Time to Build & Unanticipated Tariff Changes 
S. Schreiber, 08/08/2019 version
This industry-specific partial equilibrium (PE) model of tariff changes applies to industries where products take time to build or grow. After goods are grown, producers of imports face an unexpected tariff shock and must either sell their product at a lower price, or store some goods to future periods to partially avoid the negative effects of a tariff. The model has three periods. In period 1, producers decide how much to produce for the period 2 market and begin growing their goods. In period 2, an unexpected tariff enters into force and producers decide how much of their grown product to sell in the market and how much to store to period 3, subject to a storage cost. Producers also decide how much to produce for the period 3 market and begin growing those goods. In period 3, producers sell their period 2 goods plus any stored goods. 

This version of the model has THREE sources of supply: domestic production, imports subject to the tariff policy change, and non-subject imports. The model only considers the possibility of storing the subject imported good, not the other two sources of supply. The user inputs initial expenditures on the foreign and domestic products in the initial period, the initial and new tariff rate, elasticity parameters, time cost of money, and ad valorem storage costs. The user can modify data inputs in the simulation by changing the values in the ORANGE - shaded lines in the notebook below. The spreadsheet will  update the estimated changes in economic outcomes that are reported in the GREEN - shaded cells once the user selects "Evaluate Notebook" under "Evaluation" in the Menu above.
   
This model is provided as a generic analytical tool, and the data and parameter values are fictional and illustrative. Actual data and parameter values should be supplied by the user based on the industry and market to which the model is applied. The model is the result of ongoing professional research of USITC staff and may be updated. The model is not meant to represent in any way the view of the U.S. International Trade Commission or any of its individual Commissioners. The model is posted to promote the active exchange of ideas between USITC staff and experts outside the USITC and to provide useful economic modeling tools to the public.
In[1]:= ClearAll[f];
Parameter Inputs
Elasticity of Substitution 
In[2]:= sigma=3;
Total Price Elasticity of Demand for the Sector
In[3]:= eta=-1;
Supply Elasticity of Subject Imports
In[4]:= es=5;
Supply Elasticity of Non-Subject Imports
In[5]:= en=5;
Supply Elasticity of Domestic Shipments
In[6]:= ed=2;
Initial Ad Valorem Tariff in Period 0
In[7]:= t0=0;
New Ad Valorem Tariff in Period 1
In[8]:= t1=0.25;
Initial Expenditures
In[9]:= vdomestic=50;
In[10]:= vsubject=25;
In[11]:= vnonsubject=25;
Time Cost of Money
In[12]:= ROR=0.05;
Ad Valorem Carrying (or Storage) Costs, Subject Imports
In[13]:= cf=0.05;
Hidden Sections
In[14]:= vd0 = vdomestic/vdomestic;
In[15]:= vs0 = vsubject/vdomestic;
In[16]:= vn0 = vnonsubject/vdomestic;
Prices
In[17]:= pd0=1;
In[18]:= ps0=1;
In[19]:= pn0=1;
Quantities
In[20]:= qd0=vd0/pd0;
In[21]:= qs0=vs0/(ps0(1+t0) );
In[22]:= qn0=vn0/pn0 ;
