Durable goods user cost and utilisation rate

Good afternoon everyone,

I have a rather theoretical question. I have a durable good that is bought on the market by households and that can be rented to firms to produce a service.

If no renting market exist for the durable good, normally the user cost should be set such that marginal benefit equals marginal costs:

  • Marginal cost = (UmND(t)/UmND(t+1)) * (1/actualisation rate ‘beta’), where ND is the non durable good

  • Marginal benefit = (UmD(t+1)/UmND(t+1))+(1-delta_d), where delta_d is the depreciate rate. The first term represents the marginal utility of durable goods to the marginal utility of non durable goods. And, the second term is the reselling value of the durable good at the end of period, supposing that the price of that durable good is 1.

User cost (=(UmD(t+1)/UmND(t+1))) = Marginal cost - (1-delta_d)

Now, if I incorporate a renting market for the durable good, things get tricky:

  • Marginal cost = (UmND(t)/UmND(t+1)) * (1/beta)

  • Marginal benefit = (UmD(t+1)/UmND(t+1)) - ((1-delta_d)+r_d(t+1)*(1-u(t+1))), where r_d is the rental price of durable goods and u is the utilization of durable goods for home-produced services by households and (1-u) is the utilisation of durable goods for the producer who produces market-based services.

I have two questions:

  1. Should I assume that households rent the durable good stock to firms at a rental rate or at the user cost of durable goods ?

  2. I’m not sure about the expression/intuition of the marginal benefit of durable goods in the case a renting market for durable goods exists. I’m thinking of multiplying the relative marginal utility of durables, (UmD(t+1)/UmND(t+1)), by the utilization rate for households, u. Because, households gains utility from adding one unit of durable goods times the utilisation rate. If I do not suppose that, it is as if they where gaining utility from all of the durable good stock, while they are only gaining utility from a 'portion of time" they use the stock. Note that in the utility function of households my durable goods are multiplied by this utilisation rate u.

Thank you!

To give a reliable answer, you need to provide the full setup and equations. But generally speaking, the two setups should be equivalent, because marginal costs equal marginal benefits. The benefit should be the same regardless of whether the services are outright owned or just rented. I would recommend comparing the setup to a standard RBC model with rental markets in physical capital. If households own the capital stock and rent it to firms, firms pay the rental rate and households need to cover depreciation. Consequently, the rental rate is higher than the net return/interest rate by the amount of deprecation.

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thank you @jpfeifer

But in this case, things get tricky as the household does not simply rent its durable good stock but use some part, u*D, of it for its own consumption. It rents the rest (1-u)*D to the producer. So, I cannot compare this to capital as capital never gives utility to households and does not appear in the utility function of households. I hit the wall when I want to find the SS u and uc. If I wouldn’t have a renting market, it would be straight forward, the user cost (uc) from the Durable Euler equation would only depend on parameters. While if I have a renting market for durable goods, uc depends on u…

        [name='Aggregate consumption function']

        C                   =   (X^alppha_x)*(ES_home^(1-alppha_x));

        [name='Home-produced energy services function']

        ES_home             =   (alppha_d*(u*D)^(siggma_home-1)/siggma_home)+(1-alppha_d)*((A_e_h*E_h)^((siggma_home-1)/siggma_home)))^(siggma_home/(siggma_home-1));

        [name='Home-produced energy services price']

        p_home              =   (((alppha_d^(siggma_home))*(uc)^(1-siggma_home))+((1-alppha_d)^(siggma_home)*(p_e_h/A_e_h)^(1-siggma_home)))^(1/(1-siggma_home)); 

        [name='Relative demand of durable goods compared to non-durable goods']

        uc                  =   ((1-alppha_x)/alppha_x)*(X(+1)/ES_home(+1))*alppha_d*(u(+1))*((ES_home(+1)/((u(+1))*D(+1)))^(1/siggma_home));

        [name='Durable goods Euler equation']

        uc                  =   (1/betta)*((C/(C(+1)*(1+g+n+g*n)))^(1-siggma_ies))*(X(+1)*(1+g+n+g*n)/X)-(r_d(+1)*(1-u(+1))+1-deltta_d);

        [name='Utilization rate decision'] 

        r_d   =   Marginal utility of u; (then the rental price of durable goods r_d=uc/u following the relative demand of durable good equation)

Note that in the BC of households, they get r_d(1-u)*D from the producer that uses the portion (1-u) of the durable good stock

But there still needs to be a no-arbitrage relationship. Even in the case of capital, you could assume households use some fraction for themselves. They would still equalize costs and benefits along the two margins when deciding on the split. The tricky part may be to generate an interior, unique solution for the split.

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Thanks @jpfeifer

What do you mean by a non-arbitrage relationship? The equalization of not renting the durable good to renting the durable good ?

Yes, the household needs to have the same marginal benefit from using the durable good directly and from renting it out and consuming out of the net return.

I knew something was missing. Thanks @jpfeifer

Analytically, does this translate to this condition at steady state:

  • The marginal benefit of consuming the durable good is given by : Marginal utility of Durable good (UmD) + (1-delta), the last term being the reselling value of the durable good.

  • The marginal benefit of renting the durable good: r_d*(1-u) + (1-delta), r_d being the rental rate of the durable good and u the utilisation rate of that durable good.

If I follow this and the user cost of durable good is equal to the marginal utility of durable good UmD, then the rental rate should equal uc/(1-u). If so, then I can solve my steady state

Am I getting it right ?

It’s hard to follow the math without the full equations, but it sounds reasonable. You only have to be careful with the resale value next period. Agents usually care about the expected utility derived from this value rather than the value itself.

@jpfeifer . Here are the maths,

The only problem if I follow what I said earlier is that the user cost SS will be determined by:

        uc                  =   ((((1+g+n+g*n)^siggma_ies)/betta)-1+deltta_d_fix)/(2);

The “2” is bugging me. Without a rental market, the “2” would be “1”. This comes from the equalization of marginal benefit and marginal cost of increasing one unit of durable good (eq.1), and the equalization of the marginal benefit of renting the durable good and the marginal benefit of owning it (eq.2):

  1.    uc (=UmD(+1))                  =   (1/betta)*((C/(C(+1)*(1+g+n+g*n)))^(1-siggma_ies))*(X(+1)*(1+g+n+g*n)/X)-(r_d(+1)*(1-u(+1))+1-deltta_d_fix);
    
  2.    r_d=uc(-1)/(1-u);
    

→ 2uc = ((((1+g+n+g*n)^siggma_ies)/betta)-1+deltta_d_fix)

Note that the model is in efficient unit of labor, where n is the demographic growth rate and g the labor productivity growth rate, hence the (1+g+n+g*n) that appears for lead variables.

Does that “2” bother you too?

Can you please provide the full optimization including the computed FOCs. The user cost should endogenously arise from the problem.

Dear @jpfeifer, here is the full model. I think some mistakes were made during the resolution process.

Thank you
Sharing sector.pdf (104.5 KB)