Our final short problem involves building international trad…

Our final short problem involves building international trade into a model of supply and demand.  We consider what happens in the market for widgets when imports expand supply.  First, we describe the relationship between the domestic supply of widgets and the price with the equation qdomS = 3p – 10. The relationship between the demand for widgets and their price is given by the equation qD = 150 – 5p. First, solve for the market equilibrium before trade: we have p0* = [p0] and q0* = [q0].  Total revenue for domestic producers here is obtained by multiplying the price by the quantity of domestic widgets sold: $[tr1].  . Next, we describe the relationship between the foreign supply of widgets and the price with the equation qforS = 2p. To consider the total supply of widgets, we must have qtotS = qdomS + qforS, where each part of the right-hand side is given above.  Simply add these parts together to get qtotS.  To find the new price and quantity sold after trade, set qtotS = qD.  Here, in the new equilibrium, we have p1* = [p1] and q1* = [q1].  To calculate the quantity of domestic widgets sold after trade, plug p1* into qdomS.  Domestic suppliers are able to sell [qdom] widgets here for a total revenue of $[tr2].  Foreign suppliers are able to sell [qfor] widgets (obtained in an equivalent manner to domestic suppliers) for a total revenue of $[tr3].  Does trade increase or decrease revenue for domestic suppliers?  Does it increase or decrease total revenue?

There is a duopoly (two-firm control) over the local market…

There is a duopoly (two-firm control) over the local market for widgets.  The marginal cost of production to each firm is given by the equation MC = 2q. Average costs are given by the equation AC = 30 – 2q. Since the market for widgets is not perfectly competitive, marginal revenue decreases with the quantity of widgets sold: MR0 = 20 – 2q. Finally, market demand is given by the equation q0D = 20 – p.   How many widgets does each firm produce?  Set MC = MR0.  q0* = [q0].  What price do they charge?  p0* = $[p0]. Profit is $[prof0].  Your number may be negative, in which case you should insert a single hyphen, “-” (without the quotes), before the number.   If profits are positive, a new firm enters the market.  In this case, marginal revenue per firm decreases to MR1 = 16 – 2q and demand decreases to q1D = 16 – p.  On the other hand, if profits are negative, a firm exits the market and leaves the other with a monopoly.  Marginal revenue for the monopolist increases to MR1 = 24 – 2q and demand increases to q1D = 24 – p. Now set MC = MR1 to find the new number of widgets sold by each firm: q1* = [q1].  What price do they charge?  p1* = $[p1]. Profit is $[prof1].  Is the market stable, in the sense that no existing firms want to exit and no potential firms want to enter?

A local public university is designing a scholarship program…

A local public university is designing a scholarship program to boost enrollment.  We will model scholarships as subsidies paid to consumers (i.e., students) for pursuing degrees.  Everything below can be thought of in thousands — i.e., prices are in thousands of dollars and we consider quantities as thousands of students.  But don’t multiply anything by 1000 here!  The math is meant to be simple.  Let the relationship between supply and the price of tuition be given by the equation qS = 3p. The relationship between demand and the price of tuition is given by the equation qD = 40 – p. Let’s create a benchmark by characterizing equilibrium without scholarships.  Here, tuition is p* = $[p] with q* = [q] students enrolling.   Now let’s introduce scholarships.  These act as a subsidy by driving a wedge between the price that students pay for tuition and the amount that the university receives.  In particular, we say that pS = pD + B, where B is the “size” of the scholarship.  Our new equilibrium condition is that 3pS = 40 – pD. Substitute the identity for pS in terms of pD into the equilibrium condition (making sure to distribute the 3 correctly) and solve for equilibrium prices (with B still on the right-hand side).  Next, plug pS* into qS or pD* into qD to obtain the equilibrium number of students enrolled in terms of B.  If the university seeks to enroll 33 (thousand) students, we must have B* = $[b4].  In this case, we have pS* = $[p1] and pD* = $[p2].  If the university instead seeks to enroll 36 (thousand) students, we must have B** = $[b8].  In this case, we have pS** = $[p3] and pD** = $[p4].