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Homework answers / question archive / University of South Florida, TampaECO 4401 In this problem you're going to analyze the effects of a unit subsidy in a market
University of South Florida, TampaECO 4401
In this problem you're going to analyze the effects of a unit subsidy in a market. With a unit subsidy the government "pays" a fixed amount per unit of the good produced, s. It doesn't matter if the government pays the consumers or the producers for this, the effect in the market is the same. The system of equations that determine the equilibrium, are Qd = Q. Pa = P. - s ($ 2 0) Qa = a - bPa (a > 0,b > 0) Q. = -c+dP. (c> 0,d > 0) (a) Find out the expressions fo the equilibrium prices, Pe and Par, and quantity in the market, in terms of all the parameters of the model, and the exogenous variable s. How do these expressions differ from those in equation (4)? (In this exercise all the letters represent the same as in the class example, except for s, which is a new parameter.)
(a) Find out the expressions fo the equilibrium prices, Pr and P7, and quantity in the market, in terms of all the parameters of the model, and the exogenous variable s. How do these expressions differ from those in equation (4)? (In this exercise all the letters represent the same as in the class example, except for s, which is a new parameter.)
(b) Derive the partial effects of d in the equilibrium prices and quantity. Can you sign each one of them? Compare them to the partial effects on the equilibrium price and quantity in the class example without the unit tax? How does s, affect each partial effect if at all?
(c) Derive the partial effects of s in the equilibrium prices and quantity. Can you sign each of these partial effects?
Answer:
(a.) The given sets of equations in the question are as follows.
~by "Qd=Qs", => a - b*Pd = (-c) + d*Ps, => a + c = b*Pd + d*Ps, => a + c = b*(Ps-s) + d*Ps.
~a + c = b*Ps - bs + d*Ps, => a + c + bs = (b + d)*Ps, => (a + c + bs)/(b+d) = Ps.
~ Ps = (a+c+bs)/(b+d).............(1).
~ Qs = (-c) + d*Ps = (-c) + d*[(a+c+bs)/(b+d)] = [(-c)*(b+d) + d*(a+c+bs)]/(b+d).
~ Qd = Qs = (-cb - cd + ad+cd+bds)/(b+d) = (-cb + ad+bds)/(b+d),
~Qs = (ad+bds - cb)/(b+d).
Now, Pd = Ps - s = (a+c+bs)/(b+d) - s = (a+c+bs - bs - ds)/(b+d) = (a+c - ds)/(b+d).
~ Pd = (a+c-ds)/(b+d).
So, these are the equilibrium values of "Ps", "Pd" and "Q" in terms of the parameters of the model. Now, the equation "4" is the supply curve shows only the positive relationship between "Qs" and "Ps" and these expressions are on the supply curve.
(b).
The partial effect of "d" is given below.
= Ps = (a+c+bs)/(b+d), dPs/dd = [(b+d)*0 - (a+c+bs)*1]/(b+d)^2.
= dPs/dd = - (a+c+bs)/(b+d)^2.
Now, Pd = Ps - s, => dPd/dd = dPs/dd, since "ds/dd = 0".
=dPd/dd = dPs/dd = - (a+c+bs)/(b+d)^2.
Now, Qs = (ad+bds - cb)/(b+d), => dQs/dd = [(b+d)*(a+bs) - (ad+bds-cb)*1]/(b+d)^2.
=dQs/dd = [ab + b^2*s + cb)]/(b+d)^2.
So, here we can see that "dPd/dd" and "dPs/dd" both are negative and "dQs/dd" is positive, => as "d" increases leads to increase in "Qs" and decrease in "Ps" and "Pd".
Now, if there don't have any unit5 tax then the system of equation is given by.
=Qd=Qs, Qd = a - b*Pd, Qs = (-c) + d*Ps, => under this situation the equilibrium "Q" and "P" are given below.
= Pd=Ps=(a+c)/(b+d) and Qd = Qs = (ad+cd)/(b+d).
So, the partial effect of "d" on "P" and "Q" is given by.
=dP/dd = -(a+c)/(b+d)^2 < 0 and dQ/dd = (ab+cb)/(b+d)^2 > 0.So, if we compare both these expression we can see that these also have same sign but their absolute value are differ.
Since (a+c) < (a+c+bs), => (a+c)/(b+d)^2 < (a+c+bs) /(b+d)^2, => absolute value of "dP/dd" is more under subsidy compare to under no tax. Similarly, (ab+cb) < (ab+b^2*s+cb).
= (ab+cb) /(b+d)^2 < (ab+b^2*s+cb)/(b+d)^2, => absolute value of "dQ/dd" is more under subsidy compare to under no tax.
As "s" increases implied the supply curve shift right side, => the equilibrium "P" that is "Pd" decreases and the "Q" increases and "Ps" increases.
(c).
So, the partial effects of "s" are given below.
= Ps = (a+c+bs)/(b+d), => dPs/ds = b/(b+d) > 0, Pd = (a+c - ds)/(b+d), dPd/ds = (-d)/(b+d)^2 < 0 and "Q = (-cb + ad+bds)/(b+d)", => dQ/ds = bd/(b+d)^2 > 0.
= we can see that as "s" increases implied increase in "Q" and "Ps" but "Pd" decrease.
