9) specializes to

(10) $C_{h,t}=c_y{Y}_{h,t}+c_f{W}_{h,t}^f$

where $c_y$ is the propensity to consume out of income.²⁹

Setting $c_f=0$, from (10) we get a specification of the consumption function sometimes adopted in MABMs of a strictly Keynesian flavor

(11) $C_{h,t}=c_y{Y}_{h,t}.$

In many models of the KS family, the behavioral rule is (11) with $c_h=1$:

(12) $C_{h,t}=Y_{h,t}.$

This specification describes the behavior of “hand-to-mouth” consumers.³⁰

LAGOM adopts a rule such as (10) with $c_y=0$. Moreover, since financial wealth coincides with money holding ($W_{h,t}^f=\frac{M_{h,t}}{P_t}$), the consumption budget turns out to be an increasing linear function of real money balances³¹

(13) $C_{h,t}=c_f\frac{M_{h,t}}{P_t}.$

In CATS/DD, consumption is a special case of (10) obtained by setting $c_y=c_f=c$. Therefore

(14) $C_{h,t}=c(Y_{h,t}+W_{h,t}^f).$

The expression in parentheses is one of the possible specifications of “cash-on-hand”, which can be defined in the most general way as liquid assets which can be used to carry on transactions. In small MABMs which do not feature banks such as LEN, cash-on-hand coincides with currency. In a setting with banks such as CATS/DD, cash-on-hand coincides with new deposits, which in turn amount to the sum of income and old deposits. This is in line with Deaton (1991), who defines cash-on-hand as the sum of income and financial assets.

In EUBI and EUGE, saving behavior aims at achieving a target for wealth, namely $S_{h,t}=\nu ({\omega }^f{Y}_{h,t}-W_{h,t}^f)$ where ${\omega }^f$ is the target wealth-to-income ratio and ν is the velocity of adjustment of wealth to targeted wealth. Since by definition $S_{h,t}=Y_{h,t}-C_{h,t}$, simple algebra shows that under this assumption the consumption budget can be written as

(15) $C_{h,t}=(1-\nu {\omega }^f)Y_{h,t}+\nu W_{h,t}^f$

which is a special case of (10) with $c_y=1-\nu {\omega }^f$ and $c_f=\nu$.

Carroll has shown that in an uncertain world consumption is a concave function of cash-on-hand which he defines as the sum of human capital and beginning of period wealth.³² In Carroll's framework, for low values of wealth, the propensity to consume (out of wealth) is high due to the precautionary motive: a reduction of wealth would in fact lead to a sizable reduction of consumption to rebuild wealth (buffer stock rule). In some MABMs the consumption function is adjusted to mimic Carroll's precautionary motive, i.e., to reproduce the nonlinearity of the relationship between consumption and wealth. This theoretical setting has been corroborated by a number of empirical studies (see Carroll and Summers, 1991; Carroll, 1997), such that using this approach in MABMs is consistent with the agenda to rely on behavioral rules which have strong empirical foundations.

For instance, in EUBI the specification of the consumption function discussed above gives rise to a piecewise linear heuristic based on the buffer-stock rule. Also EUGE adopt a specification of the consumption budget which adopts Carroll's rule.

In the CATS/MBU framework, consumption is specified as in (11) but $c_y$ is a nonlinear function of wealth. This allows capturing the interaction between income and wealth in the determination of consumption.

LEN explicitly models the consumption budget as an increasing concave function of money holdings by

(16) $C_{h,t}={\left(\frac{M_{h,t}}{P_t}\right)}^c$

where $c\in (0,1)$.³³

JAMEL presents a variant of Carroll's framework. The household computes “average income” (which can be considered a proxy of permanent income) as the mean of incomes received over a certain time span occurred in the past, $Y_{h,t}^a=\frac{1}{n}{\sum }_{\tau =t-n}^t{Y}_{h,\tau }$.³⁴ Moreover, JAMEL define desired or targeted cash on hand as a fraction of average income, $m_{h,t}^T=s{Y}_{h,t}^a$, where $m_{h,t}^T=M_{h,t}^T/P_t$ are real money balances and $s\in (0,1)$ is the desired propensity to save (out of average income). The consumption budget is defined as follows:

(17) $C_{h,t}=\{\begin{array}{cc}c{Y}_{h,t}^a\hfill & \text{if }{m}_{h,t}<m_{h,t}^T\text{,}\hfill \\ Y_{h,t}^a+c_m(m_{h,t}-m_{h,t}^T)\hfill & \text{if }{m}_{h,t}>m_{h,t}^T\hfill \end{array}$

where $c=1-s$ and $c_m\in (0,1)$ is the propensity to consume. In words, if liquid assets are (relatively) “low”, the household spends a fraction c of average income. If liquidity is “high”, the household consumes the entire average income and a fraction $c_m$ of excess money balances, defined as the difference between the current and targeted money holding.