In[23]:= ad=qd0 pd0^-ed;
In[24]:= as=qs0 ps0^-es;
In[25]:= an = qn0 pn0^-en;
In[26]:= bs = (vs0 (ps0/pd0)^(1-sigma))/vd0;
In[27]:= bn = (vn0 (pn0/pd0)^(1-sigma))/vd0;
In[28]:= P0=(pd0^(1-sigma)+bs (ps0(1+t0))^(1-sigma)+bn pn0^(1-sigma))^(1/(1-sigma));
In[29]:= k = qd0 P0^(-sigma-eta) pd0^sigma;
In[30]:= pd1planned = pd0;
ps1planned = ps0;
pn1planned = pn0;
qd1planned = ad pd1planned^ed;
qs1planned = as ps1planned^es;
qn1planned = an pn1planned^en;
Period 2 with no tariff
In[36]:= P2=(pd2^(1-sigma)+bs ps2 ^(1-sigma)+ bn pn2^(1-sigma))^(1/(1-sigma));
In[37]:= EqnD2=ad pd2^ed==(k P2^(sigma+eta))/pd2^sigma;
In[38]:= EqnS2=as ps2^es==(bs k P2^(sigma+eta))/ps2 ^sigma;
In[39]:= EqnN2=an pn2^en==(bn k P2^(sigma+eta))/pn2 ^sigma;
Period 1 with unanticipated tariff
In[40]:= P1=(pd1^(1-sigma)+bs (ps1(1+t1))^(1-sigma)+bn pn1^(1-sigma))^(1/(1-sigma));
In[41]:= EqnD1=ad pd1planned^ed==(k P1^(sigma+eta))/pd1^sigma;
In[42]:= EqnS1=as ps1planned^es==(bs k P1^(sigma+eta))/(ps1(1+t1))^sigma;
In[43]:= EqnN1=an pn1planned^en==(bn k P1^(sigma+eta))/pn1^sigma;
Solve
In[44]:= FindRoot[{EqnD1,EqnS1,EqnN1,EqnD2,EqnS2, EqnN2},{pd1,pd0},{ps1,ps0},{pd2,pd0},{ps2,ps0},{pn1, pn0},{pn2, pn0}]
Out[44]= {pd1->1.,ps1->0.8,pd2->1.,ps2->1.,pn1->1.,pn2->1.}
Assign
In[45]:= pd1nn=pd1/.%;
In[46]:= ps1nn=ps1/.%%;
In[47]:= pd2nn=pd2/.%%%;
In[48]:= ps2nn=ps2/.%%%%;
In[49]:= pn1nn = pn1/.%%%%%;
In[50]:= pn2nn = pn2 /.%%%%%%;
In[51]:= qd1nn=qd1planned;
In[52]:= qs1nn=qs1planned;
In[53]:= qn1nn= qn1planned;
In[54]:= qd2nn=ad pd2nn^ed;
In[55]:= qs2nn=as ps2nn^es;
In[56]:= qn2nn= an pn2nn^en;
Indicator for whether this scenario applies
In[57]:= ScenarioNN=If[ps1nn>(ps2nn (1-cf))/(1+ROR) ,1,0]
Out[57]= 0
Period 2 with no tariff
In[58]:= P2=(pd2^(1-sigma)+bs ps2 ^(1-sigma)+bn pn2^(1-sigma))^(1/(1-sigma));
In[59]:= EqnD2=ad pd2^ed==(k P2^(sigma+eta))/pd2^sigma;
In[60]:= EqnS2=as ps2^es+zs==(bs k P2^(sigma+eta))/ps2 ^sigma;
In[61]:= EqnN2=an pn2^en==(bn k P2^(sigma+eta))/pn2 ^sigma;
Period 1 with unanticipated tariff
In[62]:= P1=(pd1^(1-sigma)+bs (ps1(1+t1))^(1-sigma)+bn pn1^(1-sigma))^(1/(1-sigma));
In[63]:= EqnD1=ad pd1planned^ed==(k P1^(sigma+eta))/pd1^sigma;
In[64]:= EqnS1=as ps1planned^es-zs==(bs k P1^(sigma+eta))/(ps1(1+t1))^sigma;
In[65]:= EqnN1=an pn1planned^en==(bn k P1^(sigma+eta))/pn1^sigma;
Arbitrage condition that the expected return to selling production in period 1  is the same as the expected return from period 2 for marginal unit
In[66]:= EqnArbS=ps1==(ps2 (1-cf))/(1+ROR);
Solve
In[67]:= FindRoot[{EqnD1,EqnS1,EqnD2,EqnS2,EqnArbS,EqnN1, EqnN2},{pd1,pd0},{ps1,ps0},{pd2,pd0},{ps2,ps0},{pn1, pn0},{pn2,pn0},{zs,0}]
Out[67]= {pd1->1.03197,ps1->0.882037,pd2->0.996506,ps2->0.974884,pn1->1.03197,pn2->0.997815,zs->0.0900016}
Assign
In[68]:= pd1yn=pd1/.%;
In[69]:= ps1yn=ps1/.%%;
In[70]:= pd2yn=pd2/.%%%;
In[71]:= ps2yn=ps2/.%%%%;
In[72]:= zsyn=zs/.%%%%%;
In[73]:= pn1yn = pn1 /.%%%%%%;
In[74]:= pn2yn = pn2 /.%%%%%%%;
In[75]:= qd1yn=qd1planned;
In[76]:= qs1yn=qs1planned - zsyn;
In[77]:= qn1yn = qn1planned;
In[78]:= qd2yn=ad pd2yn^ed;
In[79]:= qs2yn=as ps2yn^es+zsyn;
In[80]:= qn2yn = an pn2yn^en;
Indicator for whether this scenario applies
In[81]:= ScenarioYN=If[zsyn>0,1,0]
Out[81]= 1
Check that only one scenario holds...
