Balancing the current account: experimental evidence on underconsumption

Abstract

How should countries in a fixed exchange rate system balance their current account? This question was at the center of the historical debate between Keynes and White in 1944 and is being debated increasingly these days for the Eurozone. Should consolidation by deficit countries be complemented by higher spending in surplus countries in order to avoid a downward bias in the level of consumption? We investigate the associated disequilibrium behavior experimentally, letting our experimental subjects act as heads of state who repeatedly seek to coordinate their policies. If only deficits are punished, as proposed by White, we observe that consumption is persistently reduced and adjustment is slower than in a treatment in which a surplus is also punished, as suggested by Keynes. We find support for the fact that underconsumption occurs for behavioral but not for rational reasons.

Appendices

Appendix 1

There are 6 symmetric countries, each inhabited by a representative household. The household of country j sets a constant level of consumption, savings, investment and labor input for 4 periods and lives only from the savings in a 5th period, which represents retirement. A household lives only for five periods, but in our experiment we want to determine optimal policy choices where subjects can learn from past mistakes. We thus assign the task of setting consumption to the head of state, labelled an adviser in our experiment. This head of state is influential for determining total consumption in the country, for example by designing relevant government policies, and he lives for 20 rounds (each round consisting of 5 periods). We assume that experience gathered by the government exceeds that of single households. This can be justified by assuming that households make an inexperienced decision on retirement once in their lifetime, while heads of state have their experts, constantly evaluate earlier decisions and confront new households with novel incentives. For the sake of simplicity we disregard the dynamic effects that decisions by households in previous rounds have on current consumption. The household seeks to avoid variations in real consumption C _j due to a decreasing marginal utility of consumption. Lifetime utility is given by

$$ U_{j} = 4\sqrt {C_{j} } + \sqrt {4S_{j} } - \rho 4Y_{j} - \gamma \left| {C_{j} + I_{j} - Y_{j} } \right| $$

with I _j being country j’s investments, S _j savings and Y _j income. The first term captures utility in the first 4 periods, the second term denotes utility in the retirement period where households rely on previous savings, labor disutility is denoted by the third term, where labor is proportional to income and ρ denotes real labor costs. Deviations between the household’s savings S _j  = Y _j  − C _j and a country’s investment I _tj must be balanced with those in 5 other countries. This recognizes that the current account must add to zero across all countries. Households thus have an incentive to arrange for retirement provisions through local investment in order to avoid current account imbalances. Financial intermediation or political pressure imposed on countries absorb the share γ < 1 of the transferred real amount, which equals the current account imbalance. Due to symmetry, such transfers can only arise outside Nash equilibrium. This will concern us later and we can leave out the last term for now.

The 6 products of the six countries are homogeneous and can be used either for consumption or investment purposes. Total demand is equal to the sum of all countries’ consumption plus all investment and it is equally split across all 6 producers. Demand for a country’s production is thus given by $Y_{j} = \frac{1}{6}\mathop \sum \nolimits_{h = 1}^{6} \left( {C_{h} + I_{h} } \right)$. Whenever a country demands goods for consumption or investment purposes 1/6 of it will end up as income in the country itself.

Households seek to maximize utility by setting C _j and I _j . Inserting for S _j  = Y _j  − C _j and leaving out the last term, we rewrite

$$ U_{j} = 4\sqrt {C_{j} } + 2\sqrt {\frac{1}{6}\mathop \sum \limits_{h = 1}^{6} \left( {C_{h} + I_{h} } \right) - C_{j} } - \frac{4\rho }{6}\mathop \sum \limits_{h = 1}^{6} \left( {C_{h} + I_{h} } \right) $$

The first-order conditions for an optimum are:

$$ \frac{{dU_{j} }}{{dC_{j} }} = 2C_{j}^{ - 0.5} + \left( {\frac{1}{6}\sum\nolimits_{h = 1}^{6} {\left( {C_{h} + I_{h} } \right) - C_{j} } } \right)^{ - 0.5} \left( {\frac{ - 5}{6}} \right) - \frac{4\rho }{6} = 0 $$

(5)

$$ \frac{{dU_{j} }}{{dI_{j} }} = \left( {\frac{1}{6}\sum\nolimits_{h = 1}^{6} {\left( {C_{h} + I_{h} } \right) - C_{j} } } \right)^{ - 0.5} \frac{1}{6} - \frac{4\rho }{6} = 0 $$