With the help of some algebra, the behavioral rule above can be written as follows:

(18) $C_{h,t}=\{\begin{array}{cc}c{Y}_{h,t}^a\hfill & \text{if }{Y}_{h,t}^a>{\overline{Y}}_{h,t}^a\text{,}\hfill \\ c^{\prime }{Y}_{h,t}^a+c_m{M}_{h,t}\hfill & \text{if }{Y}_{h,t}^a<{\overline{Y}}_{h,t}^a\hfill \end{array}$

where ${\overline{Y}}_{h,t}^a=\frac{1}{s}{m}_{h,t}$ is the cut-off value of average income and $c^{\prime }=1-c_m(1-c)=(1-c_m)(1-c)+c$. Notice that $c^{\prime }>c$. The cut-off value of average income is a multiple of current money holdings. The specification above highlights the basic tenets of Carroll's buffer stock theory in a very simple piecewise linear setting. When average income is below the threshold, i.e., when the threshold is relatively high because the household is wealthy/liquid, the marginal propensity to consume is $c^{\prime }$, i.e., it is relatively high. When average income surpasses the threshold, the marginal propensity drops to c.

JAMEL also incorporates consumer sentiment and opinion dynamics in the consumption function. The propensity to save, s, can take on two values. When “optimistic”, the household is characterized by $s^L$ while if “pessimistic” it sets s to $s^H$, with $s^H>s^L$. Pessimism leads to a reduction of the propensity to consume: households save more for rainy days.

Each household switches from a low to a high propensity to save (and vice versa) depending on its employment status. The household turns pessimistic if unemployed. In each period, the household observes the consumer sentiment (proxied by the employment status) of a finite subset of other households (a neighborhood, for short). With probability ${\pi }^m$ it adopts the majority opinion, i.e., the prevailing consumer sentiment, of the neighborhood; with probability $1-{\pi }^m$ the household relies on its own situation: if it is unemployed (employed), it will be pessimistic (optimistic) and will choose $s^H$ ($s_L$). The probability of adopting the sentiment of the majority can be interpreted as the strength of the households' “animal spirits.”

The Choice of the Goods to Buy

As to the second stage, the choice of the goods to buy is generally influenced by the relative price. EUBI and EUGE introduce a multinomial logit model, CATS posit a search mechanism.

In LEN each household is linked to a fixed number of firms from which it can buy. The network can change over time. Each period, with a certain probability, the household chooses at random a firm in the neighborhood of current trading partners and a firm outside it, compares the prices, and switches to the new firm if the price of the price set by the current partner exceeds the price of the new firm by at least a certain margin. In the end, therefore, the household searches for better trading opportunities based on relative price.³⁵

AGH posit that the household can buy only from two firms (“stores”, i.e., shops where the household purchases its consumption goods). Hence it has to choose the quantity to be demanded from each firm. AGH derives behavioral rules by solving a simple optimization problem. The household maximizes $U_{h,t}=U(C_{1,h,t},C_{2,h,t})$ subject to $p_{1,t}{C}_{1,h,t}+p_{2,t}{C}_{2,h,t}=C_{h,t}$ where $p_{i,t}=\frac{P_{i,t}}{P_t}$, $i=1,2$. Hence the quantity demanded for each variety turns out to be

(19) $C_{i,h,t}=f_i(p_{1,t},p_{2,t},C_{h,t}),i=1,2,$

which can be easily compared with (22). The household explores the goods market looking for stores supplying the goods it wants to consume. The household buys from store j only if the posted price $P_{j,t}$ is smaller than a reservation price equal to the price paid by the household in the past $P_{h,j,t-1}$ (appropriately indexed): $P_{h,i,t-1}(1+{\pi }^T)>P_{j,t}$ where ${\pi }^T$ is expected inflation, anchored to the inflation target of the central bank.³⁶

In the CATS framework, households collect information and pick goods by visiting firms chosen at random. The hth household visits $Z_c$ randomly selected firms, ranks them according to the price they charge and purchases C-goods starting from the firm charging the lowest price. If it does not spend the entire consumption budget at the first firm, it will move up to the second firm in the ranking, and so on. If it has not spent all the consumption budget after visiting $Z_c$ firms, it will save involuntarily. This implies that the demand for the good produced by the ith C-firm is implicitly decreasing with the relative price $p_{i,t}$. Notice, however, that the implicit demand curve that the ith firm is facing is shifting over time. A given firm may be visited by different sets of consumers on subsequent days. The search and matching mechanism leads to the coexistence of queues of unsatisfied consumers (involuntary savers) at some firms and involuntary inventories of unsold goods at some other firms.

LAGOM adopts a similar market protocol as CATS for the market for C-goods. In the LAGOM model also firms enter the market for C-goods as they use circulating capital together with fixed capital for production.

