The framework that we apply is the continuous time Blanchard-Yaari overlapping generation model, see Blanchard (1985). Cohorts enter the labour force at a particular date. To keep the framework as simple as possible, we do not allow for retirement as this does not affect the main argument. People work till they die. Every period, a fixed share d of a cohort dies; d is therefore the inverse lifetime of the worker or the inverse length of working career. We discuss the values of the parameters in our framework as to get an idea about the order of magnitude of various effects. As a fair compromise between life-expectancy and the length of a working career one can think of d as d = 1/50 years = 2% per year. We ignore population growth or changes in productivity. Each cohort runs a system of perfect life-insurance: the assets of the fraction d of the cohort that dies are distributed among the surviving members of the cohort. Hence, there are no bequests to younger generations and each generation fully depletes its own wealth. Apart from their date of birth, all workers are identical.
Generations spend their income on three types of consumption: tradables, domestics, and housing services. They have a Cobb Douglas intertemporal utility function, implying that they spend a fixed share of their lifetime wealth every period and that they distribute that spending among the three commodities in fixed proportions. There is no renting sector in this economy. Workers enter the market without any wealth. At the beginning of their career, they buy a house which they fund with a mortgage. We assume that people can get full funding for their purchase, hence ignoring collateral constraints. Hence, after buying a house, the balance sheet of the new cohort contains the value of their human capital (the net present value of their expected labour income) and their house on the credit side and their mortgage debt on the debit side. Since the latter two cancel, the net wealth of a newly entering cohort is equal to its human capital.
Corresponding to the three types of consumption, production takes place in three industries: tradables, domestics, and construction. Productivity is normalized to unity. The tradable industry’s output can be sold on the world market. When home demand for tradables so requires, this commodity can also be imported. The demand for tradables is infinitely elastic and its price is normalized to unity. Since productivity is equal to unity, since all markets are perfectly competitive, and since there are no other factors of production, wages in the tradable industry are equal to unity. This set up where the nominal price level is fully determined at the global market fits the context of EMU, where the price level for each member state is set by the ECB. Since wages and prices are fully flexible, this does not affect relative prices, but it does affect the denomination of the economy’s public and private debt. De Grauwe (2011) argued that this is the main difference between the UK and Spain: the former can change the real value of its debt by a devaluation of its currency, the latter cannot. Contrary to tradables, domestics can only be consumed within the country. Neither tradables nor domestics can be stored for future consumption. Finally, a house is made up of two parts, land and the building. The supply of land is fixed. Land does not require any maintenance. The building depletes at a fixed rate a, let us say a = 2% per year. Apart from housing, there is no other physical capital. In steady state equilibrium, a fall in the demand for housing services will lead to a lower price for the land and a lower quantity of building.
The capital market of the monetary union is fully unified. Both the government and the private sector have full access to financial markets at the going interest rate r. Since we abstract from growth by keeping labour supply and productivity constant, the relevant rate is the difference between nominal interest rate and nominal growth. We use r = 2% per year. For simplicity, workers use the same discount rate for the calculation of net present value of their human capital. Since productivity and hence wages are normalized to unity, the value of human capital satisfies:
Due to the perfect life insurance scheme that allows each cohort to consume the wealth that it 'inherited’ from its deceased members, a generation can consume a fraction d + r of its wealth without depleting it: r accounts for the interest revenues, d for the 'inherited’ wealth paid out by the life insurance scheme. Since this relation applies to all generations alike, total consumption is a fixed proportion of aggregate wealth:
Hence, the marginal propensity to consume from wealth is 4%, in line with the evidence reported by Case et al. (2013). As long as the economy is in steady state, aggregate wealth is equal to the value of human capital, since housing wealth and mortgage debt cancel. Hence, the value of consumption is equal the value of production. However, after a fall in house prices, aggregate wealth is less than the value of human capital. From equation  and  above, it easy to see that total annual consumption falls below the value of production by (d + r) x Fall in housing wealth. That is equal to 4% of the fall in housing wealth. Generations entering the market after the fall in house prices do not suffer from this negative wealth effect. A fraction d of the old generations dies every period and is replaced by new generations who have not suffered from the decline in house prices. Let t denote time elapsed since the financial crisis. Then, the time path of the negative effect on consumption reads:
The effect will be the largest at the moment that the crisis hits, and will decline exponentially to zero afterwards.
