# Brumm et al (2015): A GDSGE model with asset price bubble¶

Brumm, Grill, Kubler, and Schmedders (2015) study a GDSGE model with multiple assets with different degrees of collaterability. They find that a long-lived asset that never pays dividend but can be used as collateral to borrow can have strictly positive price in equilibrium. From the traditional asset-pricing point of view, this positive price is a bubble because the present discounted value of dividend from the asset is exactly zero.

The model is similar to the ones in Heaton and Lucas (1996) and Cao (2018). It features two representative agents $$h\in\{1,2\}$$ who have Epstein-Zin utility with the same intertemporal elasticity of substitution and different coefficients of risk-aversion:

$U_{h,t}=\left\{\left[c_{h,t}\right]^{\rho^h}+\beta\left[\mathbb{E}_{t}\left(U_{h,t+1}^{\alpha^h}\right)\right]^{\frac{\rho^h}{\alpha^h}}\right\}^{\frac{1}{\rho^h}}.$

The agents can trade in shares of two long-lived assets (Lucas’ trees), $$i\in\{1,2\}$$ and a risk-free bond.

In period $$t$$ (we suppress the explicit dependence on shock history $$z^t$$ to simplify the notation), asset $$i$$ pays dividend $$d_{i,t}$$ worth a fraction $$\sigma_i$$ of aggregate output and traded at price $$q_{i,t}$$. The agents cannot short sell the long-lived asset but they can short sell the risk-free bond, i.e., borrow. In order to borrow in the risk-free bond, the agents must use the assets as collateral. The agents can pledge a fraction $$\delta_i$$ of the value of their asset $$i$$ holding as collateral. In particular, the collateral constraint takes the form:

$\phi^h_{t+1}+\sum_{i\in\{1,2\}}\theta^h_{i,t+1}\delta_i\min_{z^{t+1}|z^t}(q_{i,t+1}+d_{i,t+1})\geq0,$

where $$\phi^h,\theta^h_1,\theta^h_2$$ denote the bond holding, and asset $$1$$ and asset $$2$$ holdings of agents $$h$$.

This model can be written and solved using our toolbox. This is done by Hewei Shen from the University of Oklahoma, who generously contributed the GDSGE code below. Hewei’s own research demonstrates the importance of GDSGE models in studying macro-prudential and fiscal policies in emerging market economies, such as his recent publication and his ongoing work with Siming Liu from Shanghai University of Finance and Economics.

Notice that in the GDSGE code, asset $$1$$ never pays dividend, i.e., $$\sigma_1 = 0$$. Therefore, in finite-horizon economies, by backward induction, its price is always equal to $$0$$. In order for the sequence of finite-horizon equilibria to converge to an infinite-horizon equilibrium in which asset $$1$$ has positive price, we need to assume that agents derive utility from holding the asset in the very last period in each T-period economy:

$U_{h,T} = \left[c_{h,T}\right]^{\rho^h}+\zeta \left[\theta^h_{1,T}\right]^{\rho^h}.$

This assumption is a purely numerical device and becomes immaterial when $$T\rightarrow\infty$$. To solve the last period problem, we use a model_init block that has a different system of equations than that of the main model block (see Cao and Nie (2017) for another example using a different model_init).