(6)

Jointly this implies

$$ 2C_{j}^{ - 0.5} = \left( {\frac{1}{6}\sum\nolimits_{h = 1}^{6} {\left( {C_{h} + I_{h} } \right) - C_{j} } } \right)^{ - 0.5} . $$

(7)

Imposing symmetry with C _j  = C _i we obtain 4C ⁻¹ _j  = I ⁻¹ _j  ⇒ C ^* _j  = 4I _j . Inserting this and symmetry into (5) yields

$$ \frac{{dU_{j} }}{{dC_{j} }} = 2\left( {4I_{j} } \right)^{ - 0.5} + \left( {I_{j} } \right)^{ - 0.5} \left( {\frac{ - 5}{6}} \right) - \frac{4\rho }{6} = 0 \Rightarrow I_{j}^{*} = \frac{1}{{16\rho^{2} }} . $$

(8)

Chosen levels of investment thus decrease with real labor costs. Optimal levels of investment are given exogenously in our experiment and are not determined by the head of state. According to (8), varying levels of optimal investment require values for ρ to vary across the 20 rounds accordingly. For example, a level of investment of 9, as in round 1, requires $I_{j}^{*} = 9 = \frac{1}{{16\rho^{2} }} \Leftrightarrow \rho = \frac{1}{12}$. The reported sequence of investments can thus be related to optimizing behavior with variations in ρ across rounds. ρ can be assumed to be non-stationary due to a non-stationary evolution of technology.

Is consumption complementary, as assumed in the experiment? In order to investigate this, let us assume optimal investment in all countries but excess consumption by other players −j according to C _−j  = 4(1 + ɛ)I ^* _j  > 4I ^* _j . Production will amount to $Y_{j} = \frac{1}{6}\mathop \sum \limits_{h = 1}^{6} \left( {C_{h} + I_{h} } \right) = I_{j}^{*} + \frac{1}{6}\left( {C_{j} + \mathop \sum \limits_{h \ne j}^{{}} C_{h} } \right) = I_{j}^{*} + \frac{1}{6}\left( {C_{j} + 20\left( {1 + \varepsilon } \right)I_{j}^{*} } \right) = \left( {\frac{13}{3} + \frac{10}{3}\varepsilon } \right)I_{j}^{*} + \frac{1}{6}C_{j}$. The right hand side of (7) denotes (Y _j  − C _j )^−0.5. Inserting for Y _j , we obtain

$$ 2C_{j}^{ - 0.5} = \left( {\left( {\frac{13}{3} + \frac{10}{3}\varepsilon } \right)I_{j}^{*} - \frac{5}{6}C_{j} } \right)^{ - 0.5} \Leftrightarrow C_{j} = 4\left( {1 + \frac{10}{13}\varepsilon } \right)I_{j}^{*} = \left( {1 + \frac{10}{13}\varepsilon } \right)C_{j}^{*} $$

(9)

Consumption is thus complementary across countries. If other countries increase consumption by ɛ, best responding to this implies an increase of consumption in the country in question by less due to 10ɛ/13 < ɛ. As can be seen, this is a reaction in line with Eq. (2), which requests consumption to be proportional to income, C _j  = 0.8Y _j . The fact that this proportional relationship holds true even if other players fail to optimize, becomes apparent by modifying (9): $\frac{13}{12}C_{j} = \frac{13}{12}4\left( {1 + \frac{10}{13}\varepsilon } \right)I_{j}^{*} \Leftrightarrow \frac{5}{4}C_{j} = \left( {\frac{13}{3} + \frac{10}{3}\varepsilon } \right)I_{j}^{*} + \frac{1}{6}C_{j} = Y_{j}$.