EUBI and EUGE assume that the consumer receives information about the range of products, their prices, and their availability at (local) malls. The choice of the good to buy is random and the probability of buying a certain product is determined by a multinomial logit model:

(20) $P[\text{consumer }h\text{ selects good }i]=\frac{\exp (-\gamma \log P_{i,t})}{{\sum }_j\exp (-\gamma \log P_{j,t})}$

where the sum in the denominator includes all products for which the consumer has received information.³⁷ As has been discussed in Dawid et al. (2018d), this formulation can also be interpreted as a representation of a situation in which the product offered by the different C-firms are horizontally differentiated along dimensions not explicitly captured in the model. The parameter γ governs the intensity of (price) competition between the C-firms and typically has strong influence on the emerging economic dynamics on the aggregate level. Dawid et al. (2018d) also point out that the use of logit-choice models for the representation of consumer choice has strong empirical foundations, e.g., in the Marketing literature.³⁸

The Dixit–Stiglitz Benchmark

As far as the corporate sector is concerned, NK DSGE models are based on monopolistic competition: each firm behaves as a monopolist on its own market. The firm's market power is due to product heterogeneity. Products may be different in the eyes of consumers for a number of reasons: spatial dispersion, product differentiation, or transaction costs which may generate captive markets.

In the Dixit–Stiglitz benchmark, the demand of the hth household for the ith good is represented by Eq. (5). Summing across H households and rearranging, one gets the demand function for the ith good:

(22) $C_{i,t}={\left(\frac{P_{i,t}}{P_t}\right)}^{-\epsilon }\frac{C_t}{F_c}$

where $C_t={\sum }_{h=1}^H{C}_{h,t}$ is total consumption (economywide).⁴¹

With a linear technology and only the labor input, the marginal cost will be $w_t/\alpha$ where $w_t$ is the wage and α is the marginal productivity of labor. Optimal pricing yields

(23) $P_{i,t}^{⁎}=(1+\mu )\frac{w_t}{\alpha }$

where $\mu ={(\epsilon -1)}^{-1}$ is the mark-up.

Substituting (23) into (22), one gets the optimal scale of production

(24) $Y_{i,t}^{⁎}=C_{i,t}^{⁎}={\left(\frac{1+\mu }{\alpha }\frac{w_t}{P_t}\right)}^{-\epsilon }\frac{C_t}{F_c}.$

Since the manager of the firm knows the production function and input costs, and therefore can compute the marginal cost, if she knows also the demand curve she will choose an optimal mark-up (i.e., ratio of the individual price to marginal cost) and an optimal scale of activity. In other words, in the absence of uncertainty on market demand, the firm will choose a point on the demand curve in such a way as to maximize profits. In this case supply will always be equal to demand: the firm will neither accumulate involuntary inventories of unsold goods nor face queues of unsatisfied consumers; output will always be sold and no potential customer will leave the firm's premises empty handed.

Moreover, in the absence of frictions on the labor market, the firm will always be able to carry out planned production. In the end therefore, planned output will be equal to actual output and actual output will be equal to sales.

From (23) and (24) we can predict the following: the firm will react to (i) an increase of elasticity of demand by reducing the mark up and therefore the individual price and increasing production; (ii) an increase of the real wage by increasing the price and reducing production; (iii) an increase in productivity by reducing the price and increasing production; (iv) an increase in total consumption by increasing production.

Since firms have the same technology, incur the same labor cost, and face the same demand function, a symmetric equilibrium in which all firms charge the same price, is considered: $P_{i,t}=P_t$, so that the real wage will be equal to ${(w_t/P_t)}^{⁎}=\alpha /(1+\mu )$. At that wage, employment is determined by labor supply, the labor market clears, and there is full employment.⁴² Total output is equal to the full employment level of GDP.⁴³ In the absence of investment, output is equal to consumption. Each firm therefore will produce the same quantity, i.e., a fraction $1/F_c$ of total consumption, which is equal to total full employment output. In the presence of nominal rigidities (e.g., Calvo pricing), this level of output is unattainable, at least in the short run.

Price/Quantity Decisions when Demand Is Unknown

In order to understand the price/quantity strategies of firms in MABMs and compare them with firms' decisions in the standard model, in this “intermezzo” we will keep the basic structure of the Dixit–Stiglitz framework, but will relax the assumption of perfect information as far as market demand is concerned. We will assume instead that, at the moment decisions on price and quantity must be made, the firm does not know the position of the demand curve on the price/quantity space.

Uncertainty on market demand may be due to the fact that transactions do not occur at the same time on all markets. There is a sequence of events which take place at different times on different markets. For instance, we can envisage the following 4 steps: (1) At the beginning of period t (before the market for C-goods opens), the firm decides the price and the quantity. On the basis of the production plan, it decides whether to hire or fire workers. Suppose the workforce must be expanded to fulfill the production plan. (2) The firm enters the labor market by posting vacancies at a certain wage. These vacancies may or may not be filled depending on the number of matches between demand (job offers) and supply (unemployed workers) of labor. Suppose all the vacancies are filled. (3) The firm can therefore implement the desired production plan. (4) The quantity produced is then sold on the market of C-goods. Production must be decided in step (1) but actual demand will be revealed to the firm only in step (4).