In steady state equilibrium, the cost of housing services are equal to interest rate times the total value of the estate (land + building) plus the depreciation rate times the value of the building. This cost should be equal to share of housing services in total consumption in the Cobb Douglas utility function. A financial crisis will move the economy of its equilibrium path. House prices are temporarily below their new equilibrium value. When bidding for a house, people form expectations about the future path of house prices. If there were no capital market constraints that limit the availability of credit, buyers would pay higher house prices when house prices are expected to rise in the future, as future wealth gains offsets current excess expenditure. In theory, this effect should dampen house price fluctuations, making it hard to rationalize the fall in house prices observed in the Netherland in the first place. The availability of credit seems therefore to be a binding constraint in the aftermath of a financial crisis. The initial effect on house prices due to the combination of the fall in productivity, the reform of tax deductibility of mortgage interest, and austerity policies might therefore be ameliorated by a financial multiplier: lower house prices lead to lower credit, which puts further pressure on house prices. We leave this issue open here, and just take the fall in house prices as given.
The government does four things in this framework. First, it employs part of the workforce as civil servant, e.g. in health care and education. Second, it pays interest on its debt. Third, it collects consumption taxes. Taxes lead to a loss of efficiency due to tax evasion, e.g. by reducing labour supply. Like any Harberger triangle, these losses increase quadratically with the tax rate. Finally, the government runs a mortgage subsidy scheme.
Since people make their lifetime consumption-plan conditional on their expectations on future productivity and future taxes, there has to be a credible commitment to a path of future taxes. The simplest case is when the economy is in a steady state. Then, the tax rate can be maintained at its current level without public debt either im- or exploding. Steady states can be easily calculated for this economy. There are many, depending on the level of public debt. The higher public debt, the higher the interest payments a country has to make. This leaves less means for consumption. Hence, the higher public debt, the higher taxes and the share of the workforce that has to work in the tradable industry.
We apply this framework to analyse the adjustment path of a country to a one time shock, a financial crisis. Though we shall use the risk of future shocks as an argument for imposing constraints on the adjustment process, this risk is not modelled explicitly. Hence, after the unexpected shock, there is no uncertainty left about the future path of productivity. Moreover, policymakers can commit to their future policy. After the crisis, workers and policymakers live therefore in an ideal world of perfect foresight about future productivity and future fiscal policy.
Most macro-economic models require some form wage or price rigidity that prevents markets from clearing. This seems to be a prerequisite for a model with unemployment. Yet, in the framework applied here, wage or price rigidities are not strictly required. Instead, unemployment is interpreted as the time needed to acquire new industry specific human capital when entering a new industry. Mostly, workers acquire this human capital at the start of their career and never change industries. Hence, a new cohort starts unemployed. Each period, a share l of the unemployed manages to acquire this industry specific human capital. Hence, l is the inverse duration of unemployment. However, after a financial crisis, product demand shifts away from domestics to tradables. If this shift is large enough, workers prefer quitting their current industry and retraining for another industry above staying in their current job at a lower wage. Unemployed are free to choose in what industry they want to specialize. Reinhart and Rogoff’s evidence on the persistence of unemployment after a financial crisis suggest l to be 0.25: it takes the average worker about 4 years to retrain for another industry. In practice, the Cobb Douglas utility function allows too much substitution in consumption between industries. Hence, accepting a lower wage in your current job is usually a more attractive strategy than switching to another industry. Hence, we work with a model where wages are fixed. Hence, the economy responds to a fall in the demand for an industry’s output by a reduction in employment, not in wages.
This set up of an economy with three industries captures the main mechanisms at work after a financial crisis has wiped out part of a country’s wealth. Current cohorts have seen their wealth evaporating, partly directly by the financial crisis, partly by the endogenous response of house prices and fiscal policy. Hence, they reduce their consumption as to satisfy their lifetime budget constraint. This reduces domestic demand for all three industries’ output. For tradables, this is no problem, since it can sell its output on global markets at a constant price. For construction and domestics, however, this leads to a fall in demand. Wages in construction are determined by a non-arbitrage condition. Since productivity in all industries is normalized to unity, house-prices are equal to the gross wage in construction. Hence, the value of the specific human capital of workers employed in construction and domestics falls, leading to a fall in employment by not hiring replacements for the workers in the industry that die (=retire). If the fall in consumption is big, construction and domestics will even fire a share of its workforce, which then becomes unemployed to acquire the new specific human capital needed for finding employment in the tradable industry. Only when these cohorts who have experienced a wealth loss due to the financial crisis gradually die out, consumption recovers and hence employment in the domestic and construction industry.