## The gmod File¶

BGKS2015.gmod

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 %Options SolMaxIter=20000; PrintFreq=200; SaveFreq =200; USE_SPLINE=1; INTERP_ORDER=2; % Parameters parameters beta alpha1 alpha2 rho1 rho2 e1 e2 sigma1 sigma2 d1 d2 delta1 delta2 zeta zeta = 10; %utility from holding asset 1 % which only applies in the very last period of finite horizon economies beta=0.977; %subjective discount factors alpha1=0.5; % RA of agent 1 = 0.5 alpha2=-6; % RA of agent 2 = 7 rho1=0.5; % IES=2 rho2=0.5; % IES=2 e1=0.085; % agent 1 income share e2=0.765; % agent 2 income share sigma1=0.0; % Tree 1 dividend share, here =0. Tree 1 has no intrinsic value sigma2=0.15; % Tree 2 dividend share d1=sigma1; % Tree 1 dividend d2=sigma2; % Tree 2 dividend delta1=1; % Tree 1 collateralizability delta2=0; % Tree 2 collateralizability % Shock variables var_shock g; %aggregate endowment growth rate % Shocks and transition matrix shock_num = 4; g = [0.72, 0.967, 1.027, 1.087 ]; shock_trans = [ 0.022, 0.054, 0.870, 0.054 0.022, 0.054, 0.870, 0.054 0.022, 0.054, 0.870, 0.054 0.022, 0.054, 0.870, 0.054 ]; % State variables var_state w1; % wealth share w1 = linspace(0,1,240); % Variable for the last period var_policy_init c1 c2 q1 theta11; inbound_init theta11 0 1; %agent 1: tree 1 holding, constrained by no-short selling condition inbound_init q1 0 100; inbound_init c1 1e-10 1; inbound_init c2 1e-10 1; var_aux_init v1 v2; model_init; budget_1 = c1 + theta11*q1 - e1 - w1*(q1+ d1 +d2); FOC1 = c1^(rho1-1)-q1*zeta*theta11^(rho1-1); FOC2 = c2^(rho2-1)-q1*zeta*(1-theta11)^(rho2-1); v1 =(c1^(rho1)+ zeta*theta11^(rho1))^(1/(rho1)); v2 =(c2^(rho2)+ zeta*(1-theta11)^(rho2))^(1/(rho2)); equations; budget_1; c1+c2-1; FOC1; FOC2; end; end; % Endogenous variables and bounds var_policy c1 c2 theta11 theta21 nphi1 nphi2 mu_theta11 mu_theta21 mu_theta12 mu_theta22 muphi1 muphi2 q1 q2 p w1n[4]; inbound c1 1e-10 1; inbound c2 1e-10 1; inbound theta11 0 1; %agent 1: tree 1 holding, constrained by no-short selling condition inbound theta21 0 1; %agent 1: tree 2 holding, constrained by no-short selling condition inbound nphi1 0 100; inbound nphi2 0 100; inbound mu_theta11 0 100; %multipliers on no-short selling constraint on trees inbound mu_theta21 0 100; inbound mu_theta12 0 100; inbound mu_theta22 0 100; inbound muphi1 0 100; %multipliers on the bond position inbound muphi2 0 100; inbound q1 0 100; inbound q2 0 100; inbound p 0 100; inbound w1n -0.05 1.05 adaptive(2); % Extra output variables var_aux phi1 phi2 theta12 theta22 v1 v2 collat_premium; % Interpolation objects var_interp c1p c2p v1p v2p q1p q2p pp; % Initialize using model_init initial c1p c1; initial c2p c2; initial v1p v1; initial v2p v2; initial q1p q1; initial q2p 0; initial pp 0.0; % Time iterations update c1p = c1; c2p = c2; v1p = v1; v2p = v2; q1p = q1; q2p = q2; pp = p; % Variables to be used in simulation if SIMU_RESOLVE=1 var_output c1 c2 v1 v2 theta11 theta21 theta12 theta22 phi1 phi2 q1 q2 p collat_premium w1n; model; % Interpolation [c1p',c2p', v1p', v2p', q1p', q2p', pp'] = GDSGE_INTERP_VEC'(w1n'); % Expectations ev1 = GDSGE_EXPECT{ (g'*v1p')^(alpha1)}; ev2 = GDSGE_EXPECT{ (g'*v2p')^(alpha2)}; expuc1 = GDSGE_EXPECT{ (g'^alpha1)*(v1p'^(alpha1-rho1))*((c1p'/c1)^(rho1-1))/g'}; expuc2 = GDSGE_EXPECT{ (g'^alpha2)*(v2p'^(alpha2-rho2))*((c2p'/c2)^(rho2-1))/g'}; expucq1h= GDSGE_EXPECT{ (g'^alpha1)*(v1p'^(alpha1-rho1))*((c1p'/c1)^(rho1-1))*(q1p' + d1)}; expucq1f= GDSGE_EXPECT{ (g'^alpha2)*(v2p'^(alpha2-rho2))*((c2p'/c2)^(rho2-1))*(q1p' + d1)}; expucq2h= GDSGE_EXPECT{ (g'^alpha1)*(v1p'^(alpha1-rho1))*((c1p'/c1)^(rho1-1))*(q2p' + d2)}; expucq2f= GDSGE_EXPECT{ (g'^alpha2)*(v2p'^(alpha2-rho2))*((c2p'/c2)^(rho2-1))*(q2p' + d2)}; q1p_min = GDSGE_MIN{q1p'}; q2p_min = GDSGE_MIN{q2p'}; g_min = GDSGE_MIN{g'}; % bond transformation theta12=1-theta11; %calculate the tree holding for agent 2 theta22=1-theta21; collat1= delta1*theta11*(q1p_min+d1) + delta2*theta21*(q2p_min+d2); %collateral values collat2= delta1*theta12*(q1p_min+d1) + delta2*theta22*(q2p_min+d2); phi1 = (nphi1 - collat1)*g_min; %back out the bond holdings phi2 = (nphi2 - collat2)*g_min; %Euler Equations EE1_q1= beta*(ev1^((rho1-alpha1)/(alpha1)))*expucq1h + muphi1*delta1*g_min*(q1p_min + d1)+ mu_theta11-q1; EE1_q2= beta*(ev1^((rho1-alpha1)/(alpha1)))*expucq2h + muphi1*delta2*g_min*(q2p_min + d2)+ mu_theta21 - q2; EE1_p= beta*(ev1^((rho1-alpha1)/(alpha1)))*expuc1 + muphi1 -p ; EE2_q1= beta*(ev2^((rho2-alpha2)/(alpha2)))*expucq1f + muphi2*delta1*g_min*(q1p_min + d1)+ mu_theta12-q1; EE2_q2= beta*(ev2^((rho2-alpha2)/(alpha2)))*expucq2f + muphi2*delta2*g_min*(q2p_min + d2)+ mu_theta22- q2; EE2_p= beta*(ev2^((rho2-alpha2)/(alpha2)))*expuc2 + muphi2-p; % Budget constraint agent 1 budget_1 = c1 + p*phi1 + theta11*q1 + theta21*q2 - e1 - w1*( q1+ q2 +d1 +d2); % Consistency w1_consis' = ( theta11*(q1p' + d1) + theta21*(q2p' + d2) + phi1/g' )/(q1p' + q2p' + d1 +d2) - w1n'; % Aux variables v1 =( c1^(rho1)+ beta* (ev1^((rho1)/(alpha1))))^(1/(rho1)); v2 =( c2^(rho2)+ beta* (ev2^((rho2)/(alpha2))))^(1/(rho2)); collat_premium = (q1 - q2*sigma1/sigma2)/(q1+q2); equations; EE1_q1; EE2_q1; EE1_q2; EE2_q2; EE1_p; EE2_p; phi1+phi2; nphi1*muphi1; nphi2*muphi2; mu_theta11*theta11; mu_theta21*theta21; mu_theta12*theta12; mu_theta22*theta22; budget_1; c1+c2-1; w1_consis'; end; end; simulate; num_periods = 10000; num_samples = 100; initial w1 0.5; initial shock 1; var_simu c1 c2 theta11 theta21 theta12 theta22 phi1 phi2 q1 q2 p collat_premium; w1' = w1n'; end; 

## Results¶

In this example, agents $$1$$ ($$\alpha_1 = 0.5$$) are less risk-averse than agents $$2$$ ($$\alpha_2 = -6$$) so they tend to borrow to invest in the risky assets $$1$$ and $$2$$. Asset $$1$$ nevers pays dividend ($$\sigma_1=0$$) but can be used as collateral to borrow ($$\delta_1 = 1$$). While asset $$2$$ pays dividend ($$\sigma_2>0$$) but cannot be used as collateral ($$\delta_2=0$$). In equilibrium, asset $$1$$ still commands a positive price. Therefore, this is an asset price bubble arising from collaterability. The following figure shows the price of asset $$1$$ relative to aggregate output as a function of agents $$1$$’s wealth share

$\omega^1_t = \frac{\theta^1_{1,t}(q_{1,t}+d_{1,t})+\theta^1_{2,t}(q_{2,t}+d_{2,t})+\phi^1_t}{q_{1,t}+d_{1,t}+q_{2,t}+d_{2,t}}.$

The price of asset $$1$$ be as high as several times aggregate output and tends to be higher when agents $$1$$ are relatively richer.