The information set of the firm at the moment the scale of activity must be decided may contain precise information on the pricing behavior of competitors and the total number of firms but information may be incomplete and imperfect on the price elasticity of demand and/or total households' consumption. In order to maximize expected profits, the firm must estimate the elasticity of demand and total consumption planned by households. The market demand that the ith firm expects is

(25) $C_{i,t}^e={\left(\frac{P_{i,t}}{P_t}\right)}^{-{\epsilon }_t^i}\frac{C_t^i}{F_c}$

where ${\epsilon }_t^i$ and $C_t^i$ are price elasticity and total consumption expected or estimated by firm i. In order to maximize profits in this informational scenario, the firm could set the price as a mark up over marginal cost, the mark-up being a function of the estimated price elasticity:

(26) $P_{i,t}=\frac{1+{\mu }_t^i}{\alpha }{w}_t$

where ${\mu }_t^i={({\epsilon }_t^i-1)}^{-1}$. Plugging (26) into (25), one gets

(27) $C_{i,t}^e={\left(\frac{1+{\mu }_t^i}{\alpha }\frac{w_t}{P_t}\right)}^{-{\epsilon }_t^i}\frac{C_t^i}{F_c}.$

The firm uses expected demand to plan production, $Y_{i,t}^{⁎}=C_{i,t}^e$. For the sake of discussion, let's assume that actual production is equal to desired production, $Y_{i,t}=Y_{i,t}^{⁎}$. Hence⁴⁴

(28) $Y_{i,t}=Y_{i,t}^{⁎}=C_{i,t}^e={\left(\frac{1+{\mu }_t^i}{\alpha }\frac{w_t}{P_t}\right)}^{-{\epsilon }_t^i}\frac{C_t^i}{F_c}.$

At the beginning of time t (step (1) in the sequence above) the status quo for the ith firm is the point $(P_{i,t},Y_{i,t})$ on the price/quantity space. The coordinates are determined by (26) and (28). Once production has been carried out, transactions can take place. Once transactions are completed, the ith firm can observe the amount of goods actually sold $Q_{i,t}=\text{min}(Y_{i,t},C_{i,t})$ where $C_{i,t}$ is actual demand.

Since sales occur only after the firm has carried out production, actual demand can differ from current production. If $Q_{i,t}=C_{i,t}<Y_{i,t}$, there will be involuntary inventories whose size is ${\mathrm{\Delta }}_{i,t}=Y_{i,t}-C_{i,t}>0$. This is a signal of excess supply, i.e., of a positive forecasting error (demand has been overestimated).

If $Q_{i,t}=Y_{i,t}<C_{i,t}$, all the output will be sold and therefore there will not be involuntary inventories, ${\mathrm{\Delta }}_{i,t}=0$. The difference $Y_{i,t}-C_{i,t}>0$ measures the demand of unsatisfied consumers, who may turn to other producers (goods are substitutable, albeit only imperfectly) or save involuntarily. This is a signal of excess demand. The firm has made a negative forecasting error (demand has been underestimated).

It can also happen, of course, that demand equals production, $Y_{i,t}=C_{i,t}$. There will be no forecasting error (no inventories and no queues) in this case.

Inventories are defined as

(29) ${\mathrm{\Delta }}_{i,t}=\min (Y_{i,t}-C_{i,t},0).$

In the case of excess demand or excess supply the firm is underperforming,⁴⁵ and has an incentive to choose a different price/quantity in the future, say $(P_{i,t+1},Y_{i,t+1})$.

One possible strategy consists in setting the direction of the change in price and quantity in line with the sign of the forecasting error. For instance, if ${\mathrm{\Delta }}_{i,t}>0$, the firm has an incentive to scale down production and cut the price. The size of the change may be linked to the magnitude of the error.

Notice that by making mistakes the firm collects new information on market conditions. As an alternative strategy, the firm can therefore make new estimates of demand and set the price and the quantity accordingly (see Eqs. (26) and (28)).

In the end, uncertainty and imperfect knowledge induce the firm to adapt period by period to evolving market conditions. This adaptation takes the form of price and quantity updating rules. There is no guarantee that this adaptation will lead to the type of equilibrium discussed above.

The Agent-Based Approach to Price/Quantity Decisions

The MABMs we are surveying generally conceive firms as production/trading units immersed in a market environment which is utterly uncertain. Uncertainty is due to the sequential nature of the market economy (which is easily replicated in an agent based framework) and to the ever changing conditions of demand and supply.

Different MABMs deal with price/quantity strategies in different ways. For most of the models we can detect a general pattern, however, in the description of the behavioral rules the firm adopts.⁴⁶ The firm starts with an initial pair of price and quantity, ($P_{i,t},Y_{i,t}$), which we define the status quo. At the end of period t, once transactions have been carried out, the firm observes actual sales $Q_{i,t}=\text{min}(Y_{i,t},C_{i,t})$ and learns also the average price $P_t$, which is a proxy of the price charged on average by the firms' competitors.

