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].
Read DetailsWe now examine a production problem. Consider two firms, ea…
We now examine a production problem. Consider two firms, each producing widgets, who sell in two different types of output markets. In particular, firm PC sells widgets in a perfectly competitive output market, where there are so many firms and widgets sold that a single firm’s quantity produced does not affect the market price. In a separate market, firm M acts as a monopolist and sees diminishing marginal revenue with higher levels of production. Each firm produces widgets according to the same production technology. The table below shows the number of widgets produced given some number of workers employed. # of workers widgets produced 1 12 2 23 3 33 4 42 5 50 6 57 7 63 8 68 9 72 10 75 Marginal revenue is constant for firm PC — they obtain $5 per widget sold. Marginal revenue is decreasing for firm M — they obtain $5 per widget sold for their first 4 workers; $4 per widget for the next 3 workers; and $3 per widget for the final 3. If the market wage is $48, firm PC should hire [pc1] workers and firm M should hire [m1] workers. If the market wage is $36, firm PC should hire [pc2] workers and firm M should hire [m2] workers. Finally, if the market wage is $18, firm PC should hire [pc3] workers and firm M should hire [m3] workers.
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