In[82]:= ScenarioNN +ScenarioYN
Out[82]= 1
Which One?
In[83]:= ScenarioNN  
Out[83]= 0
In[84]:= ScenarioYN
Out[84]= 1
Summary of Economic Effects
Fraction of  subject imports stored from period 1 to period 2
In[85]:= ScenarioNN *0+ScenarioYN*( zsyn*100)/qs1planned
Out[85]= 18.0003
Effects in period 1 with storage, planned versus actual
Percent change in price of domestic shipments (planned versus actual)
In[86]:= ScenarioNN*((pd1nn-pd0) 100)/pd0+ScenarioYN*((pd1yn-pd0) 100)/pd0
Out[86]= 3.19717
Percent change in producer price of subject imports (planned versus actual)
In[87]:= ScenarioNN*((ps1nn-ps0) 100)/ps0+ScenarioYN*((ps1yn-ps0) 100)/ps0
Out[87]= -11.7963
Percent change in consumer price of subject imports (planned versus actual)
In[88]:= ScenarioNN*((ps1nn(1+t1)-ps0) 100)/ps0+ScenarioYN*((ps2yn(1+t1)-ps0) 100)/ps0
Out[88]= 21.8604
Percent change in producer price of non-subject imports (planned versus actual)
In[89]:= ScenarioNN*((pn1nn-pn0) 100)/pn0+ScenarioYN*((pn1yn-pn0) 100)/pn0
Out[89]= 3.19717
Percent change in quantity of domestic shipments (planned versus actual)
In[90]:= ScenarioNN*((qd1nn-qd1planned) 100)/qd0+ScenarioYN*((qd1yn-qd1planned) 100)/qd0
Out[90]= 0
Percent change in quantity of subject imports (planned versus actual)
In[91]:= ScenarioNN*((qs1nn-qs1planned) 100)/qs0+ScenarioYN*((qs1yn-qs1planned) 100)/qs0
Out[91]= -18.0003
Percent change in quantity of non-subject imports (planned versus actual)
In[92]:= ScenarioNN*((qn1nn-qn1planned) 100)/qn0+ScenarioYN*((qn1yn-qn1planned) 100)/qn0
Out[92]= 0
Effects in period 2 with storage
Percent change in price of domestic shipments
In[93]:= ScenarioNN*((pd2nn-pd1nn) 100)/pd1nn+ScenarioYN*((pd2yn-pd1yn) 100)/pd1yn
Out[93]= -3.43673
Percent change in producer price of subject imports
In[94]:= ScenarioNN*((ps2nn-ps1nn) 100)/ps1nn+ScenarioYN*((ps2yn-ps1yn) 100)/ps1yn
Out[94]= 10.5263
Percent change in producer price of non-subject imports
In[95]:= ScenarioNN*((pn2nn-pn1nn) 100)/pn1nn+ScenarioYN*((pn2yn-pn1yn) 100)/pn1yn
Out[95]= -3.30989
Percent change in quantity of domestic shipments
In[96]:= ScenarioNN*((qd2nn-qd1nn) 100)/qd1nn+ScenarioYN*((qd2yn-qd1yn) 100)/qd1yn
Out[96]= -0.697668
Percent change in quantity of subject imports
In[97]:= ScenarioNN*((qs2nn-qs1nn) 100)/qs1nn+ScenarioYN*((qs2yn-qs1yn) 100)/qs1yn
Out[97]= 29.3386
Percent change in quantity of non-subject imports
In[98]:= ScenarioNN*((qn2nn-qn1nn) 100)/qn1nn+ScenarioYN*((qn2yn-qn1yn) 100)/qn1yn
Out[98]= -1.08796