If $Q_{i,t}=C_{i,t}<Y_{i,t}$, there will be an inventory equal to ${\mathrm{\Delta }}_{i,t}=Y_{i,t}-C_{i,t}>0$. Notice that the firm may want to hold a buffer stock of finished goods (desired or voluntary or target inventories). Desired inventories may be an interval, ${\mathrm{\Delta }}_{i,t}\in [{\mathrm{\Delta }}_{i,t}^m,{\mathrm{\Delta }}_{i,t}^M]$, where the extremes are the minimum and maximum levels of desired inventories, or collapse to a single point, ${\mathrm{\Delta }}_{i,t}^{⁎}$. In the first case there will be a shortage of inventories if ${\mathrm{\Delta }}_{i,t}<{\mathrm{\Delta }}_{i,t}^m$ and excess (or involuntary) inventories ${\mathrm{\Delta }}_{i,t}^u={\mathrm{\Delta }}_{i,t}-{\mathrm{\Delta }}_{i,t}^M$ if ${\mathrm{\Delta }}_{i,t}>{\mathrm{\Delta }}_{i,t}^M$. In the second case there will be a shortage of inventories (resp. involuntary inventories) if ${\mathrm{\Delta }}_{i,t}<{\mathrm{\Delta }}_{i,t}^{⁎}$ (resp. ${\mathrm{\Delta }}_{i,t}>{\mathrm{\Delta }}_{i,t}^{⁎}$). It is reasonable to assume that planned production for period t will take into account expected demand in the same period and desired inventories.

Of course, desired inventories may be zero, ${\mathrm{\Delta }}_{i,t}^{⁎}=0$. In this case, planned production is equal to expected demand, $Y_{i,t}=C_{i,t}^e$, the difference $e_{i,t}=C_{i,t}-Y_{i,t}=C_{i,t}-C_{i,t}^e$ therefore is the forecasting error. A positive inventory will always be involuntary and coincide with the absolute value of the (negative) forecasting error, $e_{i,t}<0$ and ${\mathrm{\Delta }}_{i,t}=|e_{i,t}|>0$. If $Q_{i,t}=Y_{i,t}<C_{i,t}$, the forecasting error is positive ($e_{i,t}>0$), there are no inventories (${\mathrm{\Delta }}_{i,t}=0$) but a fringe of unsatisfied consumers.

Desired inventories are generally determined by applying a desired or target inventory-to-sales ratio ${\delta }^{⁎}$ to current production ${\mathrm{\Delta }}_{i,t}^{⁎}={\delta }^{⁎}{Y}_{i,t}$. This ratio represents the number of periods of production at the current rate necessary to build desired inventories. Hence the minimum and maximum levels of voluntary inventories are determined as follows: ${\mathrm{\Delta }}_{i,t}^m={\delta }_m{Y}_{i,t}$ and ${\mathrm{\Delta }}_{i,t}^M={\delta }_M{Y}_{i,t}$, $0<{\delta }_m<{\delta }_M$. The parameters ${\delta }_m$ and ${\delta }_M$ are the minimum and the maximum inventory-to-sales ratios. If there are no desired inventories, then ${\delta }^{⁎}=0$.

The firm receives two types of signals of market conditions: (i) an excess or shortage of inventories (with respect to the target, which may be an interval, a positive value or zero); (ii) the relative price, i.e., the ratio or difference between the individual price and average price. From signals of type (i) the firm infers that it has underestimated or overestimated the strength of demand: a shortage of inventories reveals stronger than expected demand; an involuntary inventory signals weaker than expected demand. From signals of type (ii) the firm infers its position relative to competitors: if the firm's price is lower than the average price, the firm is undercutting competitors (and its market share is bigger than the average market share); if the opposite holds true, the good produced by the firm is overpriced with respect to competitors (and its market share is smaller than the average market share).⁴⁷ The sign and size of signals determine the direction of the price and quantity change to be implemented. The magnitude of the price and quantity change is stochastic and/or governed by the signals. The change in price and quantity therefore will be governed by behavioral rules which take the form of updating algorithms.

The most general form of a quantity updating algorithm we can detect in the MABMs under scrutiny is

(30) $Y_{i,t+1}^{⁎}=\{\begin{array}{cc}{Y}_{i,t}(1+{\eta }^y)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}<{\mathrm{\Delta }}_{i,t}^m\text{ and }{P}_{i,t}>P_t\text{,}\hfill \\ Y_{i,t}(1-{\eta }^y)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}>{\mathrm{\Delta }}_{i,t}^M\text{ and }{P}_{i,t}<P_t\text{.}\hfill \end{array}$

In words, the firm revises production plans up (resp. down) by ${\eta }^y$ if there is a shortage (resp. excess) of inventories and the current price is greater (resp. smaller) than the competitors' price. The assumption of an upper and a lower bound on the level of desired inventories introduces inertia in quantity updating: if actual inventories fall in the desired interval, the firm does not change the scale of activity.