How costly is a deviation from the optimum $C_{j} = C_{j}^{*} + \varepsilon = 4I_{j}^{*} + \varepsilon$ when all other players choose the optimum? Utility then amounts to: $U_{j} = 4\sqrt {4I_{j}^{*} + \varepsilon } + 2\sqrt {\frac{1}{6}\left( {30I_{j}^{*} + \varepsilon } \right) - 4I_{j}^{*} - \varepsilon } - \frac{4\rho }{6}\left( {30I_{j}^{*} + \varepsilon } \right) = 4\sqrt {4I_{j}^{*} + \varepsilon } + 2\sqrt {I_{j}^{*} - \frac{5}{6}\varepsilon } - \left( {20\rho I_{j}^{*} + \frac{2\rho }{3}\varepsilon } \right)$. This functional form does not allow for a simple solution. Simulations reveal an increasing marginal disutility the more ɛ differs from zero, the curve coming close to a quadratic loss function. Contrary to this, in our payoff function (2), deviations from optimal values are sanctioned linearly. We introduced this modification for the sake of simplicity and can justify it by pointing out that incentives to heads of state need not be identical to those of households.

An additional sanction for missing the Nash equilibrium is determined by costs for financial intermediation and financial pressure imposed on countries with current account imbalances, γ|C _j  + I _j  − Y _j |. In case of excess consumption by other players −j we have C _−j  = 4(1 + ɛ)I ^* _j  > 4I ^* _j . If player j best responds according to (9), he will run a current account surplus. This is due to $Y_{j} - C_{j} = \left( {\frac{13}{3} + \frac{10}{3}\varepsilon } \right)I_{j}^{*} - \frac{5}{6}C_{j} = \left( {\frac{13}{3} + \frac{10}{3}\varepsilon } \right)I_{j}^{*} - \frac{10}{3}\left( {1 + \frac{10}{13}\varepsilon } \right)I_{j}^{*} = I_{j}^{*} + \frac{10}{13}\varepsilon I_{j}^{*}$. The current account surplus thus amounts to $\frac{10}{13}\varepsilon$ and generates a disutility of the amount $\gamma \frac{10}{13}\varepsilon$. Due to $\gamma < 1$ heads of state will strictly prefer the lower sanction imposed for a current account imbalance to the higher one from missing the consumption target.

Appendix 2

While Hypothesis H2 is intuitive (see explanation in the main text), it is not straightforward to derive even with standard level-k models. Nagel (1995) suggests that L1 players assume L0 players to pick an average value of 50. L1 players then best respond to this non-random number. But in this case, no difference would arise between the Keynes and the White treatment because the uncertainty about other players, in particular the potential failure to correctly predict other player’s behavior, is disregarded. We thus base our level-k simulation on a method suggested by Ho et al. (1998). We assumed that L0 players are uniformly distributed between the decision space of 0 and 100. We then calculated the best response of an L1 player for each value of i _t by randomly drawing 5 values for the 5 other players from the L0 distribution. We hereby build on the mental model in Ho et al. (1998), where “level-L players are assumed to believe that all other players (beside themselves) choose from the level L-1 density (…). Believing this, they mentally simulate n-1 draws from the level L-1 distribution.” (p. 960). Then we calculated the consumption level¹⁸ which best responds to this subsample. In contrast to Ho et al. (1998) we did not let L1 best respond to the average of the five numbers drawn but to each of the five draws individually, this way considering the heterogeneity among L0 players. This modification was immaterial to the Keynes treatment but implied for the White treatment that an L1-player took the uncertainty of the average of the five numbers into consideration. This means that an L1 player mentally draws a value for each of her or his opponents and determines a unique best response for a variety of such draws. We did this 10,000 times to determine the distribution for L1 players, which is depicted in Fig. 10 (blue solid line). We recursively iterated this process by then drawing 5 values out of the L1 distribution, best responding to these values and thereby calculating the L2 distribution, and so forth. The L2 distribution is also shown in Fig. 10 (red solid-dotted line). To illustrate the procedure take a consumption level of e.g. c _mt  = 40. From Fig. 10 it can been seen that the probability of being L1 amounts to around 0.02, while the probability of being L2 amounts to around 0.04. Assuming only L0–L2 the relative probability can be calculated. Due to the fact that L0 is given by the uniform probability of 0.01, the probability mass for all levels L0–L2 is given by 0.07 and the chance of being L0, L1 or L2 is $\frac{1}{7},\,\frac{2}{7}$ and $\frac{4}{7}$ in this example and the player has an expected level of 1.42. In order to converge to the equilibrium value, up to around 20 levels are necessary and were therefore simulated. We did the simulation separately for both treatments. Figure 10 also plots the L1 distribution for the White treatment, which is biased downwards (blue dashed line).

Fig. 10

Kernel density plot level 1 and 2 distributions for $i_{t} = 10$