The most general form of a price updating algorithm is

(31) $P_{i,t+1}^{⁎}=\{\begin{array}{cc}{P}_{i,t}(1+{\eta }^p)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}<{\mathrm{\Delta }}_{i,t}^m\text{ and }{P}_{i,t}<P_t\text{,}\hfill \\ P_{i,t}(1-{\eta }^p)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}>{\mathrm{\Delta }}_{i,t}^M\text{ and }{P}_{i,t}>P_t\text{.}\hfill \end{array}$

In words, the firm revises the price up (resp. down) by ${\eta }^p$ if there is a shortage (excess) of inventories and the current price is smaller (greater) than the competitors' price.

We will now survey the specifications of these general rules adopted in some of the MABMs under scrutiny. Let's start with a small MABM. LEN posits the following quantity updating rule:

(32) $Y_{i,t+1}^{⁎}=\{\begin{array}{cc}{Y}_{i,t}^{⁎}+1\hfill & \text{if }{\mathrm{\Delta }}_{i,t}<{\mathrm{\Delta }}_{i,t}^m\text{,}\hfill \\ Y_{i,t}^{⁎}-1\hfill & \text{if }{\mathrm{\Delta }}_{i,t}>{\mathrm{\Delta }}_{i,t}^M\text{.}\hfill \end{array}$

In words, the firm scales up production by one unit if inventories are “too low”, i.e., smaller than the minimum. On the other hand, the firm scales down production if inventories are “too high”, i.e., greater than the maximum. LEN assumes the firm has a one-to-one technology which uses only labor as input, $Y_{i,t}=N_{i,t}$. Therefore, in order to scale up production by one unit, the firm has to post a vacancy. The firm scales down production by one unit by firing a worker chosen at random in the employed workforce.⁴⁸

The rule (32) is a special case of (30), in which the signal coming from the relative price is ignored. Moreover, the change in quantity is a discrete step of size one.

As to pricing, LEN assumes that the firm wants the actual mark-up ${\mu }_{i,t}=\frac{P_{i,t}}{M{C}_{i,t}}-1$ to fall into a prespecified interval ${\mu }^m<{\mu }_{i,t}<{\mu }^M$ where $0<{\mu }_m<{\mu }_M$. The introduction of an upper and a lower bound on mark-ups mitigates price volatility. In fact, the current price is deemed adequate if $P_{i,t}^m<P_{i,t}<P_{i,t}^M$ where $P_{i,t}^m=(1+{\mu }_m)M{C}_{i,t}$ and $P_{i,t}^M=(1+{\mu }_M)M{C}_{i,t}$ are the minimum and maximum prices. LEN posits the following price updating rule:

(33) $P_{i,t+1}=\{\begin{array}{cc}{P}_{i,t}(1+{\eta }^p)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}<{\mathrm{\Delta }}_{i,t}^m\text{ and }{P}_{i,t}<P_{i,t}^M\text{,}\hfill \\ P_{i,t}(1-{\eta }^p)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}>{\mathrm{\Delta }}_{i,t}^M\text{ and }{P}_{i,t}>P_{i,t}^m\hfill \end{array}$

where ${\eta }^p$ is a random draw from a uniform distribution. In words, the firm raises the posted price if inventories are “too low” provided the current price is smaller than the maximum. On the other hand, the firm cuts the price if inventories are “too high” provided the current price is higher than the minimum. The size of the rate of change is stochastic.

LEN introduces inertia in price changes which may result in stickiness. In fact, there is an interval of values for inventories such that the firm does not change the price.⁴⁹

Expression (33) is a special case of (31), in which the signal coming from the relative price has been replaced by an upper and lower bound on the range of the mark-up (with no reference to the prices charged by the competitors).

In AGH, before deciding to enter the market and produce, the firm conducts market research by sending out a message to people who are prospective buyers. This message contains the “normal” posted price $P_{i,t}=(1+\mu )w_{i,t}$ where $w_{i,t}$ is the posted wage.⁵⁰ The mark-up μ is drawn from a uniform distribution.

As in LEN, AGH assume that the firm wants to keep a buffer stock of finished goods. Contrary to LEN, however, the desired size of inventories is unique, ${\mathrm{\Delta }}_{i,t}^{⁎}={\delta }^{⁎}{Y}_{i,t}$. AGH assume a fairly simple price updating rule:

(34) $P_{i,t+1}=\{\begin{array}{cc}{P}_{i,t}(1+{\eta }^p)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}<{\mathrm{\Delta }}_{i,t}^{⁎}\text{,}\hfill \\ P_{i,t}(1-{\eta }^p)\hfill & \text{if }{\mathrm{\Delta }}_{i,t}>{\mathrm{\Delta }}_{i,t}^{⁎}\hfill \end{array}$

where ${\eta }^p$ is a random draw from a uniform distribution. Rule (34) is a special case of (31), in which ${\delta }_m={\delta }_M={\delta }^{⁎}$ and the relative price has no role to play.

The prospective customer (household h) decides to buy if the price posted by firm i is smaller than the price the household previously paid – by purchasing goods at firm j – updated by means of expected inflation, equal to the inflation target of the central bank. In symbols, $P_{j,t-1}(1+{\pi }^T)>P_{i,t}$.

Transactions occur in the following way. The household visits two stores, 1 and 2. Consider one of the two, say, store 1, which has inventories ${\mathrm{\Delta }}_{1,t}$. The household can place an order for $C_{1,h,t}$⁵¹ if a cash-in-advance constraint is satisfied, $P_{1,t}{C}_{1,h,t}<M_{h,t}$ where $M_{h,t}$ are the household's money holdings. The store offers an amount $Y_{1,t}=\min (C_{1,h,t},{\mathrm{\Delta }}_{1,t})$. Suppose the household previously purchased goods at firm 2. The households decides to buy from firm 1 if $P_{1,t}<P_{2,t-1}(1+{\pi }^T)$.

Upon entering the market, according to AGH, the firm sets a target for the labor input $N_{i,t}^{⁎}$, which must satisfy the requirement to produce desired output $Y_{i,t}^{⁎}$, given the fixed labor input (overhead costs) F and the need to move closer to desired inventories:

(35) $N_{i,t}^{⁎}=Y_{i,t}^{⁎}+F+{\lambda }_{\mathrm{\Delta }}({\delta }^{⁎}{Y}_{i,t}^{⁎}-{\mathrm{\Delta }}_{i,t})$

where ${\lambda }_{\mathrm{\Delta }}$ is the inventory adjustment speed.

In CATS, firms explore the price-quantity space around their status quo $(P_{i,t},Y_{i,t})$ in order to maximize profits by adapting to the market environment. For simplicity, in this overview of firms' behavior in CATS, we will assume that planned production can be always carried out, $Y_{i,t}^{⁎}=Y_{i,t}$.⁵² Notice that desired inventories are zero in this setting, i.e., ${\delta }^{⁎}=0$.

Once transactions have been carried out, the information set of the firm ${\mathrm{\Omega }}_{i,t}=(P_t,e_{i,t})$ consists of two signals: the average price $P_t$ and the forecasting error $e_{i,t}=C_{i,t}-Y_{i,t}$. These signals capture – albeit imprecisely – the distance between the firm's actual position (the status quo) and the “benchmark”, i.e., a situation in which all firms charge the same price and demand equals supply: $P_{i,t}=P_t$ and $Y_{i,t}=C_{i,t}$.⁵³ There are four cases: (a) $P_{i,t}>P_t,e_{i,t}>0$; (b) $P_{i,t}<P_t,e_{i,t}>0$; (c) $P_{i,t}>P_t,e_{i,t}<0$; (d) $P_{i,t}<P_t,e_{i,t}<0$. On the basis of this information, the firm decides (i) whether to change the price or the quantity in period $t+1$ (with respect to the status quo in period t), (ii) in which direction, and (iii) the size of such a change.

As to quantity, the firm sets in period t desired production $Y_{i,t+1}^{⁎}$ for period $t+1$ at the level of expected future demand, i.e., $Y_{i,t+1}^{⁎}=C_{i,t+1}^e$. Expectations of future demand are based on the forecasting error $e_{i,t}$. CATS/ADG assume that the firm decides to update the desired scale of activity as follows:

(36) $Y_{i,t+1}^{⁎}=C_{i,t+1}^e=Y_{i,t}(1+\rho e_{i,t})\text{if}\{\begin{array}{c}{e}_{i,t}>0\text{ and }{P}_{i,t}>P_t\text{,}\hfill \\ e_{i,t}<0\text{ and }{P}_{i,t}<P_t\hfill \end{array}$

where $\rho \in (0,1)$. The firm revises expected demand (and desired production) up if there is a queue of unsatisfied consumers ($e_{i,t}>0$) and the price is higher than the average price (case (a)). The change in the scale of activity is proportional to the size of the error. If the firm's price is smaller than the average price, the firm does not adjust production even in the presence of a queue (case (b)). As we will see below, in case (b) the firm will revise the price up.

The firm revises expected demand and desired production down if $e_{i,t}=-{\mathrm{\Delta }}_{i,t}<0$ and the price is lower than the average price (case (d)). The decrease in activity in this case is proportional to the level of involuntary inventories, $Y_{i,t+1}-Y_{i,t}=-\rho {\mathrm{\Delta }}_{i,t}$. If $e_{i,t}<0$ but the price is higher than the average price, on the contrary, the firm will cut the price, keeping the quantity constant (case (c)).

Expression (36) is a special case of (30) with ${\eta }^y=\rho (C_i-Y_i)$ in case of excess demand and ${\eta }^y=-\rho {\mathrm{\Delta }}_{i,t}$ in case of excess supply.

This updating rule is based implicitly on an adaptive scheme to compute expected future demand, $C_{i,t+1}^e=C_{i,t}^e+\rho (C_{i,t}-C_{i,t}^e)$. Recalling that $C_{i,t}^e=Y_{i,t}$, by iterating, it is easy to see that expected future demand is a weighted average of current and past demand with exponentially decaying weights:

(37) $Y_{i,t+1}=C_{i,t+1}^e=\rho \sum _{s=0}^{\infty }{(1-\rho )}^s{C}_{i,t-s}.$

As to the price, CATS/ADG assume the following updating rule:

(38) $P_{i,t+1}=\{\begin{array}{cc}{P}_{i,t}(1+{\eta }^p)\hfill & \text{if }{e}_{i,t}>0\text{ and }{P}_{i,t}<P_t\text{,}\hfill \\ P_{i,t}(1-{\eta }^p)\hfill & \text{if }{e}_{i,t}<0\text{ and }{P}_{i,t}>P_t\hfill \end{array}$

where ${\eta }^p⩾0$ is drawn from a uniform distribution. The firm revises the price up if there is a queue of unsatisfied consumers and the price is lower than the average price (case (b)): an increase of the price will mitigate excess demand for the good in question. If the firm's price is higher than the average price, in the presence of a queue the firm adjusts not the price but quantity (case (a)).

It is easy to see that (38) is a special case of (31) with ${\delta }^{⁎}=0$.

The firm revises the price down if there is an involuntary inventory and the price is higher than the average price (case (c)): a reduction of the price will boost demand (and will allow increasing the market share). If the firm's price is lower than the average price, in the presence of inventories the firm adjusts not the price but quantity (case (d)).

The mark-up therefore is determined residually by the joint dynamics of the price – governed by the stochastic adjustment mechanism described in (38) – and the average cost, which is determined endogenously.

It is worth noting that once the direction of the price change has been decided on the basis of the signals, the size of the price change is determined by a random draw. Contrary to the updating rule for the quantity, in the updating rule for the price the size of the adjustment does not depend on the size of the error. In CATS/MBU, there is symmetry in the updating rules because the sizes of both the price and quantity adjustments are stochastic.

In this MABM an (unexpected) upward shift of the demand function (due, for instance, to an increase of total output, which will boost the consumption budget of each and every household) will create a queue and bring about an increase of production if the price is above the average price (case (a)); an increase of the price if the latter is below the average price (case (b)).

Financial factors seem to play no role in the price and quantity decisions of firms. This is not true. First, CATS assume there is a lower bound of the firm's price: the firm will never set the price below the average cost, which consists of operating costs (incurred to hire workers and purchase capital goods) and debt commitments. If interest payments increase, the average cost goes up and this may affect price and quantity decisions.

Second, once the scale of production has been decided, the firm goes to the bank asking for a loan to fill the financing gap (see below, Section 2.4.5). But the bank may refuse to extend a loan of the requested size. In this case the firm is rationed and has to scale down production. In other words, the firm is subject to an ex post borrowing constraint.

In a couple of CATS models characterized by a network based financial accelerator, planned production is financially constrained ex ante. For instance, Riccetti et al. (2013, 2016b) assume that the firm takes explicitly into account its balance sheet in deciding production. The production function is $Y_{i,t}=K_{i,t}^{\beta }$ where $K_{i,t}$ is the firm's capital stock and $\beta \in (0,1)$. The firm sets a leverage target ${\lambda }_{i,t}^{T}t$ where leverage is defined as the ratio of debt (bank loans) to net worth.⁵⁴ Hence the target level of debt is $L_{i,t}^T={\lambda }_{i,t}^T{E}_{i,t}$ where $E_{i,t}$ is net worth. Since $K_{i,t}=L_{i,t}^T+E_{i,t}$ by definition, it turns out that

(39) $Y_{i,t}={[(1+{\lambda }_{i,t}^T)E_{i,t}]}^{\beta }.$

In a previous model, Delli Gatti et al. (2010) posit a financially constrained output function $Y_{i,t}=E_{i,t}^{\beta }$ which is a special case of Riccetti et al. (2013, 2016b).

In EUBI, following the “Management Science” approach, the procedure determining a firm's price and quantity is based on heuristics put forward in the literature on Operations Management and strategic pricing. A given firm (say firm i) in a given period $t_0$ knows the market size (total consumption expenditure) in the past n periods, i.e., $C_{t_0-\tau }={\sum }_{h=1}^H{C}_{h,t_0-\tau }$ for $\tau =1,2,\dots ,n$. This means that the firm has a time series of aggregate consumption consisting of n data points. The firm estimates total consumption by simple linear regression, the regressor being time: $C_t^e=\hat{C}+\hat{g}t$ where $\hat{C}$ and $\hat{g}$ are the estimated intercept and slope of the regression line. Hence the estimated current market size will be $C_{t_0}^e=\hat{C}+\hat{g}{t}_0$ and the future market size at time $t_0+T$ can be estimated as follows: $C_{t_0+T}^e=\hat{C}+\hat{g}(t_0+T)=C_{t_0}^e+\hat{g}T$. From this simple statistical exercise, the firm derives a way of updating expectations of aggregate demand for C-goods.